Brigitte Bidégaray-Fesquet

Laurent Fesquet
email: laurent.fesquet@imag.fr

A fully non-uniform approach to FIR filtering

We propose a FIR filtering technique which takes advantage of the possibility of using a very low number of samples for both the signal and the filter transfer function thanks to non-uniform sampling. This approach leads to a summation formula which plays the role of the discrete convolution for usual FIR filters. Here the formula is much more complicated but it can be implemented and the evaluation of more elaborate expressions is compensated by the very low number of samples to process.

Introduction

Reducing the power consumption of mobile systemssuch as cell phones, sensor networks and many others electronic devices -by one to two orders of magnitude is extremely challenging but will be very useful to increase the system autonomy and reduce the equipment size and weight. In order to reach such a goal, this paper proposes a solution applicable to FIR filtering which completely rethinks the signal processing theory and the associated system architectures. Today the signal processing systems uniformly sample analog signals (at Nyquist rate) without taking advantage of their intrinsic properties. For instance, temperature, pressure, electro-cardiograms, speech signals significantly vary only during short moments. Thus the digitizing system part is highly constrained due to the Shannon theory, which fixes the sampling frequency at least twice the input signal frequency bandwidth. It has been proved in [START_REF] Mark | A nonuniform sampling approach to data compression[END_REF] and [START_REF] Sayiner | A level-crossing sampling scheme for A/D conversion[END_REF] that Analog-to-digital Converters (ADCs) using a non equi-repartition in time of samples leads to interesting power savings compared to Nyquist ADCs. A new class of ADCs called A-ADCs (for Asynchronous ADCs) based on level-crossing sampling (which produces non-uniform samples in time) [START_REF] Filipp Akopyan | A level-crossing flash asynchronous analog-to-digital converter[END_REF][START_REF] Allier | A new class of asynchronous A/D converters based on time quantization[END_REF] and related signal processing techniques [START_REF] Aeschlimann | Asynchronus fir filters, towards a new digital processing chain[END_REF][START_REF] Saeed Mian Qaisar | Adaptive rate filtering for a signal driven sampling scheme[END_REF] have been developed. This work suggests an important change in the FIR filter design. As sampling analog signals is usually performed uniformly in time, sampling the filter transfer function is also done in a regular way with a constant frequency step. Non-uniform sampling leads to an important reduction of the weight-function coefficients. Combined with a nonuniform level-crossing sampling technique performed by an A-ADC, this approach drastically reduces the compu-tation load by minimizing the number of samples and operations, even if they are more complex.

Principle and notations

For a large class of signal, non-uniform sampling leads to a reduced number of samples, compared to a Nyquist sampling. This feature has already been used in [START_REF] Aeschlimann | Asynchronus fir filters, towards a new digital processing chain[END_REF] to design non-uniform filtering techniques based on interpolation. In this work the authors however used a classical (uniform) filter, that is a usual discretization in time of the impulse response.

Here we want to go further and take advantage of the fact that the filter transfer function (the Fourier transform of the impulse response) is a very smooth function with respect to frequency. It can therefore be well approximated by the linear interpolation of quite few samples.

Level crossing sampling

The initial signals are supposed to be analog ones. The signal which we want to filter is given in the time domain and is denoted by s(t). The filter transfer function is given in the frequency domain and is denoted by H(ω). The result of the filtering process x(t) is then theoretically the convolution of s(t) with the impulse response h(t) which is the inverse Fourier transform of H(ω):

x(t) = +∞ -∞ h(t -τ )s(τ )dτ, h(t) = 1 2π +∞ -∞ H(ω)e -iωt dω.
These signal are sampled in their initial domain using a level crossing scheme. This technique has to be adapted for the filter transfer function. Indeed level crossing has a sense if an order can be defined, for example for a real valued function. The filter transfer function is complex valued, therefore we can choose to sample either when the amplitude crosses some predefined values, or the phase, or both. The samples read (s n , δt n ) for the signal and (H k , δω k ) for the filter transfer function. These samples are formed of a value and the (time or frequency) interval length "elapsed" since the last sample. To give results or describe algorithms we will use the sample times or frequencies defined as t n = t 0 + n 1 δt n ′ and ω k = ω 0 + k only the time and frequency intervals δt n and δω k . We will also denote by

I n = [t n-1 , t n ] and J k = [ω k-1 , ω k ]
the time and frequency intervals.

Linear interpolation

To derive the FIR algorithm and approximate the theoretical integral formula, we form new analog functions from the previously described samples. To this aim we choose linear interpolation and we have

s(t) = n [a n + b n t]χ In , H(ω) = k (α k + β k ω)e i(γ k +δ k ω) χ J k ,
where χ denotes the indicator function of the set given in index. The coefficients a n and b n can be expressed in terms of s n , s n-1 , t n and δt n . The coefficients α k , β k , γ k and δ k can be expressed in terms of H k , H k-1 , ω k and δω k .

In fact these formulae cover the piecewise constant case (only take b n = β k = δ k = 0) in three possible forms: constant on intervals I n or nearest neighbor interpolation, with a possible need to modify the definition of t n and δt n in the algorithms. They also cover two ways to linearly interpolate the complex valued filter transfer function: interpolate separately the amplitude and the phase (α k and β k are real) or interpolate in the complex plane (α k and β k are complex, γ k and δ k are zero).

The digital filter then consists in computing (possibly) for all time

x(t) = +∞ -∞ h(t -τ )s(τ )dτ, h(t) = 1 2π +∞ -∞
H(ω)e -iωt dω.

Deriving a filtering formula in the general context 3.1 A summation formula

The impulse response h(t) can be split in contributions for each frequency sample

h(t) = k h k (t) with h k (t) = 1 2π ω k ω k-1 (α k + β k ω)e i(γ k +δ k ω) e -iωt dω
for which we will give an explicit expression in Section 3.2. Although the piecewise linear function H(ω) has a compact support (we only have a finite number of samples), the functions h k (t) have an infinite support. This is not a problem since the convolution will involve s(t)

which has a compact support. The convolution reads

x(t) = +∞ -∞ h(t -τ )s(τ )dτ = n tn tn-1 h(t -τ )s n (τ )dτ = n k tn tn-1 h k (t -τ )(a n + b n τ )dτ = n a n k h 0 nk (t) + b n k h 1 nk (t)
where

h 0 nk (t) = tn tn-1 h k (t -τ )dτ, h 1 nk (t) = tn tn-1 h k (t -τ )τ dτ.
We obtain a summation formula as in the classical FIR filtering case where it takes the form of a discrete convolution. To be closer to this classical case, we should write this as

x(t) = n s n k h nk (t),
which is possible but the effective expression depends on the type of interpolation used (piecewise constant or linear).

There remains to make explicit these two types of elementary contributions.

Elementary impulse responses

A straightforward computation of the integral formulation for h k (t) yields

h k (t) = α k e iγ k 2π ω k ω k-1 e i(δ k -t)ω dω + β k e iγ k 2π ω k ω k-1 e i(δ k -t)ω ωdω = α k e iγ k e i(δ k -t)ω k -e i(δ k -t)ω k-1 2πi(δ k -t) + β k e iγ k ω k e i(δ k -t)ω k -ω k-1 e i(δ k -t)ω k-1 2πi(δ k -t) + β k e iγ k e i(δ k -t)ω k -e i(δ k -t)ω k-1 2π(δ k -t) 2 .
These formulae seem singular when t = δ k . This is not the case and has no reason to be since the function we integrate is smooth with respect to all parameters and variables. The limiting value for t = δ k is clearly

h k (δ k ) = α k e iγ k 2π ω k ω k-1 dω + β k e iγ k 2π ω k ω k-1 ωdω = e iγ k 2π δω k (α k + β k 1 2 (ω k-1 + ω k )).

Elementary summation coefficients

A quick glance at the explicit expression of h k (t) clearly provides the impression that the explicit formulae for h 0 nk (t) and h 0 nk (t) will not fit in the columns here. We will give only their flavor. Indeed we want to compute the time integrals of of h k (tτ ) and h k (tτ )τ for τ ∈ I n . This leads to integrate the product of a rational function with a complex exponential function. The results cannot be given in terms of simple functions but only in terms of the exponential integral function

Ei(ix) = - ∞ x e iy dy y + i π 2 .
We give in the next section a simple example of elementary summation coefficient calculation in the piecewise linear context.

A simple and ideal example 4.1 Computation of the coefficients

Our sampling for the filter transfer function yields a particularly simple formulation for the ideal low-pass filter which is 1 on the frequency interval [-ω c , ω c ] and zero elsewhere. This yields a single sample (1, 2ω c ) and linearly interpolated coefficients α 1 = 1, β 1 = 0, γ 1 = 0 and δ 1 = 0. The expression for the elementary impulse response is

h 1 (t) = e -iωct -e iωct -2πit = ω c π sinc(ω c t).
Then we have to compute

h 0 n1 (t) = tn tn-1 h 1 (t -τ )dτ = - t-tn t-tn-1 h 1 (τ )dτ = - 1 π (Si(ω c (t -t n )) -Si(ω c (t -t n-1 )),
where Si is the special function known as sine integral and defined by

Si(x) = x 0 sin(y) dy y = 1 2i (Ei(ix) -Ei(-ix)) + π 2 ,
and

h 1 n1 (t) = tn tn-1 h 1 (t -τ )τ dτ = - t-tn t-tn-1 h 1 (τ )(t -τ )dτ = t h 0 n1 (t) + 1 π t-tn-1 t-tn sin(ω c τ )dτ = t h 0 n1 (t) - 1 πω c (cos(ω c (t -t n )) -cos(ω c (t -t n-1 ))).
This case is simple due to its minimal number of samples in the frequency domain, but it displays all the difficulties of the general case, i.e. the need to evaluate special functions. These functions are built in many libraries in view of a numerical implementation of these algorithms. Moreover these functions are however very smooth: the Si function for example is almost linear in the neighborhood of 0 and tends to ±π/2 at ±∞ with very gentle oscillations. This feature makes possible the construction of efficient lookup tables in view of a hardware implementation.

Numerical results

To illustrate this simple example we filter the signal s(t) = 0.45 sin(2πt) + 0.45 sin(10πt) + 0.9 with the ideal low pass filter with the cutoff frequency ω c = 4π. The theoretical result is therefore supposed to be x(t) = 0.45 sin(2πt) + 0.9. This is not the typical sort of signal which is supposed to be addressed by our technique since it is not a sporadic one and a relatively large number of samples are taken. We perform the computations within the MATLAB SPASS (Signal Processing for ASynchronous Systems) framework (http://ljk.imag.fr/membres/Brigitte.Bidegaray/SPASS/). This signal is sampled with a M -bit Asynchronous A/D Converter (AADC) which leads to a level crossing sampling over the amplitude range [0, 1.8]. We can choose as we want the times at which the filtered signal is computed. To display the results we choose the sequence of times t m = .17m (m integer) to have sampling points dispatched irregularly over the obtained solution. On Figure 1, you can see the result for a linear interpolation of the signal non-uniform samples and a 3-bit AADC. We plot continuous functions with lines: the initial signal s(t) (dashed line) and the theoretical filtered signal x(t) (solid line). We plot the sampled results with markers: the non-uniformly sampled initial signal s n (asterisk markers) and the computed filtered samples x m (circle markers) at times t m . In the case of the 2-bit AADC, there are 2.8 points per wavelength for the highest frequency part of the signal. This is a very low rate, and we are however able to have only 6% error on the filtered result which is quite sufficient for a large range of applications. The other results all show less than 1% error. The values displayed on Table 1 are very dependent on the choice of the function to filter. Finer results (allowing less than .45% error) should certainly be obtained by using a higher order interpolation for the signal.

Conclusions

We have presented a novel approach to FIR filtering based on the non-uniform sampling of the signal but also the non-uniform sampling in frequency of the filter transfer function. The final result is complex but is nonetheless possible to implement in hardware devices and of course in numerical codes. This complexity is balanced by the very low number of samples and the relatively low number of operations needed for each evaluation. This approach is very promising to achieve a lower power consumption in mobile systems.

Table 1 :

 1 This very simple test case has quite a low number of parameters compared to the full problem for which we can finely tune the filter transfer function sampling for example. We compare here the results obtained for a zeroth and a first order interpolation of the signal and for different values (2, 3, 4 and 5) of the AADC resolution. On Table1we give the relative l 1 error between computed filtered samples x m at times t m = .01m (m integer) and the theoretical values x(t m ). l 1 error of the filtering method for 0th and first order interpolation of the signal and and M bit resolution of the AADC (M = 2, 3, 4, 5).

	1.8
	1.6

m Figure 1: Filtering result. Initial signal (dashed line), theoretical filtered signal (solid line), non-uniformly sampled initial signal (asterisk markers) and computed filtered samples (circle markers).

Acknowledgments

This work has been supported by a funding from the Joseph Fourier-Grenoble 1 University: MSTIC project TATIE.


Andriy Ya Olenko
email: a.olenko@latrobe.edu.au

Tibor K Pogány
email: poganj@pfri.hr

On average sampling restoration of Piranashvili-type harmonizable processes

Keywords: 42C15, 60G12, 94A20 WKS sampling theorem, time shifted sampling, Piranashvili-, Loève-, Karhunen-harmonizable stochastic process, weakly stationary stochastic process, local averages, average sampling reconstruction

The harmonizable Piranashvili -type stochastic processes are approximated by a finite time shifted average sampling sum. Truncation error upper bound is established; various consequences and special cases are discussed.

Introduction and preparation

Given a probability space Ω, F, P and the related Hilbert-space L 2 (Ω) := {X : E|X| 2 < ∞}. Let us consider a non-stationary, centered stochastic L 2 (Ω)-process ξ : R × Ω → R having covariance function (associated to some domain Λ ⊆ R with some sigma-algebra σ(Λ)) in the form:

B(t, s) = Λ Λ
f (t, λ)f * (s, µ)F ξ (dλ, dµ), [START_REF] Belyaev | Analytical random processes[END_REF] with analytical exponentially bounded kernel function f (t, λ), while F ξ is a positive definite measure on R 2 provided the total variation F ξ (Λ, Λ) of the spectral distribution function F ξ such that satisfies

F ξ (Λ, Λ) = Λ Λ F ξ (dλ, dµ) < ∞.
(We mention that the sample function ξ(t) ≡ ξ(t, ω 0 ) and f (t, λ) possess the same exponential types [START_REF] Belyaev | Analytical random processes[END_REF]Theorem 4], [START_REF] Zurab | On the problem of interpolation of random processes[END_REF]Theorem 3]). Then, by the Karhunen-Cramér theorem the process ξ(t) has the spectral representation as a Lebesgue integral

ξ(t) = Λ f (t, λ)Z ξ (dλ); (2)
in ( 1) and ( 2)

F ξ (S 1 , S 2 ) = EZ ξ (S 1 )Z * ξ (S 2 ) S 1 , S 2 ⊆ σ(Λ).
Such a process will be called Piranashvili process in the sequel [START_REF] Zurab | On the problem of interpolation of random processes[END_REF], [START_REF] Tibor | Almost sure sampling restoration of bandlimited stochastic signals[END_REF].

Being f (t, λ) entire, it possesses the Maclaurin expansion

f (t, λ) = ∞ n=0 f (n) (0, λ)t n /n!. Put γ := sup Λ c(λ) = sup Λ lim n n |f (n) (0, λ)| < ∞ . (3)
As the exponential type of f (t, λ) is equal to γ, for all w > γ there holds

ξ(t) = n∈Z ξ nπ w sin(wt -nπ) wt -nπ , (4)
uniformly in the mean square and in the almost sure sense [START_REF] Zurab | On the problem of interpolation of random processes[END_REF]Theorem 1]. This result we call Whittaker-Kotel'nikov-Shannon (WKS) stochastic sampling theorem [START_REF] Tibor | Almost sure sampling restoration of bandlimited stochastic signals[END_REF].

Specifying F ξ (x, y) = δ xy F ξ (x) in (1) we conclude the Karhunen-representation of the covariance function

B(t, s) = Λ f (t, λ)f * (s, λ)F ξ (dλ).
Also, putting f (t, λ) = e itλ in (1) one gets the Loèverepresentation:

B(t, s) = Λ Λ e i(tλ-sµ) F ξ (dλ, dµ).
Here is c(λ) = |λ|. Therefore, WKS-formula (4) holds for all w > γ = sup |Λ|. Then, the Karhunen process with the Fourier kernel f (t, λ) = e itλ we recognize as the weakly stationary stochastic process having covariance

B(τ ) = Λ e iτ λ F ξ (dλ), τ = t -s.
Deeper insight into different kind harmonizabilities present [START_REF] Kakihara | Multidimensional Second Order Stochastic Processes[END_REF][START_REF] Priestley | Non-linear and Nonstationary Time Series[END_REF][START_REF] Malempati | Harmonizable processes: structure theory[END_REF] and the related references therein. Finally, using Λ = [-w, w] for some finite w in this considerations, we get the band-limited variants of the same kind processes.

By physical and applications reasons the measured samples in practice may not be the exact values of the measured process ξ(t), or its covariance B(t, s) itself, near to the sample time t n , but only the local average of the signal ξ near to t n . So, the measured sample values will be

ξ, u n U = U ξ(x)u n (x)dx, U = supp(u n ) (5)
for a sequence u := u n (t) n∈Z of non-negative, normalized, that is 1, u n ≡ 1, averaging functions such that

supp(u n ) ⊆ t n -σ n , t n + σ n . (6)
The local averaging method was introduced by Gröchenig [START_REF] Gröchenig | Reconstruction algorithms in irregular sampling[END_REF] and developed by Butzer and Lei. Recently Sun and Zhou gave some results in this direction, while the stochastic counterpart of this average sampling was intensively studied in the last three-four years by He, Song, Sun, Yang and Zhu in a set of articles [START_REF] Song | Error estimate on non-bandlimited random signals by local averages[END_REF], [START_REF] Song | Approximation of weak sense stationary stochastic processes from local averages[END_REF] and their references therein; see for example the exhaustive references list in [START_REF] He | Truncation error estimate on random signals by local average[END_REF]. The listed, recently considered stochastic average sampling results are restricted to weakly stationary stochastic processes, while the approximation average sampling sums are used around the origin.

Our intentions are to extend these results to time shifted average sampling, considered for the very wide class of Piranashvili processes.

Time shifted average sampling

Now, instead to follow the approach used in [START_REF] Song | Approximation of weak sense stationary stochastic processes from local averages[END_REF] we take time shifted [START_REF] Ya | Time shifted aliasing error upper bounds for truncated sampling cardinal series[END_REF], [START_REF] Ya | On sharp bounds for remainders in multidimensional sampling theorem[END_REF] finite average sampling sum in approximating the initial stochastic signal ξ. First, we consider weighted average over J n (t) := nπ/wσ n (t), nπ/w + σ n (t) for the measured value of ξ(t) at nπ/w, n ∈ I N (t) where

I N (t) := {n ∈ Z : |tw/π -n| ≤ N }, N ∈ N.
Let N t be the integer nearest to tw/π.

By obvious reasons we restrict the study to σ := max

I N (t) sup R max σ n (t), σ n (t) ≤ π 2w .
Let us define the time shifted average sampling approximation sum in the form

A u (ξ; t) = Z ξ, u n Jn(t) • sin(wt -nπ) wt -nπ ,
and its truncated variant

A u,N (ξ; t) = I N (t) ξ, u n Jn(t) • sin(wt -nπ) wt -nπ .
One defines mean-square, time shifted, average sampling truncation error T u,N (ξ; t)

:= E ξ(t) -A u,N (ξ; t) 2 .
Now, we are interested in some reasonably simple efficient mean square truncation error upper bound appearing in the approximation ξ(t) ≈ A u,N (ξ; t).

Let us introduce some auxiliary results. As N x stands for the integer nearest to xw/π, x ∈ R, let

Γ N (x) := z ∈ C : |z -N x | ≤ N + 1 2 π w , N ∈ N.
In what follows denote int(R) the interior of some R, while the series

λ(q) := ∞ n=1 1 (2n -1) q
stands for the Dirichlet lambda function.

Theorem 1 Let f (z) be entire, bounded on the real axis and exponentially bounded having type γ < w. Denote

L f := sup R f (x) , L 0 (z) := 2wL f | sin(wz)| π(w -γ) 1 -e -π .
Then for all z ∈ int Γ N (x) and N ∈ N enough large it holds

Z\I N (x) f n π w sin(wz -nπ) wz -nπ ≤ L 0 (z)e -(N +1/2)π(w-γ)/w (N + 1/2) 1 -|z-Nx|w (N +1/2)π < L 0 (z) N . (7)
The proving method is contour integration, following Piranashvili's traces [START_REF] Zurab | On the problem of interpolation of random processes[END_REF]. Denote here and in what follows By simple use of ( 1), ( 2) and the Theorem 1 one deduces the following modest generalization of [START_REF] Zurab | On the problem of interpolation of random processes[END_REF]Theorem 2] to time shifted case of sampling restoration procedure.

Y N (ξ; t) := I N (t)
Theorem 2 Let ξ(t) be a Piranashvili process with exponentially bounded kernel function f (t, λ) and let

L f := sup R sup Λ |f (t, λ)|, L 0 (t) := 2 L f w | sin(wt)| π(w -γ) 1 -e -π . (8)
Then for all t ∈ int Γ N (t) , we have

E ξ(t) -Y N (ξ; t) 2 < L 2 0 (t) N 2 F ξ (Λ, Λ) . (9)
Remark 1 Let us point out that the straightforward consequence of (9) is not only the exact L 2 -restoration of the initial Piranashvili-type harmonizable process ξ by a sequence of approximants Y N (ξ; t) when N → ∞, but since

E ξ(t) -Y N (ξ; t) 2 = O N -2 ,
the perfect reconstruction is possible in the a.s. sense as well (by the celebrated Borel-Cantelli Lemma).

Second, the first order difference ∆ x,y B [3] of B(t, s) on the plane satisfies

∆ x,y B (t, s) = B(t + x, s + y) -B(t + x, s) -B(t, s + y) + B(t, s) = x 0 y 0 ∂ 2 ∂u∂v B t + u, s + v dvdu . ( 10
)
Theorem 3 Let ξ(t) be a Piranashvili process with the covariance B(t, t) ∈ C 2 (R). Let (p, q) be a conjugated Hölder pair of exponents:

1 p + 1 q = 1, p > 1 .
Then we have

E Y N (ξ; t) -A u,N (ξ; t) 2 ≤ C q π 2 4w 2 sup R B (t, t) • (2N + 1) 2/p , (11)
where

C q = 1 + 2 q+1 | sin(wt)| q π q λ(q) 2/q . ( 12
)
PROOF. Having on mind (1), the properties of averaging functions sequence u and ( 10), we clearly derive

E Y N (ξ; t) -A u,N (ξ; t) 2 = E I N (t) ξ nπ w -ξ(x), u n Jn(t) • sin(wt -nπ) wt -nπ 2 = I 2 N (t) σ n (t) -σ n (t) σ m (t) -σ m (t) u n (x + n π w )u m (y + m π w ) • x 0 y 0 ∂ 2 ∂u∂v B u + n π w , v + m π v dvdu • sin(wt -nπ) wt -nπ sin(wt -mπ) wt -mπ ≤ I 2 N (t) sin(wt -nπ) wt -nπ sin(wt -mπ) wt -mπ • sup x,y≤σ x 0 y 0 ∂ 2 ∂u∂v B u + n π w , v + m π v dvdu
being u normalized. For the sake of brevity let us denote H σ (n, m) the sup-term in the last display. Then, by the Hölder inequality with conjugate exponents p, q; p > 1, we get

E Y N (ξ; t) -A u,N (ξ; t) 2 ≤ I 2 N (t) H p σ (n, m) 1/p I N (t) sin(wt -nπ) wt -nπ q 2/q .
It is not hard to see that for all n, m ∈ I N (t) there holds

H σ (n, m) ≤ σ 2 sup R 2 ∂ 2 B(t, s) ∂t∂s ≤ π 2 4w 2 sup R 2 ∂ 2 B(t, s) ∂t∂s .
Applying now the Cauchy-Bunyakovsky-Schwarz inequality to the covariance

∂ 2 B, we deduce sup R 2 ∂ 2 B(t, s) ∂t∂s ≤ sup R ∂ 2 B(t, t) ∂t 2 = sup R |B (t, t)| .
It remains to evaluate the sum of qth power of the sincfunctions. As sin(wt

-N t π) wt -N t π ≤ 1
we conclude

I N (t) sin(wt -nπ) wt -nπ q ≤ 1 + C N n=1 1 (n -∆) q + 1 (n + ∆) q < 1 + 2C ∞ n=1 1 (n -1/2) q < 1 + 2 q+1 C λ(q) , where C = | sin(wt)| q π q .
Collecting all these estimates, we deduce (11).

Main result

We are ready to formulate our upper bound result for the mean square, time shifted average sampling truncation error T u,N (ξ; t). The almost sure sense restoration procedure has been treated too.

As we use average sampling sum A u,N (ξ; t) instead of Y N (ξ; t) to obtain asymptotically vanishing T u,N (ξ; t), it is not enough letting N → ∞ as in Remark 1. For average sampling we need additional conditions upon w or σ to guarantee smaller average intervals for larger/denser sampling grids.

Theorem 4 Assume the conditions of Theorems 2 and 3 have been fulfilled. Then, we have

T u,N (ξ; t) ≤ 2 L 2 0 (t) N 2 F ξ (Λ, Λ) + C q π 2 2w 2 sup R B (t, t) • (2N + 1) 2/p , (13)
where L 0 , C q are described by [START_REF] Ya | On sharp bounds for remainders in multidimensional sampling theorem[END_REF], [START_REF] Zurab | On the problem of interpolation of random processes[END_REF] respectively.

Moreover, when w = O N 1/2+1/p+ε , ε > 0, we have

P lim N →∞ A u,N (ξ; t) = ξ(t) = 1 (14)
for all t ∈ R.

PROOF. By direct calculation we deduce

T u,N (ξ; t) = E ξ(t) -A u,N (ξ; t) 2 = E ξ(t) -Y N (ξ; t) + Y N (ξ; t) -A u,N (ξ; t) 2 ≤ 2E ξ(t) -Y N (ξ; t) 2 + 2E Y N (ξ; t) -A u,N (ξ; t) 2 .
Now, we get the asserted upper bound by ( 9) and [START_REF] Zurab | On the problem of interpolation of random processes[END_REF].

To derive [START_REF] Malempati | Harmonizable processes: structure theory[END_REF], we apply the Chebyshev inequality to evaluate the probability

P N := P ξ(t) -A u,N (ξ; t) ≥ η ≤ η -2 T u,N (ξ; t) .
Accordingly, since L 0 (t) = O(1) as N → ∞, we have

N P N ≤ K N 1 N 2 + (2N + 1) 2/p w 2 < ∞,
K being a suitable absolute constant. Therefore, by the Borel-Cantelli Lemma, the the a.s. convergence result (14) holds true.

Remark 2 Theorem 4 ensures the perfect time shifted average sampling restoration in the mean square sense when

w = O N 1/p+ε , ε > 0: lim N →∞ T u,N (ξ; t) = 0 .
The a.s. sense restoration (14) requires stronger assumption, it holds when w = O N 1/2+1/p+ε .

Remark 3

In both cases we use the so called approximate sampling procedure, that is, when in the restoration procedure w → ∞ in some fashion. The consequence of these results is that we have to restrict ourselves to the case Λ = R, such that we recognize as the non-bandlimited Piranashvili type harmonizable process case.

The importance of approximate sampling procedures for investigations of aliasing errors in sampling restorations and different conditions on joint asymptotic behaviour of N and w have been discussed in detail in [START_REF] Ya | Time shifted aliasing error upper bounds for truncated sampling cardinal series[END_REF].

Conclusions

We have analyzed upper bounds on truncation error for time shifted average sampling restorations in the stochastic initial signal case. The convergence of the truncation error to zero was discussed. However, certain new questions immediately arise:

• to derive sharp upper bounds in Theorems 3 and 4;

• to obtain new results for L p -processes using recent deterministic findings [START_REF] Ya | Universal truncation error upper bounds in sampling restoration[END_REF], [START_REF] Ya | Universal truncation error upper bounds in irregular sampling restoration[END_REF];

• to obtain similar results for irregular/nonuniform sampling restoration using methods exposed in [START_REF] Ya | Direct Lagrange-Yen type interpolation of random fields[END_REF] and [START_REF] Ya | Universal truncation error upper bounds in irregular sampling restoration[END_REF].

  -nπ) wt -nπ the time shifted truncated WKS restoration sum.

Acknowledgements

The recent investigation was supported in part by Research Project No. 112-2352818-2814 of Ministry of Sciences, Education and Sports of Croatia and in part by La Trobe University Research Grant-501821 "Sampling, wavelets and optimal stochastic modelling".


J R Higgins
email: rowlandhiggins@yahoo.com

Concrete and discrete operator reproducing formulae for abstract Paley-Wiener space

The classical Paley-Wiener space possesses two reproducing formulae; a 'concrete' reproducing equation and a 'discrete' analogue, or sampling series, and there is a striking comparison between them. It is shown that such analogies persist in the setting of Paley-Wiener spaces that are more general than the classical case. In fact, there are 'operator' versions of the reproducing equation and of the sampling series that are also comparable, not 'exactly' but nearly so. Reproducing kernel theory and abstract harmonic analysis are brought together to achieve this, then the special case of multiplier operators with respect to the Fourier transform is considered. The Riesz transforms provide a two-dimensional example, with possibilities of extension to higher dimensions and to further classes of operators.

Introduction

It has often been remarked that the classical Paley-Wiener space possesses two reproducing formulae; a 'concrete' reproducing equation

f (s) = R f (t) sinc(s -t)dt, (s ∈ R), (1)
and a 'discrete' reproducing equation, or sampling series,

f (s) = n∈Z f (n) sinc(s -n), (s ∈ R), (2)
and that there is a striking analogy between the two (see, e.g., [3, p. 58]). Here, sinc denotes the standard function sinc x := (sin πx)/πx. The purpose of the present lecture is to point out that concrete and discrete reproducing formulae and analogies between them persist in the setting of Paley-Wiener spaces that are more general than the classical case. It will be shown that for suitably chosen operators there are 'operator' versions of the reproducing equation and of the sampling series that are also comparable, in the same way as in the classical case described above.

The setting

Abstract theories that lead to reproducing formulae are outlined in §2.1 and §2.2, and are brought together in §2.3.

The reproducing kernel theory

The basic setting of this paper is that of the reproducing kernel theory of Saitoh [8, Ch.

f R = Kα R ≤ α H , (3)
and there exists a unique member, g say, of the class of all α's satisfying (3) such that

f (t) = g, K t H , (t ∈ E), and f R = g H . The reproducing equation for f ∈ R is f (t) = f, k(•, t) R (4)
The following theorem is simple but very useful.

Theorem 2

The convergence of a sequence in the norm of R implies that it converges pointwise over E, and the convergence is uniform over any subset of E on which

k(t, t) = k(•, t) 2 is bounded.
The following Theorem is to be found in [START_REF] Saitoh | Integral transforms, reproducing kernels and their applications[END_REF].

Theorem 3 With notations as above, let {s n }, (n ∈ X), be points of E such that {K sn } is an orthonormal basis for H. Then the sampling series representation

f (t) = n∈X f (s n )k(s n , t), (5)
holds, convergence being in the norm of R; and then of course Theorem 2 applies.

Abstract harmonic analysis

A very brief introduction (mostly just notations) to the abstract harmonic analysis that will be needed is now given. All necessary background, and much more, is to be found in [START_REF] Dodson | Groups and the sampling theorem[END_REF], [START_REF] Dodson | Abstract harmonic analysis and the sampling theorem[END_REF].

Let G be a locally compact abelian (LCA) group (written additively). Let (t, γ) be a character of G, that is, a continuous homomorphism of G into the circle group. Let G ∧ = Γ denote the group of continuous characters on G, usually called the dual group of G. We assume that Γ has a countable discrete subgroup Λ.

Haar measures on the various groups are normalised in the standard way [START_REF] Dodson | Groups and the sampling theorem[END_REF], and this means in particular that there is a measurable transversal (i.e., a complete set of coset representatives) Ω ⊂ Γ of Γ/Λ, and it has finite Haar measure. Now

H = Λ ⊥ := {t ∈ G : (t, λ) = 1, (λ ∈ Λ)}.
is a subgroup of G and is called the 'annihilator' of Λ. We assume that H is discrete; it follows that the quotient group Γ/Λ is compact. The Fourier transform on L 2 (G) is defined in the usual way:

f ∧ (γ) = (Ff )(γ) := G f (t)(t, γ) dt,
in the L 2 sense, where dt denotes the element of Haar measure on G (likewise, dγ denotes the element of Haar measure on Γ). The inverse Fourier transform will be denoted by ∨ or by F -1 . We shall need the 'shift' property of the Fourier transform:

f (• -x) ∧ (γ) = (-x, γ)f ∧ (γ).
Abstract Paley Wiener space P W Ω (G) is defined as follows:

P W Ω (G) := {f : f ∈ L 2 (G) ∩ C(G), f ∧ (γ) = 0 (Haar) a.a. γ ∈ Ω} (6)

Combining harmonic analysis with the reproducing kernel theory

The abstract set E of §2.1 is often taken to be R or C.

Here, however, we take it to be an LCA group G thus combining two abstract theories, harmonic analysis and the reproducing kernel theory. In the notations of §2.1 and §2.2 we also take

K t = (t, •), H = L 2 (Ω) and Kg = F -1 g, g ∈ L 2 (Ω). Then we have k(s, t) = Ω (t, γ)(s, γ) dγ = Ω (t -s, γ) dγ = χ Ω ∨ (t -s) =: ϕ Ω (t -s), (7)
where χ S denotes the characteristic function of a set S. It does not seem to have been recognised that ϕ Ω (t -s) is the reproducing kernel for P W Ω (G), and that this allows a close association between sampling in the harmonic analysis setting and Saitoh's theory. The space R of §2.1 is now seen to be the Paley-Wiener space defined in [START_REF] Riesz | Functional analysis[END_REF], and its reproducing equation is

f (t) = f, ϕ Ω (t -•) L 2 (G) (8)
Kluvánek's sampling theorem [4, p. 45] is a consequence:

Theorem 4 Let f ∈ P W Ω . With the assumptions of §2.2, f (t) = h∈H f (h)ϕ Ω (t -h) (9)
in norm, etc., (see Theorem 2).

Our concrete -discrete comparison is beween ( 8) and ( 9).

Operator kernels and operator reproducing formulae

The presence of kernels and reproducing equations associated with operators on a reproducing kernel Hilbert space add greatly to the richness of its structure, as we shall see in this section.

Operator kernels and operator reproducing equations

Let R be the separable Hilbert space of functions defined on E with reproducing kernel k(s, t), as we have discussed it in §2.1. Let B be a bijection on R, and let B * denote the adjoint operator. The action of B on R is governed by the action of B * on the reproducing kernel k, because for f ∈ R,

Bf (t) = Bf, k(•, t) R = f, B * k(•, t) R . (10)
See, e.g., [START_REF] Meschkowski | Hilbertsche Räume mit Kernfunktion[END_REF].

Definition 1

The kernel

h(s, t) := B * k(•, t) (s), s, t ∈ E
will be called the operator kernel of B.

In this notation (10) is

Bf (t) = f, h(•, t) . (11)
Now from Definition 1 above, ((B * ) -1 h(•, t))(s) = k(s, t), so that, using the ordinary reproducing formula (4), we have

f (t) = f, k(•, t) = f, (B * ) -1 h(•, t) = (B * ) -1 * f, h(•, t) .
Now using standard properties of operators and their adjoints (e.g., [6, p. 202]) we can summarise these calculations as:

f (t) = (B -1 f )(•), h(•, t) . ( 12
)
This formula tells us that f can be reproduced, not from its own values as in the ordinary reproducing kernel theory, but from the result of acting on it with an operator. We can call this an operator reproducing equation in analogy with the ordinary reproducing equation ( 4). Similar formulae for B * can be obtained in the same way. First, we make the following

Definition 2 h * (s, t) := h(t, s) ((t, s) ∈ E × E)
will be called the adjoint operator kernel of B.

Kernels and their adjoints occur in important areas of study such as the theory of integral equations (see, e.g., [6, p. 170] for basic information). We shall find series expansions for such kernels and identify the action of h * explicitly in Theorem 5 below. First, let {ϕ n }, n ∈ X, be an orthonormal basis for R.

Lemma 1

h(s, t) = n∈X Bϕ n (t) ϕ n (s), (s, t ∈ E). ( 13
)
Convergence is in the norm of R for each t ∈ E, and the pointwise convergence is governed by Theorem 2.

Proof The coefficients for the expansion of h(•, t), t fixed, in the basis {ϕ n } are

h(•, t), ϕ n = ϕ n , h(•, t) = Bϕ n (t)
by (11), thus (13) is obtained.

It will be recalled that if we put

Bϕ n = ψ n (B * ) -1 ϕ n = ψ * n , (14)
then {ψ n } is a Riesz basis for R with dual basis {ψ * n }. In this notation (13) can be written

h(s, t) = n∈X ψ n (t)ϕ n (s). (15)
Hence by Definition 2 we have

h * (s, t) = n∈X ϕ n (t)ψ n (s). ( 16
)
in the norm of R for each t ∈ E. By uniqueness the coefficients {ϕ n (t)} are such that

ϕ n (t) = h * (•, t), ψ * n = (B * ) -1 ϕ n , h * (•, t) , (17)
Since this relationship is true for every member ϕ n of a basis for R, it holds for every f ∈ R by the usual density argument. This argument runs as follows: Let N c n ϕ n be the Nth partial sum of the expansion for f in the basis ϕ n . Then taking linear combinations in (17),

N c n ϕ n (t) = (B * ) -1 N c n ϕ n (t), h * (•, t) . (18) Consider f (t) -(B * ) -1 f, h * (•, t) . ( 19
) Put f (t) -N c n ϕ n (t) = F n (t)
. Now inserting the right and left hand sides of (18) we find from (19) that

f (t) -(B * ) -1 f, h * (•, t) (20) = F n (t) -(B * ) -1 (f - N c n ϕ n ), h * (•, t) ≤ F n (t) + (B * ) -1 (f - N c n ϕ n ), h * (•, t) ≤ F n (t) + (B * ) -1 (f - N c n ϕ n ) h * (•, t) ≤ F n (t) + B f - N c n ϕ n h * (•, t) (21)
for a constant B which is consequent upon the fact that, since B is bounded, B * is bounded and by Banach's 'bounded inverse' theorem (B * ) -1 is bounded. Now N can be made to approach ∞. Since F n (t) converges to 0 both in norm and pointwise on E (see Theorem 2), the expression in (21) approaches 0 for each fixed t ∈ E. Finally, from (20) we obtain the following Theorem 5 Let R, B and E be as above. Then we have the adjoint operator reproducing formula

f (t) = (B * ) -1 f, h * (•, t) .
This shows the basic property of h * ; it reproduces f from (B * ) -1 f .

Operator sampling series

There are connections here to the theory of single channel sampling (see, e.g., [3, Ch. 12]), but the present approach is much more general.

In order to match the operator reproducing equation ( 12) with a discrete analogue, some further assumption will have to be made. In fact we shall assume the existence of a sequence (s n ) ⊂ E, n ∈ X such that {h(s n , t)} is an orthogonal basis for R with normalising factors ν n , so that {ν n h(s n , t)} is orthonormal. This can sometimes be traced back to the condition that {K sn } be an orthogonal basis for H. Again, we could assume that {h(s n , t)} is just a basis for R, or just a frame. However, weaker assumptions demand more technicalities and we will not pursue this kind of generality here. Let f ∈ R. Its expansion in our assumed orthonormal basis is

f (t) = n∈X c n ν n h(s n , t) (22)
where

c n = f, ν n h(s n , •) = ν n f, h * (•, s n = ν n (B * f )(s n ) by Theorem 5. So (22) is f (t) = n∈X |ν n | 2 (B * f )(s n ) h(s n , t). ( 23
)
Then ( 12) and ( 23) are concrete -discrete analogues of each other.

Multiplier operators with respect to the Fourier transform

Take E to be an LCA group G with dual Γ (for notations and references, see §2.2), and let R be a Paley-Wiener space P W Ω . Let µ(γ) be a non-nul complex valued function on Γ such that

0 < α ≤ |µ(γ)| ≤ β < ∞, (Haar) a.a. γ ∈ Ω; µ(γ) = 0, γ ∈ Ω. ( 24
)
Let M denote the operation of multiplication by χ Ω (γ)µ(γ).

Definition 3 Let f ∈ P W Ω . The operator T is defined by

(T f )(s) := (F -1 MFf )(s) Lemma 2
The operator T of Definition 3 is a bijection on P W Ω Proof Clearly T is linear. Furthermore it is one-to-one, since the null space of T is

{f : T f = θ} = {f : µ(γ)f ∧ (γ) = θ} which implies that f = θ. Again, T is "onto". Let g ∈ P W Ω . Then if M -1 denotes multiplication by [µ(γ)] -1 , f = F -1 M -1 Fg ∈ P W Ω .
Then from Definition 3, T f = g. The boundedness of T follows from two applications of Plancherel's Theorem. Indeed, let f ∈ P W Ω . Then

T f L 2 (G) = F -1 MFf L 2 (G) = MFf L 2 (Γ) ≤ |µ| Ff L 2 (Γ) = |µ| f L 2 (G) .

The operator kernel for T

First we need to know the adjoint

T * . Let f 1 , f 2 ∈ P W Ω .
The defining equation is

T f 1 , f 2 = f 1 , T * f 2 .
Suppose that T * is of the same form as T of Definition 3, that is,

T * f = F -1 M * Ff, (25)
where M * denotes multiplication by the multiplier µ * which is to be determined. In the integral notation, and using the 'hat' notation for the Fourier transform, the criterion is:

G µ(•)f 1 ∧ (•) ∨ (t)f 2 (t) dt = G f 1 (t) µ * (•)f 2 ∧ (•) ∨ (t) dt.
By Plancherel's theorem this is:

Γ µ(γ)f 1 ∧ (γ)f 2 ∧ (γ) dγ = Γ f 1 ∧ (γ)µ * (γ)f 2 ∧ (γ) dγ,
from which we may choose µ * (γ) = µ(γ).

It may be noted that T is self-adjoint if µ is real-valued.

It is now evident that the assumption (25) leads to

(T * f )(s) = Γ µ(γ)f ∧ (γ)(s, γ) dγ (26)
The operator kernel for T can now be calculated. From Definition 1 and (7) we have

h(s, t) = T * ϕ Ω (• -t) (s), s, t ∈ G.
Therefore from (26), and using the 'shift' property of the Fourier transform,

h(s, t) = Γ µ(γ) χ Ω ∨ (• -t) ∧ (γ)(s, γ) dγ = Γ µ(γ)(-t, γ) χ Ω (γ)(s, γ) dγ = Ω µ(γ)(s -t, γ) dγ = µ(•) ∨ (s -t). Hence h(s, t) = µ(•) ∨ (s -t) = µ ∧ (s -t).

Now (12) becomes

f (t) = (T -1 f )(•), µ ∨ (• -t) , (27)
and (23) becomes Here and in subsequent Examples the choice of Haar measure on G, Γ, etc., accounts for apparent anomalies in the normalising constants in the formulae (e.g., Haar measure on R is taken to be (2π) -1/2 times Lebesgue measure. See [2, p. 257]). With these choices, (27) becomes [START_REF] Dodson | Groups and the sampling theorem[END_REF]. The classical sampling series (2) now follows the textbook proof. Since {e -iny / √ 2π : n ∈ Z} is an orthonormal (ON) basis of L 2 (-π, π), Plancherel's theorem shows that the inverse Fourier transforms {sinc(n-t)} : n ∈ Z} form an orthonormal basis of P W [-π,π] . Coefficients in the expansion of f in this basis are obtained from (1) and so, with s n = n, our choices for T and µ show that (28) becomes [START_REF] Dodson | Abstract harmonic analysis and the sampling theorem[END_REF].

f (t) = n∈X |ν n | 2 (T * f )(s n ) µ ∧ (s n -t). (28) 5. Examples

Example 2

The Hilbert transform Another well-known example illustrates the present theory; a member of P W [-π,π] can be sampled and reconstructed from samples of its Hilbert transform (see, e.g., [3, p. 126] and references there). This idea can be fitted it into the theme of the present lecture by taking G = R, Ω = [-π, π], T = H := F -1 MF where M denotes multiplication by -i sgn(y). H is the Hilbert transform on P W [-π,π] .

For (27) we need

µ ∨ (s -t) = 1 √ 2π i √ 2π π -π sgn(y)e i(s-t)y dy = -sinc 1 2 (s -t) sin π 2 (s -t) (29)
after a simple calculation. Also we have H

-1 = -H = H * , therefore (27) is f (t) = - R Hf (τ ) sinc 1 2 (τ -t) sin π 2 (τ -t) dτ. (30)
For (28) we need to find {s n } such that {µ ∧ (s n -t)}, n ∈ Z, is an ON basis of P W π . We can start with the ON basis {e iny / √ 2π}, (n ∈ Z), of L 2 (-π, π), then multiply each member by -i sgn(y). The result is again an ON basis, as a consequence of | -i sgn(y)| = 1 a.e. on [-π, π]. The inverse Fourier transform of a typical one of these basis elements is

-i 2π F -1 sgn(•)e -in• (t) = -sinc 1 2 (n -t) sin π 2 (n -t)
by the same calculation as in (29). But, taking account of Haar measure, this also gives µ ∧ (n -t). Hence (28) becomes

f (t) = n∈Z Hf (n) sinc 1 2 (n -t) sin π 2 (n -t) (31)
Our concrete -discrete comparison is between (30) and (31).

Example 3 The Riesz transforms

For background on the Riesz transforms see [9, p. 223] Take G to be R d , (d ∈ N). Let t = (t 1 , . . . , t d ) and let y = (y 1 , . . . , y d ) etc. Let the scalar product in R d be denoted by , .

Definition 4 Let f ∈ L 2 (R d ), and define

R j f := F -1 M j Ff, j = 1, . . . , d, (32)
M j denoting multiplication by -iy j /|y| χ [-π,π] d (y).

We note that this multiplier is not bounded away from zero when d ≥ 2 and y ∈ [-π, π] d and therefore does not always satisfy the criterion (24). However, it is possible to define operators involving the Riesz transforms which do satisfy the criterion (24). First we consider the case d = 2, and define the operator

R := R 1 + iR 2 (
f (t) = 1 2π R 2 R -1 f (s)m ∨ (s -t) ds. ( 34
)
Since the multiplier is of unit modulus, a two dimensional version of the construction that we used in the previous example shows that -i 2π

y 1 |y| + i y 2 |y| e -i k,y , (k ∈ Z 2 ),
is an ON basis of L 2 ([-π, π] 2 ). Then (28) becomes

f (t) = k∈Z 2 R * f (k)m ∧ (k -t). (35)
The comparison for this example lies between the concrete (34) and the discrete (35).

Other combinations of the Riesz transforms are possible, in two and higher dimensions, whose multipliers satisfy (24) but are not always of unit modulus.

Conclusions

The multiplier transforms treated in this study form a rather restricted class of operators; nevertheless, the methods can be used in connection with the very important Riesz transforms. It remains to investigate extensions to other types of operator. Likely candidates are, for example, multiplier transforms with less restrictive conditions on the multiplier, the singular integral operators of Calderón-Zygmund type (a class containing the Riesz transforms, see, e.g., [9, Ch. VI]), and operators of the Hankel and Toeplitz type (e.g., [START_REF] Rochberg | Toeplitz and Hankel operators on the Paley-Wiener space[END_REF]).

Example 1

 1 The classical caseNaturally, we expect to recover the case of the classical reproducing equation and sampling formula as special cases of the theory. To do this we pick G = R, Ω = [-π, π], T = I = T * = T -1 and µ = χ [-π,π] (y). Therefore we have µ ∨ (s -t) s-t)y dy = √ 2π sinc(s -t).

y 1 |y| + i y 2

 2 33) acting on P W [-π,π] 2 . Its multiplier is m(y) := (-i) |y| and clearly we have |m(y)| = 1 a.e. Hence m satisfies the criterion (24) with respect to two-dimensional Lebesgue measure. Now (27) becomes

  2, §1]. Very briefly the background is as follows. Let E be an abstract set. For each t belonging to E let K t belong to H (a separable Hilbert space with inner product denoted by ,

H ). Then k(s, t) := K t , K s H is defined on E ×E and is called the kernel function of the map K t . This kernel function is a positive matrix [8, Ch. 2, §2] and as such it determines one and only one Hilbert space for which it is the reproducing kernel. This Hilbert space is denoted by R(K) since it turns out to be the set of images of H under the transformation (Kg)(t) := g, K t H , (g ∈ H).

Theorem 1

(Saitoh)

With the notations established above, R(K) (which is now abbreviated to just R) is a Hilbert space which has the reproducing kernel k(•, •), and is uniquely determined by this kernel k. For f ∈ R there exists α ∈ H such that


José Luis Romero
email: jlromero@dm.uba.ar

Explicit localization estimates for spline-type spaces

We give some explicit decay estimates for the dual system of a basis of functions that are polynomially localized in space.

Introduction

A spline-type space S is a closed subspace of L 2 (R d ) possessing a Riesz basis of functions well localized is space. That is, there exists a family of functions {f k } k ⊆ S and constants 0 < A ≤ B < +∞ such that

A c ℓ 2 ≤ k c k f k L 2 ≤ B c ℓ 2 , (1)
holds for every c ∈ ℓ 2 , and the functions {f k } k satisfy an spatial localization condition. In a spline-type space any function in f ∈ S has a unique expansion f = k c k f k . Moreover the coefficients are given by c k = f, g k , where {g k } k ⊆ S is the dual basis, a set of functions characterized by the relation g k , f j = δ k,j . These spaces provide a very natural framework for the sampling problem. The general theory of localized frames (see [START_REF] Gröchenig | Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator[END_REF], [START_REF] Fornasier | Intrinsic localization of frames[END_REF] and [START_REF] Radu | Density, overcompleteness, and localization of frames I: Theory[END_REF]) asserts that the functions forming the dual basis satisfy a similar spatial localization. This can be used to extend the expansion in (1) to other spaces, so that the family {f k } k becomes a Banach frame for an associated family of Banach spaces (see [START_REF] Hans | Banach spaces related to integrable group representations and their atomic decompositions[END_REF] and [START_REF] Gröchenig | Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator[END_REF]). In the case of a splinetype space S, this means that the decay of a function in S can be characterized by the decay of its coefficients and, in particular, that the functions {f k } k form a so called p-Riesz basis for its L p -closed linear span, for the whole range 1 ≤ p ≤ ∞. We derive, in some concrete case, explicit bounds for the localization of the dual basis. We will work with a set of functions satisfying a polynomial decay condition around a set of nodes forming a lattice. By a change of variables, we can assume that the lattice is Z d . So, we will consider a set of functions

{f k } k ⊆ L 2 (R d ) satisfying the condition, |f k (x)| ≤ C (1 + |x -k|) -s , x ∈ R d and k ∈ Z d ,
for some constant C. This type of spatial localization is specifically covered by the results in [START_REF] Fornasier | Intrinsic localization of frames[END_REF], but the constants given there are not explicit. We will derive a polynomial decay condition for the dual basis {g k } k , giving explicit information on the resulting constants. This yields some qualitative information, like the dependence of theses constants on A, C and s and the corresponding p-Riesz basis bounds for the original basis.

Main result

|f k (x)| ≤ C (1 + |x -k|) -s , (x ∈ R d ).
Suppose that {f k } k is a Riesz basis for its L 2 closed linear span S, with bounds 0 < A ≤ B < ∞. Let {g k } k ⊆ S be its dual basis. Then, the dual functions satisfy,

|g k (x)| ≤ D (1 + |x -k|) -t , (x ∈ R d ).
where D is given by,

D = E st C 2t+1 (s -t -d) t 1 + A t-1 A t+1 ,
for some constant E > 0 that only depends on the dimension d.

Remark 1

The constant E can be explicitly determined from the proof.

The results in [START_REF] Gröchenig | Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator[END_REF] prescribe polynomial decay estimates for the dual basis similar to those possessed by the original basis. As a trade-off for the explicit constants we will not obtain the full preservation of these decay conditions. Nevertheless, any degree of polynomial decay on the dual system can be granted, provided that the original basis has sufficiently good decay. Finally observe that, although the basis {f k } k is assumed to be concentrated around a lattice of nodes, the functions f k are not assumed to be shifts of a single function. In particular, Theorem 1 below allows for a basis of functions whose 'optimal' concentration nodes do not form a lattice but are comparable to one. The 'eccentricity' of the configuration of concentration nodes is, however, penalized by the constants modelling the decay.

Sketch of a proof and comments

Now we sketch the proof of the main result, for a complete proof see [START_REF] Luis | Explicit localization estimates for spline-type spaces[END_REF].

Consider the gram matrix of the basis {f k } k given by,

M ≡ (m k,j ) k,j∈Z d , m k,j := f k , f j .
Since {f k } k is a Riesz sequence, M , as an operator on ℓ 2 , has an inverse N ≡ (n k,j ) k,j∈Z d . Moreover, N ℓ 2 →ℓ 2 ≤ A -1 and n k,j = g k , g j , where

{g k } k ⊆ S is the dual basis of {f k } k .
The localization assumptions on the basis {f k } k yield a polynomial decay estimate on the entries of M,

|m k,j | (1 + |k -j|) -s .
If we can establish a similar estimate for the entries of N ,

|n k,j | (1 + |k -j|) -t .
with all the constants given explicitly, then, using calculations similar to those in [START_REF] Fornasier | Intrinsic localization of frames[END_REF], we obtain the desired polynomial concentration conditions for the dual functions.

Let us first consider the case where the basis {f k } k consists of integer shifts of a single generator f (that is,

f k = f (• -k), k ∈ Z d ).
In this case, the matrix M is constant on its diagonals. That is,

m k,j = a k-j ,
for some sequence a. Similarly, N is given by

n k,j = b k-j ,
where the sequence b satisfies a * b = δ. Therefore, in this special case, M and N are convolution operators. The off-diagonal decay of their entries is equivalent to the decay of their kernels a and b. Since the decay of a sequence x can be characterized by the smoothness of its Fourier transform x, the problem can be reformulated as the preservation of the smoothness of the function â under pointwise inversion. This reasoning is present, for example, in [START_REF] Aldroubi | Nonuniform sampling and reconstruction in shift-invariant spaces[END_REF]. We can measure the smoothness of â by considering weakderivatives and use repeatedly a chain-rule argument for Sobolev spaces to obtain similar smoothness conditions for b.

In the general case, where M and N need not be convolution operators, we try to imitate this reasoning, but we avoid using the Fourier transform. Given a matrix L ≡ (l k,j ) k,j∈Z d and 1 ≤ h ≤ d, we consider the new matrix,

D h (L) k,j := (k h -j h )l k,j .
Observe that, up to some multiplicative constant, the map D h acts on a convolution operator by taking a partial derivative of its symbol (that is, the Fourier transform of its kernel.) The domain of D h consists of those matrices L such that D h (L) defines a bounded operator on ℓ 2 . We call D h (L) the derivative of L (with respect to x h .) D h is a derivation in the sense that it satisfies the equation

D h (AB) = D h (A)B + AD h (B), provided that D h (A)
and D h (B) are both defined. Derivations are a wellknown tool in operator-algebras theory (see [START_REF] Bratteli | Unbounded derivations of c*-algebras[END_REF], [START_REF] Kissin | Dense qsubalgebras of banach and c*-algebras and unbounded derivations of banach and c*-algebras[END_REF] and [START_REF] Kissin | Differential properties of some dense subalgebras of c*-algebras[END_REF].) Since M N = I and D h (I) = 0, we can formally express the high-order derivatives of N in terms of its lower-order ones and all the derivatives of M ,

D u h (N ) = - u-1 l=0 u l D l h (N )D u-l h (M )N. (2)
Using the polynomial off-diagonal decay bounds on M and the bound N ℓ 2 →ℓ 2 ≤ A -1 we can obtain bounds for the ℓ 2 → ℓ 2 norms of some derivatives of N . These imply polynomial off-diagonal decay estimates for N , and hence yield the desired spatial localization bounds for the dual basis.

In the argument above we related the off-diagonal decay of a matrix with the ℓ 2 → ℓ 2 norm of its derivatives. The ℓ 2 → ℓ 2 norm of a matrix is not determined by the size of its entries. However, there are some necessary and (other) sufficient conditions on the size of the entries of a matrix for it to be bounded on ℓ 2 . This "gap" in the conditions accounts for the loss of some decay information in Theorem 1, when passing from the original basis to its dual system. Finally we point out that the formal computations in the above argument are not sufficient to prove the theorem. Consider again the simple case of a basis of integer shifts.

With the notation of the discussion above, we have the relation

a * b = δ, (3)
we have some decay estimate on a (that can be reformulated as a smoothness condition on â) and we want to prove a similar decay condition for b. There may be various sequences x satisfying the relation a * x = δ; b can be singled out as the only one of them having a bounded Fourier transform. For example, when a is finitely supported, equation 3 is a linear difference equation which has other solutions besides b (that grow exponentially). The decay of the sequence b can be rigorously proved by resorting to some Sobolev-space smoothing argument. In the general case, to derive equation [START_REF] Radu | Density, overcompleteness, and localization of frames I: Theory[END_REF], one needs to use the associativity of the product of matrices. This is justified only if all the matrices involved represent bounded operators. In other words, we need to know a priori that the derivatives of N that are involved in equation ( 2) define bounded operators. This can be proved using the general results on derivations on Banach algebras (see [START_REF] Bratteli | Unbounded derivations of c*-algebras[END_REF], [START_REF] Kissin | Dense qsubalgebras of banach and c*-algebras and unbounded derivations of banach and c*-algebras[END_REF]) or Jaffard's Theorem [START_REF] Jaffard | Propriétés des matrices "bien localisées" près de leur diagonale et quelques applications[END_REF]. The use of derivations is somehow implicit in Jaffard's paper [START_REF] Jaffard | Propriétés des matrices "bien localisées" près de leur diagonale et quelques applications[END_REF]. Recently, Gröchenig and Klotz [START_REF] Gröchenig | Noncommutative approximation: Inverse-closed subalgebras and off-diagonal decay of matrices[END_REF] have systematically studied the use of derivations in connection to various problems including the preservation under inversion of various kinds of off-diagonal decay conditions.

Application

From Theorem 1 we can derive the following qualitative statement.

Theorem 2 Let F i i∈I be a family of Riesz sequences,

F i ≡ f i k k∈Z d ⊆ L 2 (R d ), (i ∈ I).
sharing a uniform lower basis bound. Suppose that the family F i i satisfies a uniform concentration condition,

f i k (x) ≤ C (1 + |x -k|) -s , (x ∈ R d , k ∈ Z d , i ∈ I),
for some constants C ≥ 1, s > d + t and t > d, with t an integer.

Then the following holds.

(a) The respective family of dual systems G i i -where G i ≡ g i k k∈Z d -satisfies a uniform concentration condition,

g i k (x) ≤ D (1 + |x -k|) -t , (x ∈ R d , k ∈ Z d , i ∈ I),
for some constant D ≥ 1.

(b) A uniform p-Riesz basis condition holds, for all 1 ≤ p ≤ ∞. More precisely, there exist constants q, Q > 0 such that for any p ∈ [1, ∞] and any i ∈ I, the relation

q c ℓ p ≤ k c k f i k L p ≤ Q c ℓ p
holds for all finitely supported sequences (c k ) k∈Z d .

Statement (a) follows directly from Theorem 1. Examining the proofs in [START_REF] Fornasier | Intrinsic localization of frames[END_REF] we see that the uniformity of the constants given in (a) yields statement (b). This qualitative conclusion on Theorem 2 was the original motivation for Theorem 1. Finally, observe that the arguments given above are applicable to a generel intrinsically localized basis in the sense of [START_REF] Fornasier | Intrinsic localization of frames[END_REF].

  Let C ≥ 1, and let t > d be integers. Let s > d + t be a real number. For k ∈ Z d let f k : R d → C be a measurable function such that

Theorem 1

Acknowledgements

The author wishes to thank Karlheinz Gröchenig and Andreas Klotz for their comments and for sharing an early draft of [7], and is indebted to Hans Feichtinger and Ursula Molter for some insightful discussions. The author holds a fellowship from the CONICET and thanks this institution for its support. His research is also partially supported by grants: PICT06-00177, CONICET PIP N 5650, UBACyT X149. This note was partially written during a long-term visit to NuHAG in which the author was supported by the EUCETIFA Marie Curie Excellence Grant (FP6-517154, 2005-2009).


Kunio Yoshino
email: yoshinok@tcu.ac.jp.

Tamazutsumi

Daubechies Localization Operator in Bargmann -Fock Space and Generating Function of Eigenvalues of Localization Operator

We will express Daubechies localization operators in Bargmann -Fock space. We will prove that the Hermite functions are eigenfunctions of Daubechies localization operator. By making use of generating function of eigenvalues of Daubechies localization operator, we will show some reconstruction formulas for symbol function of Daubechies localization operator with rotational invariant symbol.

Introduction

Daubechies localization operator was introduced in I. Daubechies : A Time Frequency Localization Operator: A Geometric Phase Space Approach, IEEE. Trans. Inform. theory. vol.34, pp.605-612(1988) She obtained following results.

Theorem(Daubechies)([2])

Suppose that symbol function of Daubechies localization operator is rotational invariant. Then (i) Eigenfunctions of Daubechies localization operator are Hermite functions. (ii) Eigenvalues are given by Mellin transform of symbol function.

In this paper we realize Daubechies localization opeartor in Bargamann -Fock space. We will consider the eigenvalue problem of Daubechies localization opeartor in Bargmann -Fock space. By making use of Bargamann -Fock space we will give a new proof of above theorem. We will establish reconstruction formula of symbol function of Daubechies localization operator with rotational invariant symbol by generating function of eigenvalues of Daubechies localization operator. For the simplicity, we will confine ourselves to 1-dimensional case.

Bargmann Transform

Put A(z, x) = π -1/4 exp - 1 2 (z 2 + x 2 ) + √ 2z • x ,
where z ∈ C and x ∈ R.

Bargmann transform B(ψ) is defined as follows :

B(ψ)(z) def = R ψ(x)A(z, x)dx, (ψ ∈ L 2 (R)).
Put

BF = {g ∈ H(C) : C |g(z)| 2 e -|z| 2 dz ∧ dz < ∞}
where H(C) denotes the space of entire functions in the complex plane.

BF is called Bargmann-Fock space.

Theorem 1([1])

Bargmann transform is a unitary mapping from L 2 (R) to Bargmann-Fock space BF .

For the details of Bargmann transform and Bargmann -Fock space, we will refer the reader to [START_REF] Bargmann | On a Hilbert Space of Analytic Functions and an Associated Integral Transform Part I[END_REF] and [START_REF] Folland | Harmonic Analysis in Phase Space[END_REF].

Hermite Functions Definition 1([1],[3])

Hermite functions h m (x) is defined by :

h m (x) = (-1) m (2 m m! √ π) -1/2 exp(x 2 /2) d m dx m exp(-x 2 ), (m ∈ N).
Hermite functions has following generating function expansion :

π -1/4 exp - 1 2 (z 2 + x 2 ) + √ 2z • x = ∞ m=0 z m √ m! h m (x), (z ∈ C, x ∈ R).
We recall some well known facts about Hermite functions.

Proposition 1([1],[3]

)

(i) {h m (x)} ∞ m=0 is complete orthonormal basis in L 2 (R). (ii) (- ∂ 2 ∂x 2 + x 2 -1)h m (x) = mh m (x), (iii) B(h m )(z) = z m √ m! , (z ∈ C) (iv) F(h m )(x) = (-i) m h m (x),
where F is Fourier transform.

Proposition 2([1],[3]) (i) (B • L • B -1 )g(z) = z ∂ ∂z g(z)
,

where L = - ∂ 2 ∂x 2 + x 2 -1. (ii) (B • F • B -1 )g(z) = g(-iz),
where F is Fourier transform and g(z) ∈ BF .

Daubechies Localization Operator

Put φ p,q (x) = π -1/4 e ipx e -(x-q) 2 /2 . < φ p,q , f >= R φ p,q (x)f (x)dx.
This is so called Short time Fourier transform (or Windowed Fourier transform, or Gabor transform).

Definition 2([2])

Suppose that

F (p, q) ∈ L 1 (R 2 ) and f (x) ∈ L 2 (R).
We put

P F (f )(x) = 1 2π R 2 F (p, q)φ p,q (x) < φ p,q , f > dpdq,
We call P F (Daubechies) localization operator F (p, q) is called symbol function.

Daubechies obtained following results.

Theorem([2]

). Suppose that F (p, q) ∈ L 1 (R 2 ) and

F (p, q) is rotational invariant function, i.e. F (p, q) = F (r 2 ), (r 2 = p 2 + q 2 ).
Then (i) Hermite functions h m (x) are eigenfunctions of Daubechies operator P F .

P F (h m )(x) = λ m h m (x), (m ∈ N), (ii) λ m = 1 m! ∞ 0 e -s s m F (2s)ds, (m ∈ N).

A Realization of Daubechies Localization Operator in Bargmann Fock space

In this section we will express Daubechies Localization Operator in Bargmann -Fock space. First we need following lemmas.

Lemma 1 B(φ p,q )(z) = e zw-1/2|w| 2 +1/2ipq , (w = p + iq √ 2 ) Lemma 2([1]) g(z) = 1 2πi C e wt g(t)e -|t| 2 dt ∧ dt, (g ∈ BF )
Theorem 2 Under the same assumptions in Prop. 3, we have

(B • P F • B -1 )(g)(z) = 1 2πi C F (w, w)e zw g(w)e -|w| 2 dw ∧ dw, (∀g ∈ BF ) (Proof)
Since Bargmann transform is unitary operator, we have

P F (f )(x) = 1 2π F (p, q)φ p,q (x) < φ p,q , f > dpdq, = 1 2π F (p, q)φ p,q (x) < Bφ p,q , Bf > dpdq,
So by lemma 1,

B • P F (f )(x) = 1 2π
F (p, q)Bφ p,q (z) < Bφ p,q , Bf > dpdq, = 1 2π F (p, q)e zw-1/2|w| 2 +1/2ipq < Bφ p,q , Bf > dpdq, Hence we have

(B • P F • B -1 )(g)(z) = 1 2π F (p, q)e zw-1/2|w| 2 +1/2ipq < Bφ p,q , g > dpdq,
On the other hand

< Bφ p,q , g > = 1 2π e t w-1/2|w| 2 -1/2ipq g(t)e -|t| 2 dtd t, By Lemma 2, = e -1/2|w| 2 -1/2ipq g( w)
Thus we obtained our desired result.

Proposition 3([2]

). Suppose that F (p, q) ∈ L 1 (R 2 ) and F (p, q) is rotational invariant function, i.e. F (p, q) = F (r 2 ), (r 2 = p 2 + q 2 ). Then (i) Functions z m are eigenfunctions of operator

B • P F • B -1 . (B • P F • B -1 )(z m ) = λ m z m , (m ∈ N), (ii) λ n = 1 n! ∞ 0 e -s s n F (2s)ds, (n ∈ N). (Proof)
By Theorem 2, we have

(B • P F • B -1 )(z m ) = 1 2πi C F (2|w| 2 )e zw w m e -|w| 2 dw ∧ dw,
Employing polar coordinte transform w = re iθ and s = r 2 , we have

= z m 1 m! ∞ 0 e -s s m F (2s)ds.
As a corollary of Proposition 3, we obtained following Daubechies's results in section 4.

Proposition 4([8])

Let {λ m } be eigenvalues of P F . Then there exists a positive constant C such that

|λ m | ≤ C |m| , (m ∈ N). Put Λ(w) = ∞ m=0 λ m w m .
We call Λ(w) generating function of eigenvalues of Daubechies Localization Operator. Theorem 3 Under the same assumptions in Prop. 3, we have

(B • P F • B -1 )(g)(z) = (2πi) -n g(t)Λ( z t ) dt t , (∀g ∈ BF ) (Proof)
Suppose that g(z) ∈ BF . We consider Taylor expansion of g(z) at the origin. Put

g(z) = ∞ m=0 a m z m
By Proposition 3, we have

(B • P F • B -1 )(z m ) = λ m z m . So (B • P F • B -1 )(g)(z) = (B • P F • B -1 )( ∞ m=0 a m z m ) = ∞ m=0 a m λ m z m = (2πi) -n g(t)Λ( z t ) dt t
Hence we have

(B • P F • B -1 )(g)(z) = (2πi) -1 g(t)Λ( z t ) dt t .

An Example of Daubechies Localization Operator

In this section we will consider following special Daubechies localization operators.

Put F a (p, q) = e a-1 2a (p 2 +q 2 ) = e a-1 2a r 2 , (0 < a < 1). Then λ m = a m+1 , Λ(w) = a 1 -aw . P F a (h m )(x) = a m+1 h m (x). P Fa = ∞ m=0 a m+1 h m (x)h m (y).
valids in operator sense.

(P F a = ∞ m=0 a m+1 |m >< m|, in Dirac's Notation.)
If a = 2 -1 , this is Schatten decomposition of P Fa and P F a is called density operator in quantum statistical mechanics.

Proposition 5 (Mehler's formula [START_REF] Folland | Harmonic Analysis in Phase Space[END_REF], [START_REF] Wong | Weyl Transforms[END_REF])

∞ m=0 a m+1 h m (x)h m (y) = a π(1 -a 2 ) e -1 4 ( 1-a 1+a (x+y) 2 + 1+a 1-a (x-y) 2 ) , (|a| < 1). Corollary 3 (i) P Fa (f ) = R a π(1 -a 2 ) e -1 4 ( 1-a 1+a (x+y) 2 + 1+a 1-a (x-y) 2 ) f (y)dy, (f ∈ L 2 ). (ii) If a ∈ C, |a| < 1, then P F a : L 2 -→ L 2 is bounded linear operator. (Proof) If a ∈ {a ∈ C : |a| < 1}, then real part of 1 -a 1 + a + 1 + a 1 -a is positive. So P F a is bounded linear operator from L 2 to L 2 .
Namely, we obtained analytic continuation of P F a under the condition (a ∈ C, |a| < 1).

Realization of P F a in Bargmann -Fock space

In this section we will consider P Fa in Bargmann -Fock space.

Proposition 6

(i) B • P F a • B -1 = ∞ m=0 a m+1 z m √ m! wm √ m! .
valids in operator sense.

(ii

) (B • P F a • B -1 )(g)(z) = ia 2 C e az wg(w)e -|w| 2 dw ∧ d w, (g ∈ BF ) (Proof) Since z m √ m! . are eigenfunctions of B • P F a • B -1 ,
we have

B • P Fa • B -1 = ∞ m=0 a m+1 z m √ m! wm √ m! .
Proposition 7 Suppose that |a| < 1, (a ∈ C). Then we have

(B • P F a • B -1 )(g)(z) = ag (az) , (g ∈ BF ). (Proof) (B • P F a • B -1 )(g)(z) = (2πi) -1 g(t)Λ( z t ) dt t = (2πi) -1 g(t) a t -az dt = ag(az).
Proposition 8 For f ∈ L 2 , we have

(i) lim a→1 P F a (f ) = f, (ii) lim a→-i P Fa (f ) = (-i)Ff, (iii) lim a→i P F a (f ) = iF -1 f,
where F is Fourier transform.

(Proof) By Prop.7, we have

(B • P F a • B -1 )(g)(z) = ag (az) , (g ∈ BF ). (i) lim a→0 (B • P F a • B -1 )(g) = lim a→1 ag(az) = g(z).
This means that lim a→1 P F a = Identity operator.

(ii) lim

a→-i (B • P F a • B -1 )(g) = lim a→-i ag(az) = (-i)g(-iz).
By (ii) in Proposition 2, this means that lim a→-i

P F a = (-i)F .
Proof of (iii) is same as that of (ii).

Proposition 9

(i) G = {P F a : a ∈ C, |a| < 1} ∪ {I d } is semigroup. (ii) P Fa • P Fa = P F ab . (Proof) By Proposition 7, (B • P F a • B -1 )(g)(z) = ag(az), g(z) ∈ BF So, we have (B • P F a • P F b • B -1 )(g)(z) = bag (baz) Hence we have P F b • P Fa = P F ab .
In these cases, F a (p, q) / ∈ L 1 . But these operators still define bounded operators from L 2 to L 2 . As seen in Proposition 8, these operators are obtained as limit of P F a , (F a ∈ L 1 ).

Reconstruction formulas

We assume that F (p, q) is rotational invariant L 1 function. Namely, F (p, q) = F ( p 2 + q 2 ).

In section 5, we introduced following generating function:

Λ(w) = ∞ m=0 λ m w m
Λ(w) is called generating function for eigenvalues of P F Now we consider following formal power series :

∞ m=0 m!λ m t -m-1
In general this series is divergent series. We put where F is Fourier transform.

G(t) =

∞ 0 FF

 0 (2s)e -s t -s ds, (t ∈ C\[0, ∞]). m t -m-1is an asymptotic expansion of G(t).Remark Λ(w) is the Borel transform of formal power series∞ m=0 m!λ m t -m-1 .Since G(t) is Hilbert transform of F (2s)e -s , we have Theorem 5F (2s) = e s lim t→0 -1 2πi (G(s + it) -G(s -it)) (2s) = (2π) -n e s F(Λ(iv))(s),valids in distribution sense.


Hidemitsu Ogawa
email: hidemitsu-ogawa@kuramae.ne.jp

Akira Hirabayashi

Optimal Characteristic of Optical Filter for White-Light Interferometry based on Sampling Theory

White-light interferometry is a technique of profiling surface topography of objects such as semiconductors, liquid crystal displays (LCDs), and so on. The world fastest surface profiling algorithm utilizes a generalized sampling theorem that reconstructs the squared-envelope function r(z) directly from an infinite number of samples of the interferogram f (z). In practical measurements, however, only a finite number of samples of the interferogram g(z) = f (z) + C with a constant C are acquired by an interferometer. We have to estimate the constant C and to truncate the infinite series in the sampling theorem. In order to reduce both the truncation error and the estimation error for C, we devise an optimal characteristic of the optical filter installed in the interferometer in the sense that the second moment of the square of the interferogram is minimized. Simulation results confirm the effectiveness of the optimal characteristic of the optical filter.

Introduction

White-light interferometry is a technique of profiling surface topography of objects such as semiconductors, liquid crystal displays (LCDs), and so on. It is attractive because of its advantages including non-contact measurement and unlimited measurement range in principle [START_REF] Caber | Interferometric profiler for rough surfaces[END_REF][START_REF] Chim | Three-dimensional image realization in interference microscopy[END_REF][START_REF] De Groot | Surface profiling by analysis of white-light interferograms in the spatial frequency domain[END_REF][START_REF] Hirabayashi | Fast surface profiler by white-light interferometry by use of a new algorithm based on sampling theory[END_REF][START_REF] Kino | Mirau correlation microscope[END_REF][START_REF] Larkin | Efficient nonlinear algorithm for envelope detection in white light interferometry[END_REF][START_REF] Ogawa | Sampling theory in white-light interferomtery[END_REF]. From the viewpoint of sampling theory, white-light interferometry has the following two interesting features. First, a signal to be processed, a white-light interferogram, f (z), is a bandpass signal. Second, a signal to be reconstructed from sampled values of f (z) is not the interferogram itself, but its squared-envelope function r(z). This type of sampling theorem is called a generalized sampling theorem [START_REF] Grace | Sampling and interpolation of bandlimited signals by quadrature methods[END_REF][START_REF] Rice | Quadrature sampling with high dynamic range[END_REF][START_REF] Waters | Bandpass signal sampling and coherent detection[END_REF]. The present authors also derived such a sampling theorem [START_REF] Ogawa | Sampling theory in white-light interferomtery[END_REF]. Based on the theorem, the world fastest surface profiling algorithm were proposed and installed in commercial systems [START_REF] Hirabayashi | Fast surface profiler by white-light interferometry by use of a new algorithm based on sampling theory[END_REF]. The sampling theorem is expressed in a form of infinite series and uses samples of the interferogram f (z). In practical measurements, however, only a finite number of samples of the interferogram g(z) = f (z) + C with a constant C are acquired by an interferometer. Hence, in the algorithm, the constant C is estimated from the samples, and the infinite series is truncated with the number of samples. If both the truncation error and the estimation error for C were reduced, we can further improve the preciseness of the algorithm. For both error reductions, it is very effective for interferograms to have small side lobes. The waveform of interferograms can be controlled by an optical filter installed in the interferometer. Hence, in this paper, we devise an optimal characteristic of the optical filter in the sense that the second moment of the square of the interferogram is minimized with a fixed band-width. We show that the optical characteristic is given by a sine curve which has a half of the period as the fixed band-width. We also show that we have a socalled uncertainty principle between the band-width and the second moment. Simulation results confirm the effectiveness of the optimal characteristic of the optical filter.

Surface Profiling by White-Light Interferometry

Figure 1 shows a basic setup of an optical system used for surface profiling by white-light interferometry. It uses the Michelson interferometer. A beam from a white-light source passes through an optical filter. The beam is re- It is known that the envelope function m(z) shown by the solid line in Fig. 2, or its square r(z), has the same peak as the interferogram and they are much smoother than the interferogram. Hence, usually these functions are used for detection of the peak instead of the interferogram. In this paper, we use the latter r(z), which we call the squaredenvelope function.

Sampling theorem for squared-envelope functions

Since the interferogram f (z) is a bandpass signal, it can be reconstructed from its samples by using the sampling the-orem for bandpass signals [START_REF] Kohlenberg | Exact interpolation of band-limited functions[END_REF]. It is interesting that, since the squared-envelope function r(z) is the sum of squares of f (z) and its Hilbert transform, the squared-envelope function is also reconstructed from samples of f (z), not those of r(z). Indeed, the following result was established [START_REF] Ogawa | Sampling theory in white-light interferomtery[END_REF][START_REF] Hirabayashi | Fast surface profiler by white-light interferometry by use of a new algorithm based on sampling theory[END_REF]. The center wavelength and the bandwidth of the optical filter in Fig. 1 are denoted by λ c and 2λ b , respectively. Let k l and k u be angular wavenumbers defined by

k l = 2π λ c + λ b , k u = 2π λ c -λ b . ( 1
)
Two parameters ω l = 2k l and ω u = 2k u are also used.

Proposition 1 [START_REF] Hirabayashi | Fast surface profiler by white-light interferometry by use of a new algorithm based on sampling theory[END_REF] (Sampling theorem for squaredenvelope functions) Let I be an integer such that

0 ≤ I ≤ ω l ω u -ω l , ( 2
)
and ω b be any real number that satisfies

     ω u 2 ≤ ω b (I = 0), ω u 2(I + 1) ≤ ω b ≤ ω l 2I (I = 0). (3)
Let ω c be a real number defined by

ω c = (2I + 1)ω b . ( 4
)
Let ∆ be a sampling interval given by

∆ = π 2ω b , ( 5
)
and {z n } ∞ n=-∞ be sample points defined by

z n = n∆. ( 6
)
Then, it holds that

1. When z is a sample point z j , r(z j ) = {f (z j )} 2 + 4 π 2 { ∞ ∑ n=-∞ f (z j+2n+1 ) 2n + 1 } 2 . ( 7
)
2. When z is not any sample point,

r(z) = 2∆ 2 π 2   ( 1 -cos πz ∆ ) { ∞ ∑ n=-∞ f (z 2n ) z -z 2n } 2 + ( 1 + cos πz ∆ ) { ∞ ∑ n=-∞ f (z 2n+1 ) z -z 2n+1 } 2   . ( 8
)
To apply Proposition 1 for surface profiling, we have the following difficulties. In the proposition, an infinite number of sampled values {f (z n )} ∞ n=-∞ of the interferogram f (z) are used. In practical applications, however, only a finite number of sampled values {g(z n )} N -1 n=0 of the interferogram g(z) = f (z) + C are available. Hence, we have to truncate the infinite series in Proposition 1 and approximate the sampled values f (z n ) by g(z n ) -Ĉ, where Ĉ is an estimate of C. For example, the average of g(z n ) is used as Ĉ. Now, we are suffered from the truncation error as well as the estimation error for Ĉ. Both of these errors severely affect our final goal of precise estimation of z p .

Optimal characteristics of optical filter

To reduce both of the errors, the following observation is crucial. As you can see in Fig. 2, only a few number of samples are located in the main lobe of g(z) while the rest of them are in side lobes. The latter mostly vanishes once the constant C is estimated precisely. This implies that, the smaller the side lobes are, the smaller the truncation error is. Smaller side lobes also lead us to better estimations of C as shown experimentally in Section 5. Fortunately, we can control the waveform of the interferogram by the optical filter in the interferometer. Let a(k) be its characteristic in terms of an angular wavenumber k.

The support of a(k) is the interval k l < k < k u . Averaged attenuation rates of two beams along the dashed and the dotted lines in Figure 1 are denoted by q o (k) and q r (k), respectively. Let ψ(k) be

ψ(k) = { 2{a(k)} 2 q o (k)q r (k) (k > 0), 0 (k ≤ 0). (9)
It is also supported on the same interval as a(k):

ψ(k) = 0 (k < k l , k > k u ). (10)
The function ψ(k) is related to the interferogram f (z) as

f (z) = ∫ ku k l ψ(k) cos 2k(z -z p ) dk. ( 11
)
Equation ( 11) clearly shows that we can control f (z) by a(k) through ψ(k).

To have smaller side lobes, we can easily arrive at the following idea: we design ψ(k) so that it minimizes the second moment of the square of the interferogram f (z):

J[ψ] = ∫ ∞ -∞ (z -z p ) 2 {f (z)} 2 dz. ( 12
)
Now, we are at the point to show our main result in this paper. Let k a be (k u -k l )/2.

Theorem 1 Among second continuously differentiable functions

ψ(k) ∈ C 2 [k l , k u ] satisfying ψ(k) = 0 (k ≤ k l , k ≥ k u ), (13) ψ(k) ≥ 0 (k l < k < k u ), (14) ∫ ku k l {ψ(k)} 2 dk = 1, (15)
ψ(k) that minimizes the criterion J[ψ] is given by

ψ(k) = 1 √ k a sin π(k -k l ) 2k a . ( 16
)
The minimum value J 0 is given by

J 0 = ( π 2 ) 3 1 (2k a ) 2 = π∆ 2 2 . ( 17
)
The following two results are direct consequence of Theorem 1.

Corollary 1

The optimal characteristic a(k) under the criterion J[ψ] is given by

a(k) = ( sin π(k -k l )/2k a 2 √ k a q o (k)q r (k) ) 1/2 . ( 18
)
Corollary 2 The optimal waveform of the interferogram f (z) is given by

f (z) = m(z) cos(k u + k l )(z -z p ), (19)
where

m(z) = 4π √ k a cos 2k a (z -z p ) π 2 -16k 2 a (z -z p ) 2 .
(

The interferogram shown in Fig. 2 was the optimal one given by Eqs. ( 19) and (20) while that shown in Fig. 3 is generated from a rectangular ψ(k) given by

ψ(k) = { 1/ √ k u -k l (k l < k < k u ), 0 (otherwise).
Though this ψ(k) is not continuously second differentiable, the conditions (13) ∼ (15) are satisfied. In both figures, λ c = 600[nm] and λ b = 30[nm] were used. We can see that the side lobes in Fig. 2 are much smaller than those in Fig. 3. The sampling interval used in both figures is ∆ = 1.425[µm], which is the maximum among those satisfying Eqs. ( 2) ∼ (5). We have six samples in the main lobe in Fig. 2 while only four samples are located there in Fig. 3 (these samples are displayed by relatively large dots compared to samples in side lobes). In a nutshell, the optimal characteristic results in fewer samples in the small side lobes. This results in small errors on the truncation and the estimation of C, which we demonstrate in the next section through computer simulations.

Before proceeding simulations, let us make a final remark in this section.

Corollary 3 Let σ 2 be the value of J [ψ]. Then, the following uncertainty principle holds: and reconstructed functions (the solid lines) from samples of g(z) for both of the optimal and the rectangular ψ(k).

σ 2 (2k a ) 2 ≥ ( π 2 ) 3 , σ 2 ∆ 2 ≥ π 2 .

Simulations

We compare the optimal and the rectangular characteristics ψ(k) by computer simulations. We first sample the interferograms g(z) generated from both ψ(k) with the sampling interval ∆ = 1.425µm. Then, the averages for each sample values are computed for the estimation of C. Finally, we reconstruct the squared-envelope functions r(z) by using a finite number of g(z n ) -Ĉ instead of f (z n ) in Proposition 1. The reconstructed functions are shown in Fig. 4 by the solid lines as well as the original squaredenvelope functions by the dashed lines for both of the optimal and the rectangular ψ(k). The small window in the top-right side shows the magnified image around the peak. We can see that the reconstructed function for the optimal ψ(k) provides a much better result than that for the rectangular ψ(k). We also notice that the latter oscillates severely.

The normalized truncation errors for the optimal and the rectangular ψ(k) are 0.45% and 4.68%, respectively. The former is less than 10% of the latter. When C = 1.10, its estimation results are 1.10 and 1.06 for the optimal and the rectangular ψ(k), respectively. Finally, errors for the estimation of z p are 0.05µm and 0.06µm for the optimal and the rectangular ψ(k), respectively. Even though the difference is not so significant, the oscillation of the reconstructed squared-envelope function for the rectangular ψ(k) may cause difficulties for fast search of the maximum position.

We repeated the same simulations for thirty two values of z p from 10µm to 20µm. Then, averages of estimation errors were 0.0496µm and 0.0541µm for the optimal and the rectangular, respectively. They are almost the same value. However, the averages of truncation errors were 0.35% and 4.67% for the optimal and the rectangular ψ(k), respectively. The former is less than 7% of the latter. These results show the effectiveness of the optimal characteristics of the optical filter.

Conclusion

In this paper, we devised an optimal characteristic of the optical filter that minimizes the second moment of the square of the interferogram so that both of the truncation error and the estimation error for the constant in the interferogram are reduced. We showed that the optimal characteristic is given by a sine curve which has a half of the period as the band-width of the optical filter. Simulation results showed that the truncation error for the optimal characteristic is less than 7% of that for the rectangular one. The estimation error of the constant for the optimal characteristic was also smaller than the rectangular one. Even though the difference on the estimation error of the maximum position was not so significant, reconstructed functions for the optimal characteristic was much smoother than those for the rectangular one. These results showed the effectiveness of the optimal characteristic. Our future tasks include to produce a prototype of the optical filter with the optimal characteristic.

Figure 1 :

 1 Figure 1: Basic setup of an optical system used for surface profiling by white-light interferometry.

Figure 2 :

 2 Figure 2: An example of a white-light interferogram g(z) and its sampled values.

Figure 3 :

 3 Figure 3: A white-light interferogram g(z) when ψ(k) is rectangular.

Figure 4 :

 4 Figure4: Squared-envelope functions (the dashed lines) and reconstructed functions (the solid lines) from samples of g(z) for both of the optimal and the rectangular ψ(k).