Laetitia Raison-Aubry
email: laetitia.raison-aubry@loria.fr

Loïs Naudin
email: lois.naudin@gmail.com

Nange Jin
email: njin@central.uh.edu

Christophe Ribelayga

Laure Buhry
email: laure.buhry@loria.fr

Computational modeling of the secondary rod pathway contribution to the retinal output

In the retina, signals originating from rod and cone photoreceptors can reach retinal ganglion cells (RGCs) through different pathways. The secondary rod pathway is functionally important under mesopic light, such as at dawn or dusk. Electrical synapses (gap junctions) between rods and cones form the entry to the secondary rod pathway. Recent experimental observations indicate that these gap junctions are plastic, changing with the time of day and/or light/dark adaptation. However, the impact of this plasticity on the contribution of the secondary rod pathway to RGC activity remains elusive. As a first step, we model the spatial topology and connectivity of the secondary rod pathway of the mouse retina and test the light responses of OFF RGCs to dim light stimuli. Our results corroborate previously published electrical recordings thereby establishing the validity of the model and its suitability to further study rod/cone coupling plasticity.

I. INTRODUCTION

The retina is a sensory organ located in the back of the eye and whose role is to detect light stimuli and process early visual information. Light is captured by photoreceptors (rods and cones), and converted into electrical potentials through the process of phototransduction. Electrical signals originating from photoreceptors are conveyed through secondary neurons (horizontal, bipolar and amacrine cells) to retinal ganglion cells (RGCs) -the output neurons of the retina whose axons project to higher processing centers in the brain. At the circuit level, rod and cone signals can travel across the retina through several distinct pathways: three different rod pathways and the cone pathway. Recent efforts have been made to experimentally characterize the secondary rod pathway [START_REF] Jin | Genetic elimination of rod/cone coupling reveals the contribution of the secondary rod pathway to the retinal output[END_REF]. Yet, accumulating evidence that the entry of the secondary rod pathway -the rod/cone gap junction -shows a high degree of plasticity [START_REF] Ribelayga | Circadian and light-adaptive control of electrical synaptic plasticity in the vertebrate retina[END_REF] warrants further investigation of the contribution of this rod pathway to RGC activity.

To gain biological insight into the contribution of the secondary rod pathway and the impact of rod/cone gap junction *co-first authors: these authors have equally contributed to this work. plasticity to the retinal output, computational modeling could be a valuable asset. In that sense, our paper aims at developing a detailed model of the secondary rod pathway in the mouse retina with the goal to reproduce experimental data to validate the model. Here, we use a bottom-up approach consisting in connecting single cells into circuits and circuits into systems. Each cell of the secondary rod pathway is modeled using a single-point conductance-based model (CBM). In addition, and contrarily to previous studies [START_REF] Iseri | A computational model simulates light-evoked responses in the retinal cone pathway[END_REF]- [START_REF] Kamiyama | Simulation analysis of bandpass filtering properties of a rod photoreceptor network[END_REF], our model is based on a realistic representation of the spatial topology and connectivity of the mouse retina. Finally, we show that our detailed model is able to qualitatively reproduce the output of the secondary rod pathway -the behavior of RGCs -in response to a series of dim light stimuli. This shows the suitability of our model to further study rod/cone coupling plasticity that will be implemented in future works.

The remainder of the paper is organized as follows. Section II presents the spatial topology and connectivity of the secondary rod pathway in the mouse retina, as well as the model of each cell of the pathway and the photocurrent equation. Then, Section III shows the spatial topology and connectivity simulated from our modeling framework, and the simulated light responses of the RGC against the experimental ones. Finally, Section IV discusses the potential implications of our modeling workflow.

II. MATERIALS AND METHODS

In this section, we first present the connectivity map of the secondary rod pathway to the transient OFF alpha retinal ganglion cells (tOFF αRGCs). Then, we describe the models of each cell composing this pathway, as well as the synaptic model that governs the interactions between the different components of the pathway.

A. Connectivity map of the secondary rod pathway

The secondary rod pathway is composed of the rods, cones, CBCs (Cone Bipolar Cells), and RGCs -the output of the pathway. We model the OFF component of this pathway to a single tOFF αRGC. Fig. 1 summarizes the convergence and divergence relationship between each cell population to a single tOFF αRGC based on anatomical knowledge from the mouse retina. In the photoreceptor network, rods outnumber cones by about 30:1 [START_REF] Carter-Dawson | Rods and cones in the mouse retina. i. structural analysis using light and electron microscopy[END_REF]. While rod/cone gap junctions are prevalent, there is little experimental support for rod/rod gap junctions, and cone/cone gap junctions are rare [START_REF] Jin | Molecular and functional architecture of the mouse photoreceptor network[END_REF], [START_REF] Ishibashi | Analysis of rod/cone gap junctions from the reconstruction of mouse photoreceptor terminals[END_REF], so we do not consider rod/rod and cone/cone connections in our model. Moreover, each cone makes gap junctions with around 45 nearby rods, while each rod contacts approximately 1.5 cones through gap junctions [START_REF] Jin | Molecular and functional architecture of the mouse photoreceptor network[END_REF].

Also, the tOFF αRGC makes synaptic contacts with OFF CBC types 3A, 3B and 4 only [START_REF] Helmstaedter | Connectomic reconstruction of the inner plexiform layer in the mouse retina[END_REF], [START_REF] Behrens | Connectivity map of bipolar cells and photoreceptors in the mouse retina[END_REF]. According to [START_REF] Behrens | Connectivity map of bipolar cells and photoreceptors in the mouse retina[END_REF], each cone contacts approximately 1.3 CBCs of each type, and about 2.5 cones provide input to a single OFF CBC. Finally, we connect all OFF CBCs to the single tOFF αRGC.

To preserve the convergence and divergence ratio between each population of cells, we calculate their number according to the equation Npre/Npost = convergence/divergence where Npre and Npost denotes the number of cells in the preand post-synaptic groups, respectively; and convergence and divergence denote the ratio of convergence and divergence between each cell group as described above. Following this methodology, we obtain a set of cells which comprises 37,636 rods, 1,225 cones, 625 OFF CBCs, and 1 tOFF αRGC, consistent with experimental data in the mouse retina [START_REF] Behrens | Connectivity map of bipolar cells and photoreceptors in the mouse retina[END_REF].

B. Models

1) Photocurrent model: Both rods and cones respond to light but with different sensitivities. Rods can detect as few as a single photon whereas the cone threshold requires a few dozens [START_REF] Jin | Rod electrical coupling is controlled by a circadian clock and dopamine in mouse retina[END_REF]. This implies that we can omit the cone response under dim light conditions when individual rods receive one or a few photons on average. To model changes in the rod dark current caused by the dim light transduction process, we use a simulated photocurrent waveform, slightly adapted from [START_REF] Jin | Rod electrical coupling is controlled by a circadian clock and dopamine in mouse retina[END_REF], as input in the form

I photo = I amp [1 -exp(-(t -b 1 )/τ 1 ) -exp(-(t -b 2 )/τ 2 ]
with I amp the amplitude of the photocurrent (in pA), b 1 and b 2 the inflection points of the photocurrent trajectory, and τ 1 and τ 2 time constants (in ms). In this paper, we set b 1 = 1000, b 2 = 1500, τ 1 = 100 and τ 2 = 100.

2) Cell models: For each cell we use a single-point CBM that describes the evolution of the membrane potential V by a general equation of the form CdV /dt =ion I ion + I where C is the membrane capacitance, ion I ion is the total current flowing accross the cell membrane, and I is an applied current. For the rod and cone cells, we use the CBM described in [START_REF] Kourennyi | Reciprocal modulation of calcium dynamics at rod and cone photoreceptor synapses by nitric oxide[END_REF] and [START_REF] Liu | Effects of tetraethylammonium on kx channels and simulated light response in rod photoreceptorss[END_REF], respectively. For the OFF CBCs, we use the CBM reported in [START_REF] Publio | A computational study on the role of gap junctions and rod i h conductance in the enhancement of the dynamic range of the retina[END_REF], and for the RGC the CBM described in [START_REF] Fohlmeister | Modeling the repetitive firing of retinal ganglion cells[END_REF]. Slight modifications have been made to the conductance values of these models in order to reproduce the qualitative shape of their steady-state current reported in the literature. Indeed, the steady-state current plays a paramount role in the dynamics of non-spiking neurons since it determines their qualitative features [START_REF] Naudin | Systematic generation of biophysically detailed models with generalization capability for non-spiking neurons[END_REF]. All the currents and the parameter values of each model are then reported in Table I. 3) Synaptic model: Several experimental studies have shown that the mode of neurotransmitter release in the retina is graded [START_REF] Dowling | The retina: an approachable part of the brain[END_REF]. A 'graded synaptic connection' is one in which incremental changes in the membrane potential of the presynaptic neuron cause incremental changes in the membrane potential of the postsynaptic neuron [START_REF] Roberts | Neurones without impulses: their significance for vertebrate and invertebrate nervous systems[END_REF]. To model this, we use the graded synaptic model described in [START_REF] Mulloney | During fictive locomotion, graded synaptic currents drive bursts of impulses in swimmeret motor neurons[END_REF], and adjust some parameters to reproduce the qualitative experimental behavior of the RGC described in [START_REF] Jin | Genetic elimination of rod/cone coupling reveals the contribution of the secondary rod pathway to the retinal output[END_REF].

4) Connectivity:

We connect rods and cones with a probability that is modulated according to a 2-dimensional Gaussian of the distance between the cells, so that convergence and divergence between rods and cones are preserved. We proceed in the same way to connect cones to OFF CBCs. Finally, we connect all CBCs to a single tOFF αRGC.

III. RESULTS

In this section, we first implement the spatial connectivity of the secondary rod pathway, and we compare it with data from the literature. Then, we compare the behavior of our simulated tOFF αRGC with the experimental results of [START_REF] Jin | Genetic elimination of rod/cone coupling reveals the contribution of the secondary rod pathway to the retinal output[END_REF]. All this was done using the Brian2 simulation environment [START_REF] Stimberg | Brian 2, an intuitive and efficient neural simulator[END_REF].

A. Generation of the retina network

In our model, the retinal cells are distributed in a 266 µm long square grid (i.e. 70, 756 µm 2 ), which is an accurate representation of the experimental spatial topology of the mouse retina for a single tOFF αRGC. Indeed, tOFF αRGCs have large dendritic fields of about 300 µm diameter [START_REF] Krieger | Four alpha ganglion cell types in mouse retina: Function, structure, and molecular signatures[END_REF]. In other words, the set of presynaptic cells to a tOFF αRGCincluding the photoreceptor network and the OFF CBCs in the case of the secondary rod pathway -thus extends over an area of about 70, 685 µm 2 . Within this area, the cells are evenly and equidistantly distributed (Fig. 2.A), with a rod/rod distance of approximately 1.37 µm, a cone/cone distance around 7.6 µm, and a OFF CBC/OFF CBC distance of approximately 10.64 µm, which is also consistent with biological observations [START_REF] Helmstaedter | Connectomic reconstruction of the inner plexiform layer in the mouse retina[END_REF], [START_REF] Behrens | Connectivity map of bipolar cells and photoreceptors in the mouse retina[END_REF].

In addition, our model takes into account the spatial connectivity in the secondary rod pathway in the mouse retina. In particular, the average number of connections per cell is maintained (e.g., in our model the divergence and convergence from rods to cones is, respectively, 1.56 and 47 on average, as illustrated in Fig. 2.B).

B. Comparison with the experimental behavior of RGCs

Here we simulate the secondary rod pathway, made up of about 40,000 cells, following the experimental stimulation protocol described in [START_REF] Jin | Genetic elimination of rod/cone coupling reveals the contribution of the secondary rod pathway to the retinal output[END_REF]. This protocol consists of a 1-s period of darkness, followed by a dim light stimulus of 500-ms in duration, and a second period of darkness of 2.5 seconds. The simulation time using Brian2 is about 3 minutes on a regular laptop 11th Gen Intel® Core™ i5-1135G7 @ 2.40GHz × 8, which is a very short simulation time given the model complexity.

Fig. 3A shows the simulated and experimental responses of a tOFF αRGC in response to three different light stimulus intensities delivering 0.06, 2, and 20 effective isomerizations R*/rod/sec. It can be observed that our model reproduces the qualitative features of the experimental ones [START_REF] Jin | Genetic elimination of rod/cone coupling reveals the contribution of the secondary rod pathway to the retinal output[END_REF]. Indeed, at the weakest intensities, when only 1 rod out of 17 receives a single photon (2 pA), the tOFF αRGC does not elicit a noticeable response. At the brightest intensity, when each rod absorbs 20 photons (10 pA), the cell stops firing for most of the time the light stimulus is present. At the intermediate intensity, when each rod absorbs 2 photons (4 pA), a small yet significant decrease in firing occurs. Furthermore, the frequency of the spikes of our simulated tOFF αRGC is approximately 85 Hz, which is close to the experimental value [START_REF] Jin | Genetic elimination of rod/cone coupling reveals the contribution of the secondary rod pathway to the retinal output[END_REF].

Regarding the threshold of the secondary rod pathway, defined as the stimulus intensity to generate 5% of maximal response to the light stimulus, studies have shown that it is around 2 photons/rod/sec [START_REF] Jin | Genetic elimination of rod/cone coupling reveals the contribution of the secondary rod pathway to the retinal output[END_REF]. This approximately corresponds to a rod photocurrent of 4 pA. Similarly, the response threshold of our simulated tOFF αRGC is around 3.92 pA (Fig. 3B).

In addition, because there is no noise nor heterogenity in our model, the light response of our simulated tOFF αRGC, once its threshold is reached, is binary, in contrast with the experimental traces (Fig. 3A). This also explains the slight discrepancies in the spike firing times between the experimental and our simulated cell. Furthermore, the model does not capture the experimental overshoot in the tOFF αRGC -i.e. spikes arise before the photocurrent drops back to 0 pA. This phenomenon -known as "rebound excitation" -is partly due to the intrinsic properties of RGCs, in particular the biophysical properties of the I h and I Ca currents [START_REF] Mitra | Mechanism underlying rebound excitation in retinal ganglion cells[END_REF]. These currents, together with continuous noise in the circuit and mechanisms of feedback and/or inhibition will be implemented to improve the realism of the model.

IV. DISCUSSION AND CONCLUSION In this paper, we model the spatial topology and connectivity of the secondary rod pathway of the mouse retina and test the light responses of tOFF αRGCs to dim light stimuli. We are able to reproduce the light responses of a RGC with a behavior similar to what has been experimentally reported [START_REF] Jin | Genetic elimination of rod/cone coupling reveals the contribution of the secondary rod pathway to the retinal output[END_REF]. In particular, the secondary rod pathway of the mouse retina has a threshold around 2 photons/rod/sec [START_REF] Jin | Genetic elimination of rod/cone coupling reveals the contribution of the secondary rod pathway to the retinal output[END_REF], corresponding to a photocurrent amplitude of 4 pA. The model sets the threshold around 3.92 pA (Fig. 3B).

Although the model effectively replicates the essential qualitative features of the experimental data from the secondary rod pathway, it could nevertheless be improved by addressing some limitations. First, as previously highlighted, our model lacks both noise and overshoot of the tOFF αRGC response that we could take into account. Second, we will need to add the other rod pathways, namely the primary and the tertiary rod pathway. The circuit elements of these pathways have been well-characterized and their respective contributions to tOFF αRGCs have been recently established [START_REF] Jin | Genetic elimination of rod/cone coupling reveals the contribution of the secondary rod pathway to the retinal output[END_REF], [START_REF] Krieger | Four alpha ganglion cell types in mouse retina: Function, structure, and molecular signatures[END_REF]. Modeling the respective contributions of all 3 rod pathways and cone pathway to a single RGC will further validate our model. For the lowest intensity, only 1/17 rod receives the photocurrent. The experimental data are reproduced from [START_REF] Jin | Genetic elimination of rod/cone coupling reveals the contribution of the secondary rod pathway to the retinal output[END_REF] with permission. (B) Normalized spiking frequency of the tOFF αRGC during lights-on for increasing photocurrent amplitudes. The threshold of the secondary rod pathway (red) is defined as the stimulus intensity to generate 5% of maximal response to the light stimulus [START_REF] Jin | Genetic elimination of rod/cone coupling reveals the contribution of the secondary rod pathway to the retinal output[END_REF]. Experimental data are means ± SEM (n = 5 cells).

Once these limitations are addressed, our model will be useful to study the functional importance of the plasticity of the rod/cone gap junction. Indeed, these coupling changes with the time of day and light/dark environmental conditions [START_REF] Ribelayga | Circadian and light-adaptive control of electrical synaptic plasticity in the vertebrate retina[END_REF]. As the dynamic range of rod/cone spans over more than 1,000 pS [START_REF] Ishibashi | Analysis of rod/cone gap junctions from the reconstruction of mouse photoreceptor terminals[END_REF], it is expected to dramatically affect the contribution of the secondary rod pathway to the retinal output. Because the pharmacological approach is limited (modulating rod/cone gap junctions will affect other gap junctions elsewhere in retinal circuits), our model will provide valuable insights into the functional impact of the modulation of rod/cone coupling.

Fig. 1 .

 1 Fig. 1. Secondary rod pathway diagram. The numbers above each population denote their number of cells. The numbers in red denote the convergence and those in blue the divergence, between cell populations. "All" means that all OFF CBCs are connected to the tOFF αRGC.

Fig. 2 .

 2 Fig. 2. (A) Topology of the secondary rod pathway (portion of the modeled retinal cell grid centered on the tOFF αRGC). (B) Connectivity in the photoreceptor network. (Top) Convergence: all presynaptic rod contacts for one randomly-chosen cone. (Bottom) Divergence: all postsynaptic cone contacts for one randomly-chosen rod.

Fig. 3 .

 3 Fig. 3. (A) Comparison of the simulated RGC behavior with the experimental one for three different intensities of photocurrent (0.06, 2, and 20 R*/rod/s).For the lowest intensity, only 1/17 rod receives the photocurrent. The experimental data are reproduced from[START_REF] Jin | Genetic elimination of rod/cone coupling reveals the contribution of the secondary rod pathway to the retinal output[END_REF] with permission. (B) Normalized spiking frequency of the tOFF αRGC during lights-on for increasing photocurrent amplitudes. The threshold of the secondary rod pathway (red) is defined as the stimulus intensity to generate 5% of maximal response to the light stimulus[START_REF] Jin | Genetic elimination of rod/cone coupling reveals the contribution of the secondary rod pathway to the retinal output[END_REF]. Experimental data are means ± SEM (n = 5 cells).

  The first number denotes the maximal conductances (in nS), and the second number denotes the reversal potential (in mV).

			TABLE I
		BIOPHYSICAL MODEL PARAMETERS
		Rod	Cone	CBC	RGC
	I Ca	4 / 40	4.92 / 40	0.55 / 30	44 / 40
	I N a	-	-	-	1000 / 35
	I Kv	10 / -74	2 / -80	2 / -58	240 / -75
	I h	2.5 / -32	3.5 / -32.5	0.975 / -17.7	-
	I KCa	5 / -74	0.5 / -80	8.5 / -58	1 / -75
	I L	0.52 / -74	1 / -63	0.73 / -53	1 / -65
	I Cl	1.3 / -20	6.5 / -45	-	-
	I A	-	-	-	720 / -75

ACKNOWLEDGMENT

We thank the INS2I-CNRS and LORIA for their financial support on the ModERN-Psy project. The authors thank also LUE for funding L.R.-A. travel grant. Research in the Ribelayga lab is supported by NIH grants R01EY029408, R01EY032508, RF1MH127343, and P30EY007551, and an Endowed Professorship from the FERV.


Alexandre Goldsztejn

An equioscillation theorem for multivariate Chebyshev approximation

Keywords: Chebyshev approximation problem, multivariate approximation, equioscillation theorem, convex optimization

The equioscillation theorem interleaves the Haar condition, the existence and uniqueness and strong uniqueness of the optimal Chebyshev approximation and its characterization by the equioscillation condition in a way that cannot extend to multivariate approximation: Rice [Transaction of the AMS, 1963] says "A form of alternation is still present for functions of several variables. However, there is apparently no simple method of distinguishing between the alternation of a best approximation and the alternation of other approximating functions. This is due to the fact that there is no natural ordering of the critical points." In addition, in the context of multivariate approximation the Haar condition is typically not satisfied and strong uniqueness may hold or not. The present paper proposes an multivariate equioscillation theorem, which includes such a simple alternation condition on error extrema, existence and a sufficient condition for strong uniqueness. To this end, the relationship between the properties interleaved in the univariate equioscillation theorem is clarified: first, a weak Haar condition is proposed, which simply implies existence. Second, the equioscillation condition is shown to be equivalent to the optimality condition of convex optimization, hence characterizing optimality independently from uniqueness. It is reformulated as the synchronized oscillations between the error extrema and the components of a related Haar matrix kernel vector, in a way that applies to multivariate approximation. Third, an additional requirement on the involved Haar matrix and its kernel vector, called strong equioscillation, is proved to be sufficient for the strong uniqueness of the solution. These three disconnected conditions give rise to a multivariate equioscillation theorem, where existence, characterization by equioscillation and strong uniqueness are separated, without involving the too restrictive Haar condition. Remarkably, relying on optimality condition of convex optimization gives rise to a quite simple proof. Instances of multivariate problems with strongly unique, non-strong but unique and nonunique solutions are presented to illustrate the scope of the theorem.

The multivariate equioscillation theorem

Given f : X → R continuous, we consider the Chebyshev approximation problem

min a∈R n max x∈X n i=1 a i φ i (x) -f (x) , (1)
where basis function φ i : X → R are continuous. The index set X is just assumed to be a compact topological space, so that continuity is well defined and entails the existence of the maximum (typically, we will just consider boxes in R n ). The basis function vector φ : X → R n is defined by φ(x) = (φ 1 (x), . . . , φ n (x)). We define e(a, x) = a T φ(x) -f (x) and

m(a) = max x∈X |e(a, x)|, (2)
so that the Chebyshev approximation problem is min a∈R n m(a).

Introducing non-square Haar matrices will be convenient not only to express the classical Haar condition. Given x 1 , . . . , x k ∈ X, we define the Haar matrix

H(x 1 , . . . , x k ) = φ(x 1 ) φ(x 2 ) • • • φ(x k ) ∈ R n×k . (3)
When x 1 < • • • < x k and φ(x) = (1, x, . . . , x k ), a Haar matrix is a Vandermonde matrix. The Haar condition holds for X and φ if and only if for all x 1 , • • • , x n ∈ X the square Haar matrix H(x 1 , . . . , x n ) is nonsingular. When X = [x, x] ⊆ R is a compact interval, the equioscillation condition is satisfied for a ∈ R n if and only if there exist x 1 < • • • < x n+1 ∈ X such that |e(a, x i )| = m(a) for all i ∈ {1, . . . , n + 1} and e(a, x i ) e(a, x i+1 ) < 0 hold for all i ∈ {1, . . . , n}. Values of x ∈ X that satisfy |e(a, x i )| = m(a) will be called active indices at a ∈ R n . With this settings, the equioscillation theorem characterizes the solutions of the Chebyshev univariate approximation problem: if the Haar condition holds then the Chebyshev approximation problem has one strongly unique 1 solution, which is equivalently characterized by the equioscillation condition. The Haar condition in central in this statement: in Cheney's standard textbook [START_REF] Cheney | Introduction to Approximation Theory[END_REF], the Haar condition is assumed in the statement of the Alternation Theorem on page 75, as well in the Unicity Theorem on page 80 and in the Strong Unicity Theorem on page 80. As said previously, both the Haar condition and the equioscillation condition don't extend to multivariate approximation. The Haar condition cannot be assumed anymore in the context of multivariate approximation: it entails that X is either a circle or a compact interval [START_REF] Mairhuber | On Haar's theorem concerning Chebychev approximation problems having unique solutions[END_REF] (up to homeomorphism), hence it cannot hold for multivariate approximation with other index sets, e.g., boxes. We still need a similar but weaker condition, and define the weak Haar condition as follows: it holds for X and φ if and only 1 Strong uniqueness means that not only m(a) -m(a * ) > 0 for a = a * , but that m(a)m(a * ) ≥ c a -a * X,φ for some c > 0 and the norm a -a * X,φ = max x∈X (a -a * ) T φ(x) . The definition is in fact independent of the norm since all norms are equivalent in R n .

if there exist x 1 , • • • , x n ∈ X such that the square Haar matrix H(x 1 , . . . , x n ) is nonsingular. The weak Haar condition is likely to hold in most situations, e.g.,

X = [-1, 1] × [-1, 1], φ(x) = (1, x 1 , x 2 )
where H (0, 0), (1, 0), (1, 1) is nonsingular. The weak Haar condition has several consequences, among which card(X) ≥ n, the existence of a minimizer of the Chebyshev approximation problem (cf. Section 3.1), the fact that a X,φ = max x∈X |a T φ(x)| is a norm on R n (the triangular inequality and absolute homogeneity are trivial, the positive definiteness follows directly from the weak Haar condition), with the consequence that strong uniqueness implies uniqueness.

The equioscillation condition requires a total order on indices and therefore does not make any sense for multivariate approximation. We reformulate it in a way that can apply more generally. To this end, we rely on the following lemma, which generalizes the well-known fact that Vandermonde matrices with one more column than rows have kernel vectors with oscillating signs, and its subsequent observation.

Lemma 1. If φ : [x, x] → R n is continuous and satisfies the Haar condi- tion on [x, x] then all Haar matrices H(x 1 , . . . , x n+1 ) ∈ R n×(n+1) , with x 1 < • • • < x n+1 ∈ X, have a one dimensional kernel span{u} with 0 = u ∈ R n+1
having nonzero components with alternating signs, i.e., u i u i+1 < 0 for all i ∈ {1, . . . , n}.

The proof of the lemma is given in Appendix A. The connection between the Haar matrix kernel vector component oscillations u i u i+1 < 0, the equioscillation e(a, x i ) e(a, x i+1 ) < 0 and the optimality of the Chebyshev approximation problem can be seen by introducing the matrix

G(x 1 , . . . , x d ) = (a, x 1 )φ(x 1 ) • • • (a, x d )φ(x d ) , (4)
where (a, x) = sign(e(a, x)), which will be used only when e(a, x) = 0 hence sign(e(a, x)) ∈ {-1, 1}. The capitale G is for "G"radient: the columns of G(x 1 , . . . , x d ) are gradients of |e(a, x i )| = (a, x) e(a, x), provided that |e(a, x i )| > 0. When x i are active indices for a * ∈ R n , we will see that the columns of G(x 1 , . . . , x d ) are actually subgradients of m(a * ). The optimality condition of convex optimization states that a * is optimal if and only if 0 is in the convex hull of the columns of G(x 1 , . . . , x d ), or equivalently to G(x 1 , . . . , x d ) having a kernel vector λ = 0 whose components are of the same sign (see Lemma 6 below). Finally, observe that by definition we have

G(x 1 , . . . , x d ) λ = H(x 1 , . . . , x d ) u , with u i = (a * , x i ) λ i . (5)
From this we see that all components λ i have the same signs, i.e., a * is optimal, if and only if all u i e(a * , x i ) have the same sign. This condition generalizes the univariate equioscillation condition when the Haar condition is assumed: Lemma 1 proves that u i u i+1 < 0, so if in addition e(a, x i ) e(a, x i+1 ) < 0 then all u i e(a * , x i ) have the same sign. In fact, requiring u i e(a * , x i ) ≥ 0 for some kernel vector of H(x 1 , . . . , x d ) is more convenient and equivalent since the opposite of a kernel vector is also in the kernel. These observations will be formalized in the next sections. Under this form, the equioscillation condition can be used for multivariate Chebyshev approximation problems. Uniqueness is not granted anymore in the context of multivariate Chebyshev approximation, so a technical sufficient condition for uniqueness is added below for the definition of the strong equioscillation condition.

Definition 2 (Multivariate equioscillation condition). The (multivariate) equioscillation condition is satisfied at a ∈ R n if and only if there exist

x 1 , • • • , x K ∈ X, K ≥ 1, such that |e(a, x i )| = m(a)
, and a kernel vector 0 = u ∈ R K of H(x 1 , . . . , x K ) such that e(a, x i ) u i ≥ 0 hold for all i ∈ {1, . . . , K}.

The strong (multivariate) equioscillation condition holds if and only if furthermore H(x 1 , . . . , x K ) is full rank, hence K ≥ n + 1, and u has no zero components.

First note that by Lemma 1 the Haar condition and the (univariate) equioscillation condition together imply the strong multivariate equioscillation with K = n + 1. We draw several direct consequences of this definition: the active indices x 1 , • • • , x K ∈ X need not to be ordered anymore, so the definition can be applied in the context of multivariate approximation. In fact, the definition is not affected by any permutation the active indices x 1 , • • • , x K ∈ X: indeed, the columns of the Haar matrix, together with the components of its kernel vectors, permute simultaneously with permutations of the active indices, so the sign synchronization e(a, x i ) u i ≥ 0 is not affected by permutation of the active indices. Another observation is that K < n + 1 implies that H(x 1 , . . . , x K ) not full rank: indeed in this case H(x 1 , . . . , x K ) has more rows than columns and has a non trivial kernel. In particular, if K = 1 then H(x 1 ) has to be a zero matrix (see Example 11 for such a non trivial situation with infinitely many minimizers).

These conditions lead to the following generalization of the equioscillation theorem to multivariate Chebyshev approximation problems, where X is not restricted anymore to be any interval or circle.

Theorem 3 (Multivariate equioscillation theorem). Let φ : X → R n be continuous. If the weak Haar condition holds for φ and X then there exists an optimal solution to the Chebyshev approximation problem. Furthermore, a ∈ R n is a minimizer of this problem if and only it satisfies the multivariate equioscillation condition. Finally, if the strong multivariate equioscillation condition holds then the minimizer is strongly unique.

The proof is given in Section 3. The usual equioscillation theorem for univariate approximation is a simple consequence of Theorem 3: on the one hand by Lemma 1 the equioscillation implies the multivariate equioscillation, on the other hand the Haar condition implies both the weak Haar condition and, together with Lemma 1, the strong multivariate equioscillation.

The rest of the paper is organized as follows: some basic definitions and properties of convex functions are presented in Section 2. Section 3 presents the proof of the multivariate equioscillation theorem. Finally, several instances of multivariate problems are presented in Section 5 to illustrate the scope of the theorem.

Standard convex properties of m(a)

The function m : R n → R in ( 2) is called a pointwise supremum, and it's properties are well known in the context of nonsmooth convex optimization. The reader is referred to [START_REF] Nesterov | Lectures on Convex Optimization[END_REF] for an introduction to convex optimization. First not that the function |e(a, x)| is convex, and the pointwise supremum of convex functions is convex, hence m(a) is convex. Convex functions defined in R n being continuous, so is m(a). The subdifferential of m(a) is of central importance here. It is denoted by ∂m(a) and is made of all subgradients: u ∈ R n is a subgradient at ā if it gives rise to a affine under-estimator, i.e., m(a) ≥ m(ā) + u T (a -ā). The optimality condition for unconstrained convex optimization then reads a * is a minimizer of m(a) is and only if 0 ∈ ∂m(a * ).

When |e(a, x)| is defined in R n × X with values in R, and X is compact, the pointwise supremum is actually an unconstrained maximum2 and its subdifferential enjoys a simple explicit expression [3, Lemma 3.1.14]: ∂m(a) = conv{∂ a |e(a, x)| : x ∈ act(a)} with act(a) = {x ∈ X : |e(a, x)| = m(a)} is the set of active indices. Let us illustrate this formula on a simple finite pointwise maxima example, which presents a (strongly) unique minimizer.

Example 4. Let U ∈ R 2×3 with columns u 1 = (-2, 1) T , u 2 = (1, 1) T and u 3 = (1, -3) T . Define the pointwise maximum m(a) = max{u T 1 a, u T 2 a, u t 3 a} = U T a ∞ ,
which is a piecewise linear function. Figure 1 shows the level sets of m(a). From darker to lighter level sets correspond to areas where u T 1 a, u T 2 a or u t 3 a is active. Red line are the place where to linear functions are equal, and the three linear functions are equal at the origin. The subdifferential are represented in blue: when one linear constraint is active, the pointwise maximum is differentiable and the subdifferential contains only the gradient. On red lines, where two linear constraints are active, the subdifferential is the convex hull of the two corresponding gradients, hence a segment. Finally, at the origin where the three functions are actives, the subdifferential is the convex hull of the three gradients, hence a triangle. From this analysis, we see that the origin is the only point where the subdifferential contains zero, hence the only minimizer of m(a).

In the typical situation where min a∈R n m(a) > 0, the subdifferential of m(a) has a simple expression: in this case, we have that m(a) > 0 and |e(a, x)| differentiable for all a ∈ R n and x ∈ act(a). Therefore, by [3, Lemma 3.1.7] the subdifferential of |e(a, x)| contains only its gradient, i.e., ∂e(a, x) = { (a, x) φ(x)} with (a, x) = sign e(a, x) whose value is inside {-1, 1} for x ∈ act(a). Finally we have

∂m(a) = conv{ (a, x) φ(x) : x ∈ act(a)}. ( 6
)
Also note that (a, x) φ(x) for x ∈ act(â) are subgradients at â, therefore they give rise to affine lower bounds

m(a) ≥ m(â) + ( (a, x) φ(x)) T (a -â). ( 7
)
This section is ended with the following two simple lemmas. The first provides a sufficient condition for the uniqueness of the minimizer of some pointwise maximum of linear functions. It applies directly to the pointwise maximum function of Example 4. In the sequel, the maximal component of the vector a u is denoted by max u.

Lemma 5. Let U = u 1 u 2 • • • u d ∈ R n×d , d ≥ n + 1,
be full rank and suppose that λ ∈ ker(U ) with λ i > 0 for all i ∈ {1, . . . , d} = I. Then the piecewise linear function l(a) = max U T a = max{u T 1 a, . . . , u T d a} has the origin as unique minimizer.

Proof. By way of contradiction, suppose a = 0 and l(a) ≤ 0, i.e., a T u i ≤ 0 for all i ∈ I and in matrix form U T a ≤ 0. Now U T a = 0 (because U T is full rank and a = 0) therefore exists i * ∈ I such that a T u i * < 0. Finally, a T u i * = -i∈I\{i * } λi λ i * a T u i ≥ 0, since λi λ i * > 0 and a T u i ≤ 0, a contradiction. Finally, the following lemma provides a simple matrix formulation of the property that the null vector belongs to a the convex hull of some given vectors.

Lemma 6. Let U = u 1 u 2 • • • u m ∈ R n×m . Then 0 ∈ conv{u 1 , . . . , u m } if and only if 0 = λ ∈ ker(U ) with λ i ≥ 0.
Proof. The only if part is a direct application of the convex hull definition. Now suppose that 0 = λ ∈ ker(U ) with λ i ≥ 0 and define µ = 1 λ 1 λ. Then µ ∈ ker(U ), i.e., µ i u i = 0, µ i ≥ 0 and

µ i = λ 1 λ 1 = 1 hence 0 ∈ conv{u 1 , . . . , u m }.

Proof of the multivariate equioscillation theorem

First, if min a∈R n m(a) = 0 then by the weak Haar condition there is a unique solution (a T φ(x i ) = f (x i ) with n linearly independent vectors φ(x i ) uniquely defines a), which is trivially identified with a vacuous multivariate equioscillation condition. We now suppose that min a∈R n m(a) > 0, hence the subdifferential formula (6) holds true.

Existence of a minimizer

By the weak Haar condition, there exists n distinct points x 1 , . . . , x n ∈ X such that H(x 1 , . . . , x n ) is nonsingular. We build lower bounds for m(a). First

m(a) ≥ max x∈X |a T φ(x)| -|f (x)| ≥ (max x∈X |a T φ(x)|) -M
where M is an upper bound of the continuous function f inside the compact X. Finally performing the maximum over {x 1 , . . . , x n } ⊆ X we obtain the lower bound (max x∈{x1,...,xn} |a T φ(x)|) -M . The pointwise maximum in this last lower bound is

H(x 1 , . . . , x n ) T a ∞ , which is coercive since H(x 1 , . . . , x n ) is nonsin- gular (indeed √ n H T a ∞ ≥ H T a 2 ≥ λ min (HH T
) a 2 while λ min (HH T ) > 0 because H is nonsingular). Continuous coercive functions defined in R n have a minimizer.

Necessity of the multivariate equioscillation condition at the minimizer

Consider a minimizer a * ∈ R n , hence 0 ∈ ∂m(a * ) = conv A, where is A given in [START_REF] Sukhorukova | Chebyshev Multivariate Polynomial Approximation: Alternance Interpretation[END_REF]. By Carathéodory theorem, 0 is the convex hull of d ≤ n + 1 vectors of A, i.e., by Lemma 6 we have G(x 1 , . . . , x d )λ = 0 with λ = 0 , λ i ≥ 0 and x 1 , . . . , x d ∈ act(a * ).

(

) 8
Therefore from ( 5) we obtain H(x 1 , . . . , x d )u = 0 with u i = (a * , x i )λ i . Finally u i (a * , x i ) = (a * , x i ) 2 λ i ≥ 0 proving that the multivariate equioscillation condition holds.

Sufficiency of the multivariate equioscillation condition for a minimizer

Now, we prove that the multivariate equioscillation condition at a * implies 0 ∈ ∂m(a * ). This condition at a * means there exists x 1 , . . . , x d ∈ act(a * ) and a kernel vector u = 0 of H(x 1 , . . . , x d ) such that (a * , x i ) u i ≥ 0. Again from [START_REF] Powell | Approximation Theory and Methods[END_REF] we have G(x 1 , . . . , x d )λ = 0 with u i = (a * , x i ) λ i , so that λ = 0 and by the multivariate equioscillation condition 0 ≤ (a * , x i ) u i = (a * , x i ) 2 λ i = λ i . Finally, Lemma 6 proves that 0 is in the convex hull of the columns of G(x 1 , . . . , x d ) which are subgradients of m at a * . Therefore 0 ∈ ∂m(a * ) and a * is a minimizer.

Strong equioscillation implies strong uniqueness

Consider a minimizer a * ∈ R n , so 0 ∈ ∂m(a * ). We restart at (8) but with the additional assumption that the strong multivariate equioscillation condition holds: H(x 1 , . . . , x d ) is full rank, hence so is G(x 1 , . . . , x d ) and d ≥ n + 1, and u i > 0, so λ i > 0, for all i ∈ {1, . . . , d}. Since the columns of G := G(x 1 , . . . , x d ) are subgradients at a * we have a piecewise affine lower bound m(a) ≥ m(a * ) + l(a -a * ) with l(a -a * ) = max G T (a -a * ) . The maximum is homogenous for positive scalar hence we have the lower bound l(a -a * ) = a-a * 1 l (a-a * ) a-a * 1 ≥ α a-a * 1 with α = min a-a * 1=1 l(a-a * ). By Lemma 5 the piecewise linear function l has 0 as unique minimizer, therefore α > 0. Finally, m(a) ≥ m(a * ) + α a -a * 1 , which proves the strong uniqueness since a X,φ = max x∈X |a T φ(x)| is a norm by the weak Haar condition, and all norms are equivalent in R n .

Related work

Rice [START_REF] Rice | Tchebycheff approximation in several variables[END_REF] gave a necessary and sufficient condition for the optimality of multivariate Chebyshev approximation in termes of isolation of active indices3 : the active indices act(a * ) are isolable if there is a function a T φ(x) that has the same sign as e(a * , x) on act(a * ), i.e., ∀x ∈ act(a * ) , (a * , x) ((a * ) T φ(x)) > 0.

(

Informally, since (a * , x)φ(x) are the subgradients at a * , Rice's isolation condition means that the vector a * has a positive scalar product with all subgradients, therefore -a * is a descent direction and a * cannot be a minimizer. When act(a * ) = {x 1 , . . . , x d }, Equation ( 9) can be written G(x 1 . . . , x d ) T a * > 0, which is related to the multivariate equioscillation condition by Gordan's alternative theorem: the latter proves that there is no a * such that G(x 1 . . . , x d ) T a * > 0 is and only if there exists λ = 0, λ i ≥ 0, such that G(x 1 . . . , x d )λ = 0. That is, there is no descent direction is (informally) equivalent to the multivariate equioscillation condition. The non-uniqueness necessary condition given in [4, Theorem 2 page 448] is related to the failure of the strong equioscillation: we can assume that u i = 0 in the definition of multivariate equioscillation, otherwise the active indices x i with u i = 0 can be removed. If a * is an optimal solution that is not unique then H(x 1 , . . . , x K ) is not full rank, therefore neither is H(x 1 , . . . , x K ) T and there exists 0 = v ∈ R n such that H(x 1 , . . . , x K ) T v = 0. This later means v T φ(x i ) = 0 therefore the active indices x i belong to {x ∈ X : v T φ(x) = 0}, called an isolating curve by Rice.

The optimality condition for convex optimization is used and reformulated in [START_REF] Sukhorukova | Chebyshev Multivariate Polynomial Approximation: Alternance Interpretation[END_REF], giving rise to a condition for the optimality of multivariate Chebyshev approximation in terms of the nonempty intersection of two convex hulls of vectors related to gradients. The authors explain in [START_REF] Roshchina | Uniqueness of solutions in multivariate chebyshev approximation problems[END_REF] that this condition is a reformulation of Rice's isolation condition.

Some multivariate problem instances

The first example shows an instance with a strongly unique solution.

Example 7. Let X = [0, 1] n ⊆ R n , f (x) = x 2 1 +• • •+x 2 n and φ(x) = (1, x 1 , . . . , x n
), so that we approximate f by affine functions a T φ(x). The weak Haar condition is satisfied since H 0, e 1 , . . . , e n , where e i is the i th basis vector, is nonsingular, so there is an optimal solution. Let us test the approximation a * = (-n 8 , 1, . . . , 1) using the multivariate equioscillation theorem. The error absolute value |e(a * , x)| has 2 n + 1 maximizers4 , which are all corners of the box X and its midpoint, whose error are -n 8 for the corners and n 8 for the midpoint. The Haar matrix at these active indices is as follows: its first 2 n columns correspond to corners, while the last column correspond to the midpoint. For example for n = 3 we have H (0, 0, 0), (0, 0, 1), (0, 1, 0), . . . , (

   1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 2 0 0 1 1 0 0 1 1 1 2 0 1 0 1 0 1 0 1 1 2     . 1, 1, 1), ( 1 2 , 1 2 , 1 2 ) (10) equals to 
We can observe and generalize to arbitrary n that each line contains exactly

2 n
2 zeros and as many ones, and 1 2 at the end. As a consequence, the vector (-1, . . . , -1, 2 n ) is a kernel vector, which satisfies the multivariate equioscillation condition. Therefore the multivariate equioscillation theorem proves that

-n 8 + x 1 + • • • + x 2 is
one optimal solution of this Chebyshev approximation problem. Furthermore the Haar matrix is full rank and the kernel vector has only nonzero components, hence the strong multivariate equioscillation condition holds and this solution is strongly unique.

The second example shows an example where the strong multivariate equioscillation condition does not hold, but where further numerical investigations let us conjecture there is a unique solution. This is related to the singular Chebyshev approximation problems that were investigated in [START_REF] Osborne | A note on singular minimax approximation problems[END_REF][START_REF] Watson | A multiple exchange algorithm for multivariate chebyshev approximation[END_REF][START_REF] Reemtsen | Modifications of the first remez algorithm[END_REF][START_REF] Reemtsen | Discretization methods for the solution of semi-infinite programming problems[END_REF].

Example 8. Let X = [0, 1] × [0, 1], f (x) = x 2 1 + 2x 2 2 -1 2 x 1 x 2 and φ(x) = (1, x 1 , x 2 )
, so that again we approximate f by affine functions a T φ(x) = a 1 + a 2 x 1 + a 3 x 2 . Again the weak Haar condition holds and implies the existence of a ). The Haar matrix for these maximizers is

H (0, 1), (1, 0), ( 1 2 , 1 2 ) =   1 1 1 0 1 1 2 1 0 1 2   . ( 12
)
We can extract the kernel vector u = (-1, -1, 2), which satisfies the multivariate equioscillation condition. Therefore -3 16 + 3 4 x 1 + 7 4 x 2 is one optimal solution of the Chebyshev approximation problem. The Haar matrix is not full rank, so the multivariate equioscillation theorem does not allow proving the uniqueness. Let us investigate in more details this situation. The subgradient matrix is

G (0, 1), (1, 0), ( 1 2 , 1 2 ) =   -1 -1 1 0 -1 1 2 -1 0 1 2   , (13)
whose kernel vector λ = (1, 1, 2) is positive as expected. The three columns of G are three subgradient, which give rise to a piecewise linear lower bound l(a) = max G T (a -a * ) as defined in Subsection 3.4. But in the present case, l(a) does not allow to prove uniqueness because it is constant in the direction v = (-1, 1, 1), which is a kernel vector of G T . The upper row of Figure 2 shows four directional restrictions of both m(a) and its lower bound l(a). The first three directions are columns of G, hence steepest ascent directions of the lower bound, and we observe expected strong univariate minimizers. The fourth direction is the kernel vector v of G T , so as expected l(a) is flat in this direction, and we can see that a second order curvature makes the minimizer unique, but not strongly unique. The lower two graphics of Figure 2 displays a finite pointwise maximum showing the same behavior: the unique minimizer is strongly unique in one direction, and not strongly unique in the other direction.

The following example is taken from [START_REF] Roshchina | Uniqueness of solutions in multivariate chebyshev approximation problems[END_REF] and shows a case with infinitely many optimal solutions to the Chebyshev approximation problem.

Example 9. Let X = {x ∈ R 2 : x 2 1 + x 2 2 ≤ 1}, f (x) = x 6 1 + x 6 2 + 3x 4 1 x 2 2 + 3x 2 1 x 4 2 + 6x 1 x 2 2 -2x 3 1 and φ(x) = (1, x 1 , x 2 , x 2 1 , x 2 2 , x 1 x 2 ).
As shown in [START_REF] Roshchina | Uniqueness of solutions in multivariate chebyshev approximation problems[END_REF], for a * = (1, 0, 0, 0, 0, 0) the error extrema are (-1, 0), ( 12 , -

√ 3 2 ), 1 2 , √ 3 2 , (-1 2 , - √ 3 2 ), (-1 2 , √ 3
2 ) and (1, 0), with errors (2, 2, 2, -2, -2, -2). The Haar matrix is

H((-1, 0), . . . , (1, 0)) =         1 1 1 1 1 1 -1 1 2 1 2 -1 2 -1 2 1 0 - √ 3 2 √ 3 2 - √ 3 2 √ 3 2 0 1 1 4 1 4 1 4 1 4 1 0 3 4 3 4 3 4 3 4 0 0 - √ 3 4 √ 3 4 √ 3 4 - √ 3 4 0         , (14)
with kernel vector (1, 1, 1, -1, -1, -1), which satisfies the multivariate equioscillation condition. As previously, the Haar matrix is not full rank so the strong multivariate equioscillation condition does not hold. In Figure 3, we again display the directional restrictions of the piecewise linear lower bound in the direction of each column of G, where we see expected strong univariate minimizers, and in the direction of the kernel vector v = (-1, 0, 0, 1, 1, 0) of G T , which this time shows a continuum of minimizers. The kernel vector v is the direction where m(a) is constant, therefore, as one expects, (a * + tv) T φ(x) = 1 -t + tx 2 1 + tx 2 2 for t ∈ [START_REF] Cheney | Introduction to Approximation Theory[END_REF][START_REF] Nesterov | Lectures on Convex Optimization[END_REF] corresponds to the description of the minimizers given in [START_REF] Roshchina | Uniqueness of solutions in multivariate chebyshev approximation problems[END_REF]. The lower two graphics of Figure 3 displays a finite pointwise maximum showing the same behavior.

The Haar matrix usually contains a row of 1, which corresponds to a constant basis function. In this case, no kernel vector can have components with constant sign, therefore the optimal solution must have some positive and negative error extrema, which corresponds to our intuition. However, the following simple example shows that when the constant basis function is not included some optimal solution may have error extrema with constant sign.

Example 10. Let X = [-1, 1] × [-1, 1], f (x) = x 2 1 + x 2 2 and φ(x) = (x 1 , x 2
), so that we approximate f by linear functions a T φ(x) = a 1 x 1 + a 2 x 2 . The weak Haar condition is satisfied since H((1, 0), (0, 1)) is nonsingular. Let us test the approximation a * = (0, 0) using the multivariate equioscillation theorem. The error has four maximizers at the corners of the box with all the positive error 2, and no other extrema. The Haar matrix is

H (-1, -1), (-1, 1), (1, -1), (1, 1)) = -1 -1 1 1 -1 1 -1 1 (15)
with kernel vector (1, 1, 1, 1). The multivariate equioscillation condition is satisfied, therefore 0 is one best linear approximation of f (x) in X. The Haar matrix is full rank and the kernel vector has no negative components, therefore the strong equioscillation condition holds and the solution is strongly unique. Finally, the following example shows an untypical case where there is only one active index and the multivariate equioscillation theorem succeeds. It is presented to show that K = 1 actually makes sense in the definition of the multivariate equioscillation condition.

Example 11. Let X = [-1, 1] × [-1, 1], f (x) = 2 -x 2
1 -x 2 2 and φ(x) = (x 1 , x 2 ), so that we approximate again f by linear functions a T φ(x) = a 1 x 1 +a 2 x 2 . Let us test the approximation a * = (0, 0) using the multivariate equioscillation theorem. The error is 0 on the four corners of the box, and has a unique maximizer at (0, 0) with error 2. The Haar matrix is H (0, 0) = 0 0 (16)

with kernel vector 1. The multivariate equioscillation condition is satisfied, therefore 0 is one best linear approximation of f (x) in X. In fact, this seemingly strange situation is quite normal from the point of view of convex analysis: the column of the Haar matrix is a subgradient, and having a null subgradient entails being a minimizer.

Appendix A. Proof of Lemma 1

It is well known if φ : X → R n is continuous and satisfies the Haar condition on X then the determinant of all square Haar matrices H(x 1 , . . . , x n ), with x 1 < • • • < x n ∈ X, have the same sign, see [1, proof of lemma page 74]. Now, the Haar matrix H(x 1 , . . . , x n+1 ) is full rank (because Haar condition entails det H(x 1 , . . . , x n ) = 0) so its kernel is dimension 1. Let us consider one kernel element 0 = u ∈ R n+1 , i.e., n+1 i=1 u i φ(x i ) = 0. For an arbitrary i ∈ {1, . . . , n + 1}, we have u i = 0 otherwise j =i u i φ(x j ) = 0 and H i := H(x 1 , . . . , x i-1 , x i+1 , . . . , x n+1 ) would be singular contradicting Haar contidition. For i ∈ {1, . . . , n} we have φ(x i ) =j =i uj ui φ(x j ) and replacing the column φ(x i ) in H i+1 by this sum shows that det H i+1 is equal to

φ(x 1 ) • • • φ(x i-1 ) - j =i uj ui φ(x j ) φ(x i+2 ) • • • φ(x n+1 ) = -ui+1
ui det H i , (A.1) the last equality obtained using the multilinearity and alternativity of the determinant. The determinants det H i and det H i+1 having the same sign, we conclude that u i and u i+1 have opposite signs.

Figure 1 :

 1 Figure 1: Pointwise maximum of Example 4, with the subdifferential evaluated at several points.

4 Figure 2 :, 7 4 )

 424 Figure 2: The four upper graphics show four directional restrictions of m(a) in blue and its piecewise linear lower bound l(a) in orange, showing a unique non-strongly unique minimizer. The two lower graphics show a similar situation with a finite pointwise maximum.

4 Figure 3 :

 43 Figure 3: The seven upper graphics show seven directional restrictions of m(a) in blue and its piecewise linear lower bound l(a) in orange, showing infinitely many minimizers. The two lower graphics show a similar situation with a finite pointwise maximum.

In the convex analysis litterature, the wording pointwise supremum applied to infinite possibly compact index sets, and the working pointwise maximum is restricted to finite index sets.

A similar argument is used in[5, Equation (7.6) page 74].

The error function e(a * , x) = (a * ) T φ(x) -f (x) is a strictly concave quadratic function, whose global maximizer is the midpoint of the box. Since it is strictly concave, all other local maximizers are strictly lower than the global maximizer, and strictly larger than the global minimizers. These latter can only be at the corners of the box.


Patricia Garcia-Muñoz
email: patricia.gmunoz@upm.es-nicolas

Fernando Fresno

Christophe Lefevre

Didier Robert

Nicolas Keller
email: nkeller@unistra.fr

Influence of the solid titanium source on the activity of La 1-x Ti x FeO 3 photo-CWPO catalysts under UV-A light

Keywords: dual Ti-substituted LaFeO3 catalyst, photoassisted CWPO, photocatalysis, advanced oxidation processes, water treatment

In the search for more efficient low-temperature catalysts able to oxidise biorecalcitrant pollutants in water, the selective substitution of La 3+ by Ti 3+ cations was reported to boost the UV-A light driven activity of LaFeO3 orthoferrites as H2O2-mediated photo-CWPO catalysts. La1-xTixFeO3 catalysts with 0 < x < 0.11 were obtained by a modified Pechini sol-gel route via the addition of a solid source of titanium during the synthesis. By using a span of crystallized and amorphous TiO2, the influence of the titanium source on the substitution rate in the orthoferrite network and on the performances of the catalysts under UV-A light was demonstrated using the degradation of 4-chlorophenol as test reaction.

The amorphous content of the TiO2 precursor is proposed to be the key factor driving the substitution of La 3+ , the largest substitution of 11% being obtained using a dried sol-gel TiO2 precursor.

We suggest that the substitution proceeds during the thermal treatment via a solid-solid diffusion between TiO2 and the amorphous LaFeO3 before it crystallises. The catalyst robustness was influenced by the TiO2 nature. Small size Ti-LaFeO3 crystallites strongly lowered or blocked the Fe release, while no improvement was observed for large size crystallites vs. the pristine material, what was associated to a poor homogenization of the titanium in the network. We showed that the mineralization activity was proportional to the substitution rate of La1-xTixFeO3 catalysts, while a similar relationship could be drawn for the degradation activity only in the case of highly robust catalysts with pure heterogeneous surface reactions.

INTRODUCTION

The increase in the amounts of polluted water and the existence of refractory pollutants in the water streams prompt the search for new processes able to meet the regulations and the legislation requirements in force, and to reach the necessary abatement of the biorecalcitrant pollutants [START_REF] Phan | Heterogeneous photo-Fenton degradation of organics using highly efficient Cu-doped LaFeO3 under visible light[END_REF].

Advanced Oxidation Processes (AOPs) have shown their ability to oxidise recalcitrant pollutants of water that are not removed by conventional treatments, through the generation of HOx° radicalar species that can stepwise oxidise the organic matter up to CO2 and H2O [START_REF] Serpone | Light-driven advanced oxidation processes in the disposal of emerging pharmaceutical contaminants in aqueous media: A brief review[END_REF][START_REF] Malato | Decontamination and disinfection of water by solar photocatalysis: Recent overview and trends[END_REF][START_REF] Pliego | Trends in the Intensification of the Fenton Process for Wastewater Treatment: An Overview[END_REF]. There is a vast classification of AOPs combining O3, O2, UV, H2O2 and/or a catalyst into a wide span of processes such as photocatalysis, ozonation, Fenton or photo-Fenton processes (or their heterogenous counterparts as CWPO and photo-CWPO, respectively), etc.

Photocatalysis and CWPO-like processes have received attention due to their efficiency and low cost. Though photocatalysis can achieve full mineralization of a wide spectra of dissolved pollutants, it suffers from low reaction rates and UV-light dependence for the most active TiO2 references. Also photo-CWPO takes advantage of higher reaction rates, but does not reach complete mineralization.

Combining both AOPs in a single light-driven catalyst is thus considered as a worth approach for synergistically benefitting from their advantages and overcoming their respective drawbacks [START_REF] Pliego | Trends in the Intensification of the Fenton Process for Wastewater Treatment: An Overview[END_REF][START_REF] García-Muñoz | Sulfonamides photoassisted oxidation treatments catalyzed by ilmenite[END_REF][START_REF] Pliego | Improving the Fenton process by visible LED irradiation[END_REF][START_REF] Garcia-Muñoz | Ti-substituted LaFeO3 perovskite as photoassisted CWPO catalyst for water treatment[END_REF].

LaFeO3 is a versatile material that has been employed as catalyst in different AOPs at dark and illuminating conditions [START_REF] Rao | Efficient degradation of diclofenac by LaFeO3-Catalyzed peroxymonosulfate oxidation---kinetics and toxicity assessment[END_REF][START_REF] Parrino | Cu-substituted lanthanum ferrite perovskites: Preparation, characterization and photocatalytic activity in gas-solid regime under simulated solar light irradiation[END_REF][START_REF] Song | Enhanced photocatalytic decomposition of an organic dye under visible light with a stable LaFeO3/AgBr heterostructured photocatalyst[END_REF][START_REF] Afzal | High surface area mesoporous nanocast LaMO3 (M=Mn, Fe) perovskites for efficient catalytic ozonation and an insight into probable catalytic mechanism[END_REF]. It is a p-type semiconductor with a narrow band gap value (between 2.1-2.7 eV) and a orthoferrite (distorted) perovskite (ABO3) structure, whose main characteristic is that the B cation presents redox properties associated to the oxygen mobility, and that develops redox surface activity. Further, its properties and in particular its redox properties can be modified by a controlled (partial) replacement of the A or B cations [7-10, 12, 13]. In photocatalysis or photo-Fenton catalysis, works on the partial cationic A (La) or B (Fe) sub-lattice substitution of the LaFeO3 perovskite remain scarce, notably in comparison to the wide span of heteroatom-doped TiO2 systems studied.

The removal of organic dyes in water by photocatalysis was improved via the partial substitution of La by Li [START_REF] Hou | Preparation, characterization and investigation of catalytic activity of Li-doped LaFeO3 nanoparticles[END_REF] or Ca, or that of Fe by Zn or Mn [START_REF] Wei | Synthesis, magnetization and photocatalytic activity of LaFeO3 and LaFe0.5Mn0.5-xO3-δ[END_REF]. Some works were reported in heterogeneous photo-Fenton catalysis under visible or UV-A light, targeting dyes, methylparaben micropollutant or acetic acid [START_REF] Sannino | Structured catalysts for photo-Fenton oxidation of acetic acid[END_REF][START_REF] Orak | Photocatalytic and photo-Fenton-like degradation of methylparaben on monolith-supported perovskite-type catalysts[END_REF]. Substitution with cations with different radii, valence-electron numbers and coordination numbers was notably reported to enhance the redox activity by producing electron-deficient or electron-rich species at the surface [START_REF] Pan | Ca and Cu doped LaFeO3 to promote coupling of photon carriers and redox cycling for facile photo-Fenton degradation of bisphenol A[END_REF].

Our precursor work first showed the high-prospect capability of a single-phase Ti-substituted LaFeO3 orthorferrite material to behave as a dual light-driven catalyst able to take advantage of a synergistic effect between heterogeneous photo-Fenton catalysis and photocatalysis under UV-A light [START_REF] Garcia-Muñoz | Ti-substituted LaFeO3 perovskite as photoassisted CWPO catalyst for water treatment[END_REF]. It generates HOx° radicals by Fe-catalyzed H2O2 decomposition under UV-A light (photo-CWPO), while simultaneously showing photocatalytic activity thanks to its semiconductor nature with a suited band gap and well-located electronic bands. Photocatalysis was proposed here to allow the cleaning of the catalyst surface from poisoning Fe-carboxylate complex intermediates blocking the mineralization and leading to Fe leaching to the aqueous medium. The catalyst was prepared by means of a simple modified Pechini sol-gel route via the addition of a solid source of Ti heteroatom during the synthesis.

The partial substitution of La by Ti ions stabilised in the state 3+ in the A-site of the orthoferrite network was demonstrated by XRD analysis with Rietvelt structural refinement and XPS surface analysis [START_REF] García-Muñoz | Synergy Effect Between Photocatalysis and Heterogeneous PhotoFenton Catalysis on Ti-Doped LaFeO3 Perovskite for High Efficiency Light-Assisted Water Treatment[END_REF][START_REF] García-Muñoz | Ti-Modified LaFeO3/β-SiC Alveolar Foams as Immobilized Dual Catalysts with Combined Photo-Fenton and Photocatalytic Activity[END_REF]. It enhanced both the catalyst performances and its robustness compared to the pristine LaFeO3, as full mineralization of 4-chlorophenol (4-CP) as model pollutant was achieved under UV-A light, with improved degradation and mineralization rates and with no loss of activity after 5 test cycles. This was proposed to result from an enhanced photogenerated charge carriers availability at the surface and an electronic enrichment of surface iron. We however reported that the nature of the Ti precursor might impact the properties of the catalyst and thus its behavior in water treatment [START_REF] García-Muñoz | Synergy Effect Between Photocatalysis and Heterogeneous PhotoFenton Catalysis on Ti-Doped LaFeO3 Perovskite for High Efficiency Light-Assisted Water Treatment[END_REF][START_REF] Garcia-Muñoz | Highly robust La1-xTixFeO3 dual catalyst with combined photocatalytic and photo-CWPO activity under visible light for 4chlorophenol removal in water[END_REF].

Therefore, this work aimed at studying the influence of the Ti precursor source on the performances of dual Ti-modified LaFeO3 catalysts under UV-A light. To this end, a set of commercially available TiO2 and an amorphous TiO2 sample were selected as solid precursors, and the removal of the 4-chlorophenol (4-CP) under UV-A light was used as test reaction, as a example of industrial waste water refractory pollutant.

MATERIALS AND METHODS

Materials

Reactants for synthesis and photocatalytic reactions were used as purchased from Sigma-Aldrich: H2O2 (>30%), La(NO3)3•6H2O, Fe(NO3)3•9H2O, citric acid hydrate (C6H8O7 • xH2O, 99.5%), titanium tetraisopropoxide (TTIP, 97%) and 4-Cl-Phenol (>99%). NH3 (30% aqueous solution) was obtained from Carlo Erba. Aeroxide © TiO2 P25 and Aeroxide © TiO2 P90 were provided by Evonik. TiO2 UV100 Hombikat was obtained from Sachtleben Chemie GmbH while TiO2 CristalACTiVE PC500 was provided by Tronox.

Synthesis of Ti-modified LaFeO3

La1-xTixFeO3 catalysts were synthesised by a modified Pechini sol-gel route. La(NO3)3•6H2O (108.3 g/L) and Fe(NO3)3•9H2O (133 g/L) in stoichiometric ratio were dissolved in deionised water (30 mL), and the metal ions were subsequently complexed by adding the stoichiometric amount of citric acid (70 g/L). For the synthesis of the catalysts labelled as P25-LFO, P90-LFO, UV-LFO and PC-LFO, Ti was included by incorporating Aeroxide© TiO2 P25, Aeroxide© TiO2 P90, UV100 Hombikat and PC500, respectively, into the starting ferrite sol under vigorous stirring, in a quantity equivalent to 10 wt.% with respect to the theoretically obtained amount of LaFeO3. The resultant suspensions were heated up to 80 °C with a 2 °C min -1 ramp and kept at this temperature for 48 h for a total gelification to be attained. The dry gel was calcined with the heating cycle described by Gosavi et al. [START_REF] Gosavi | Pure phase LaFeO3 perovskite with improved surface area synthesized using different routes and its characterization[END_REF], namely heating up with a ramp of 5 °C min -1 , keeping the temperature at 600 °C for 2 h, and then reaching a second plateau at 800 °C for 2 h.

For the synthesis of the SG-LFO catalyst, a laboratory-made dried amorphous TiO2 was used as source of titanium. This TiO2 was prepared by means of a sol-gel procedure in alkaline conditions using TTIP as precursor. Typically, 17.8 g of TTIP was initially poured into 40 mL of ethanol, and then the same volume of water was added dropwise. Aqueous NH3 was adedd until a pH of 9 was reached, and then the solution was kept at room temperature under stirring until a dry paste resulted, which was dried further at 110 °C for 12 h to finally obtain the amorphous titania used in the procedure described above.

Keeping the titanium dopant apart, the stoichiometry of the reaction leading to the formation of LaFeO3 orthoferrite is represented by Eq.1 [START_REF] Qi | A simple way to prepare nanosized LaFeO3 powders at room temperature[END_REF]:

!"($% & ) & (6* + % + -.($% & ) & (9* + % + 0 1 * 2 % 3 (* + % → !"-.% & + 60% + + 3$ + + 6* + % [Eq.1]
Unmodified lanthanum ferrite, referred to as LFO, was obtained with the same procedure without adding titania and used as reference material.

Characterisation techniques

X-ray diffraction (XRD) patterns were recorded using a Bruker D8 Advance equipment with a monochromated copper X-ray source (Kα = 1.54056 Å), in θ/θ mode and with a scan step of 0.02°.

Fullprof software was used for Rietveld refinement, where the line shape of the diffraction peaks was created using the modified Thompson-Cox-Hasting function. Instrumental peak width was determined by measuring the scattering of corundum (NIST standard SRM 1976b) [START_REF] Mccusker | Rietveld refinement guidelines[END_REF][START_REF] Rodríguez-Carvajal | Recent advances in magnetic structure determination by neutron powder diffraction[END_REF]. The mean crystallite size (average size of the coherent diffracting domains) was estimated with the Scherrer equation using the full patterns after refinement, assuming sphere-shaped crystallites as usually done and considering instrumental broadening.

Surface areas were determined from nitrogen adsorption isotherms, measured with a Micrometrics Tristar 3000 equipment at -196 °C after outgassing the samples at 250°C for 16 h.

The metal contents were determined by ICP-OES (inductively coupled plasma optical emission spectroscopy) with an Optima 7000 DV equipment from Perkin Elmer. Fluorescence lifetimes were determined by time-correlated single photon counting (TCSPC) in a Mini-t equipment (Edinburgh Instruments). An EPL375 laser diode (445.2 nm, 90.7 ps pulse width, 1 MHz repetition rate) was used as excitation source. A band pass filter at 600 ± 25 nm was positioned between the sample and the detector.

Reaction procedure

4-chlorophenol degradation experiments were performed in batch-mode in a borosilicate glass beaker reactor under atmospheric pressure. The temperature was kept constant at 20 ± 2 °C with a thermostated bath. Typically, 100 mL of an aqueous 4-CP solution (25 mg L -1 , equivalent to 14.8 ppm total organic carbon) were placed in the reactor and then the catalyst (50 mg, 0.5 g L -1 ) was added and kept dispersed with vigorous magnetic stirring in the dark for 1h to allow the adsorption/desorption equilibrium to be attained. H2O2 was then added in an amount equivalent to the stoichiometric concentration for complete mineralization of the initial pollutant, i.e. 125 mg L -1 . Irradiation was then provided by Philips 24W/10/4P UV-A lamps (60 W/m 2 ; λmax=365 nm). During the reaction, samples (8 mL) were taken from the reactor at regular intervals by means of a syringe and filtered using a 0.2 μm membrane filters to remove the catalyst powder. The decay of the concentration of 4-CP was followed by keeping track of the intensity of the absorption peak at λ= 224 nm [START_REF] Pera-Titus | Degradation of chlorophenols by means of advanced oxidation processes: a general review[END_REF] in the filtered samples by means of a UV-vis spectrophotometer (VWR UV-1600 PC). Additionally, the total organic carbon (TOC) content of the filtered samples was measured with a Shimadzu TOC-L analyzer. The Fe leached into the reaction medium during the reaction was quantified by using the ortho-phenanthroline method, as described in Supporting Information S1 [START_REF] Sandell | Colorimetric Determination of Traces of Metals[END_REF].

RESULTS AND DISCUSSION

Catalyst characterization

Figure S1 shows the XRD patterns of the unmodified and different Ti-modified LaFeO3 catalysts, as well as their Rietveld refinement profiles. The XRD patterns of LFO displays the diffraction peaks of the (110), ( 112), (022), ( 220), (312) and (224) planes of the LaFeO3 phase (JCPDS card 070-7777) at 2θ = 22.6, 32.2, 39.7, 46.1, 57.4 and 67.3, respectively. A small fraction of α-Fe2O3 (11%) with a mean crystallite size of 35 nm was observed as an additional phase (JCPDS card 033-06664). Regardless of the employed source of titanium, all the patterns of the Ti-modified LaFeO3 samples can be indexed to the Pbnm orthorhombic unit cell of LaFeO3, with no significant variation regarding the α-Fe2O3 additional phase. Neither La2O3 nor TiO2 are detected as extra phases. ICP-OES analysis reveals the atomic titanium content in the Ti-containing LaFeO3 catalysts ranging from 2.9% when prepared using the P90-TiO2 precursor up to 4.0% for the amorphous TiO2 nanopowder (Table 1). The cation distribution in the LaFeO3 structure from Rietveld refinements, as well as the crystallographic data, are collected in Table S1 and S2, respectively.

Depending on the heteroatom atom nature, substitution has been evidenced on both the A and the B sites of the LaFeO3 structure, although cations in a high oxidation state usually substitute and locate in the Fe 3+ sites [START_REF] Jamil | Visible light activity of BaFe1-xCuxO3-d as photocatalyst for atrazine degradation[END_REF][START_REF] Rashmi | Solar light responsive Sm-Zn ferrite nanoparticle as efficient photocatalyst[END_REF]. However, many external factors have been shown to influence the cation substitution in oxides (like the synthesis temperature or the cooling rate) [START_REF] De Guire | The cooling rate dependence of cation distributions in CoFe2O4[END_REF][START_REF] Zhang | Temperature Dependence of Cation Distribution and Oxidation State in Magnetic Mnâˆ'Fe Ferrite Nanocrystals[END_REF], and no empirical rules related to steric or electronic criterion drive the location of the heteroatom. Refinement of both Ti/La and Ti/Fe site occupations demonstrated that Ti atoms were located preferentially in the lanthanum A site and substituted only the La atoms, independently of the titanium source. Considering a La1-xTixFeO3 labelling with x being the partial Ti→La substitution rate for La atoms in the A-site, the substitution rate within the orthoferrite structure was 5, 6, 7 and 8 at.% for the P25-TiO2, PC500-TiO2, UV100-TiO2, P90-TiO2 precursors, respectively, and increased up to 11 at.% for the dried TiO2 precursor (Tables 1 andS1).

The TiO2 precursors used differ notably regarding crystalline phases (anatase and anatase/rutile mixture), crystallinity, crystallite size and specific surface area, and their main physico-chemical properties are listed in Table 1. The phase composition refers only to the crystalline TiO2 phases, while the amorphous content was estimated by means of the method described in [START_REF] Cano-Casanova | One step hydrothermal synthesis of TiO2 with variable HCl concentration: Detailed characterization and photocatalytic activity in propene oxidation[END_REF]. It is worth noting that the partial x substitution rate was found to be related to the amorphous content of the precursor used: the higher the amorphous content, the higher the substitution rate, as evidenced in Figure 1 top.

Using the fully amorphous sol-gel TiO2 as solid source of titanium atoms led to get the highest substitution rate of 11 at.%. We can therefore propose that the partial Ti→La substitution proceeds in the course of the thermal treatment through a solid-solid diffusion between TiO2 and the amorphous LaFeO3 before it crystallises. The atomic titanium diffusion into the amorphous LaFeO3 network is proposed to be favoured by the amorphous nature of the solid TiO2 precursor, what in consequence would increase the substitution rate. Differences in terms of specific surface areas for the La1-xTixFeO₃ catalysts might result from the relative ratio between the kinetic of crystallization of the LFO structure and that of the diffusion of titanium atoms.

It must be said that the band gap of the catalysts is not affected by the substitution rate within the LFO structure, as the band gap was graphically estimated at 2.04 ± 0.04 eV from the Tauc plots derived from UV-vis diffuse reflectance spectra regardless of whether the catalyst is doped or not, and the Ti precursor source used (Figure S2).

The fluorescence decays of the different catalysts is shown in Figure 1 bottom. In comparison to the pristine LaFeO3 sample, a slower decay was observed for the Ti-substituted La1-xTixFeO3 catalysts, irrespective of the Ti source used and the substitution rate. Using a bi-exponential function, the fluorescence lifetimes were calculated as the average of both parameters [START_REF] Garcia-Muñoz | Highly robust La1-xTixFeO3 dual catalyst with combined photocatalytic and photo-CWPO activity under visible light for 4chlorophenol removal in water[END_REF]. Considering that the fluorescence emission results from transitions from surface electronic states, the higher fluorescence lifetime observed for the La1-xTixFeO3 catalysts can be associated to a longer lifetime of the electrons trapped in these states, that characterises slower radiative recombination kinetics and in consequence an enhanced availability of photogenerated charges for reacting with adsorbed species [START_REF] Charanpahari | Enhanced photocatalytic activity of multi-doped TiO2 for the degradation of methyl orange[END_REF][START_REF] Varma | Dependence of photocatalysis on charge carrier separation in Ag-doped and decorated TiO2 nanocomposites[END_REF]. The SG-LFO catalyst exhibited by far the longest fluorescence lifetime, associated to an enhanced charge separation. However, no straightforward relationship could be established between the lifetime of charges and the substitution rate. First, the crystallite size can have an impact on the carrier lifetime and thus should be considered. For a similar LFO crystallite size, the higher the amount of substituting titanium, the lower the recombination rate, as it is observed when comparing SG-LFO and P25-LFO a the phase composition corresponds to the composition within the crystallised TiO2 phases (without considering the content of amorphous TiO2). A and R represent the anatase and rutile phase, respectively b determined by applying the Scherrer equation to the (101) and (110) peaks of anatase and rutile, at 25.7° and 27.5°, respectively. c determined by using the method developed in ref [START_REF] Zhang | Temperature Dependence of Cation Distribution and Oxidation State in Magnetic Mnâˆ'Fe Ferrite Nanocrystals[END_REF] d cannot be determined as the outgassing under vacuum modifies the properties of the dried sol-gel TiO2 precursor

Catalytic activity

The catalytic behaviour of the full series of Ti-modified La1-xTixFeO3 catalysts has been evaluated

and compared to that of the pristine titanium-free LaFeO3 counterpart. Control experiments confirmed that no photolysis was observed under UV-A light, and that both LFO and Ti-modified LFO catalysts do not show any activity in absence of UV-A light and of H2O2. The activity of the catalysts in CWPO in the dark is shown in Figure S3. Figure 2 shows the evolution with UV-A irradiation time of both 4-CP concentration and TOC amount during the H2O2-mediated photocatalytic reaction, together with that of the Fe released. While the reference LaFeO3 catalyst was able to remove the 4-CP pollutant within 150 min, it did not fully mineralise the effluent and reached a maximum TOC conversion of 75%, with reference kinetic rate constants for 4-CP and TOC conversions of 0.020 min -1 and 0.12 mg L -1 min, respectively. It must be said that the reference LFO catalyst suffered from important Fe leaching to the aqueous solution, with a Fe released value of 0.5 mg/L being detected in water after 120 min of test.

By contrast, the full series of La1-xTixFeO3 catalysts allowed to reach complete mineralization in 90 to 120 min of reaction, with enhanced kinetic rate constants in the 0.17-0.28 mg/Lmin range. The substituted catalysts degrade the 4-CP substrate within shorter times than the pristine catalyst, namely within the 60-120 min range, with enhanced kinetic rate constants in the 0.025-0.042 min -1 .

The Fe leaching during the reaction was also influenced by the nature of the TiO2 precursor used in the modified Pechini synthesis. The leaching observed for both UV-LFO and PC-LFO catalysts was similar to that observed for the pristine LFO catalyst, namely around 0.5-0.6 mg/L, but it strongly decreased to 0.05 mg L -1 for both P25-LFO and P90-LFO catalysts, and further down to undetectable values in the experiment with the SG-LFO catalyst. In consequence, direct correlation between the kinetic rate constant for 4-CP degradation and the substitution rate cannot be drawn, as the high amount of Fe leached during the reactions with the pristine LFO and with both UV-LFO and PC-LFO catalysts can significantly contribute to the k4-CP value calculated, through a classical homogeneous photo-Fenton reaction. By contrast, in the case of the P25-LFO, P90-LFO and SG-LFO catalysts, the k4-CP values derived from experimental data could be assumed not to include any homogeneous photo-Fenton reaction, as those catalysts are characterised by a low (or the absence of) Fe release. So, in the case of pure heterogeneous surface reactions, Figure 3A reveals that the higher the substitution rate, the higher the kinetic rate constant for the 4-CP degradation. It must be said that the k4-CP values calculated in the case of a, in agreement with a significant contribution of free Fe species in solution.

The kinetic rate constant for TOC removal is assumed to be less dependent on the amount of Fe released to the reaction medium, as the homogeneous Fe species are forming complexes with shortchain carboxylic acids, consequently not contributing to a significant extent to the mineralization rate constant. Therefore, a direct relationship between the substitution rate of the La1-xTixFeO3 catalysts and the kTOC was established, and revealed that the higher the Ti à La substitution value, the higher the kinetic rate constant for mineralization (Figure 3B).

As above-mentioned, we cannot completely rule out the indirect role played by a La1-xTixFeO₃/α-Fe2O3 heterojunction through an improved charge separation, however we assume that this is not a predominant factor driving the activity of the catalysts, as no correlation was established between the wt.% content of additional α-Fe2O3 phase, the fluorescence lifetime and the catalytic indicators.

Increasing the amorphous content of the solid titanium source is an elegant way to increase the cationic substitution rate of La by Ti atoms, and in consequence to increase the amount of electronenriched surface Fe that boost the photo-CWPO activity. However, some substituted catalysts did not show an improved robustness in terms of Fe release to the solution compared to the pristine LFO system (UV-LFO and PC-LFO). This might be associated to a poor homogenization of the titanium inside the network that resulted in large LFO crystallite size (37-41 nm), as a strongly enhanced robustness was observed for lower-size LFO crystallites (P25-LFO, P90-LFO, SG-LFO). [a] a pseudo first-order kinetic was used as usually admitted for fitting the 4-CP concentration evolution curves

[b] a zero-order kinetic model was used as usually admitted for fitting the TOC concentration evolution curves

[c] the detection limit of Fe leached to the solution has been measured at 0.001 ppm.

Figure 3.

Influence of the substitution rate x on the kinetic rate constants of (A) 4-CP oxidation (k4-CP) and (B) TOC mineralization (kTOC) obtained using the La1-xTixFeO3 catalysts under UV-A light. In (A), the blue marks correspond to k4-CP values assumed not to contain any significant homogeneous photo-Fenton contribution due to the absence of (or the very low) Fe release to the aqueous media.

CONCLUSION

We have shown that the selective substitution of La 3+ by Ti 3+ cations is a promising way to increase both activity and robustness of LaFeO3 orthoferrites as H2O2-mediated photo-CWPO catalysts for the removal of biorecalcitrant pollutants in water. Controlled cationic substitution was achieved through the addition of a solid source of titanium during a facile citric acid-based Pechini sol-gel synthesis with final thermal treatment at 800°C. The amorphous content of the TiO2 precursor was reported to be the key-factor driving the substitution of La 3+ in the orthoferrite network, the use of a dried sol-gel TiO2 precursor allowing to reach the largest substitution rate of 11% in the La1-xTixFeO3 catalysts in comparison to more crystallized titanium sources. We proposed that the substitution proceeds in the course of the thermal treatment via a solid-solid diffusion between TiO2 and the amorphous LaFeO3 before it crystallizes.

We showed that the performances of the La1-xTixFeO3 catalysts under UV-A light in the removal of 4-chlorophenol in water can be related to the substitution rate in the orthoferrite network. We evidenced that the mineralization kinetic is proportional to the substitution rate, while a similar relationship can be drawn in regards to the degradation kinetic for the high-robustness La1-xTixFeO3 catalysts that operate only via pure heterogeneous surface reactions without any significant photo-Fenton reaction contribution in homogeneous phase. Indeed, the nature of the TiO2 used as titanium source influenced also the robustness of the substituted catalysts by impacting on the size of the resulting La1-xTixFeO3 crystallites. Small size crystallites were proposed to strongly lower or block the Fe release, while no improvement was observed for large size crystallites vs. the pristine material, what was associated to a poor homogenization of the titanium in the network.

Figure 1 .

 1 Figure 1. (top) Influence of the TiO2 amorphous content on the partial Ti→La substitution rate (x) in La

Figure 2 .

 2 Figure 2. Evolution of (A) the relative 4-CP concentration, (B) the relative TOC concentration and (C)

Table 1 .

 1 Main physico-chemical properties of the solid TiO2 precursor, the pristine LaFeO3 and the Ti-substituted LaFeO3 catalysts.

	Catalyst	TiO2 precursor properties

Table 2 .

 2 Kinetic rate constants of 4-CP oxidation and TOC mineralization, TOC conversion achieved after 120 min, and iron leached to the solution.

	Catalyst	k' (4-CP) (min -1 ) a	r0 TOC (mg/Lmin) b	XTOC (120 min)	Fe leached (mg/L)
	LFO	0.020	0.12	75	0.5
	P25-LFO	0.025	0.18	100	0.05
	P90-LFO	0.030	0.20	100	0.05
	UV-LFO	0.040	0.19	100	0.5
	PC-LFO	0.033	0.18	100	0.6
	SG-LFO	0.042	0.28	100	<0.001 c

Acknowledgements

The European Fund for regional development (EFRE/FEDER) is thanked for funding the project PHOTOPUR in the frame of Interreg V program and Offensive Sciences call. PGM acknowledges Comunidad de Madrid through the call Research Grants for Young Investigators from Universidad Politécnica de Madrid for funding the research project SUNCAT4PLAST (APOYO-JOVENES-21-JV4DEB-3-Q2WGKV) investigadores.

with crystallite sizes of 20-21 nm. Second, we cannot rule out an indirect role of the additional α-Fe2O3 minority phase through the formation of a type-II (staggered) La1-xTixFeO₃/α-Fe2O3 heterojunction, both valence and conduction bands of the α-Fe2O3 being more anodic than the corresponding bands of the ferrite after Fermi level alignment [START_REF] Zhang | Self-Sacrificial Template-Driven LaFeO3/α-Fe2O3 Porous Nano-Octahedrons for Acetone Sensing[END_REF]. Although beneficial charge transfer might occur in this model for improving charge separation, no direct correlation was established between the wt.% content of additional α-Fe2O3 phase and the fluorescence lifetime.