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Generalized Poisson Summation Formula for Tempered Distributions

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Generalized Poisson Summation Formula for Tempered Distributions 2015 International Conference on Sampling Theory and Applications (SampTA) Generalized Poisson Summation Formula for Tempered Distributions Ha Q. Nguyen and Michael Unser Biomedical Imaging Group, Ecole´ Polytechnique Fed´ erale´ de Lausanne (EPFL) Station 17, CH–1015, Lausanne, Switzerland fha.nguyen, michael.unserg@epfl.ch Abstract—The Poisson summation formula (PSF), which re- In the mathematics literature, the PSF is often stated in a lates the sampling of an analog signal with the periodization of dual form with the sampling occurring in the Fourier domain, its Fourier transform, plays a key role in the classical sampling and under strict conditions on the function f. Various versions theory. In its current forms, the formula is only applicable to a of (PSF) have been proven when both f and f^ are in appro- limited class of signals in L1. However, this assumption on the signals is too strict for many applications in signal processing priate subspaces of L1(R) \C(R). In its most general form, ^ that require sampling of non-decaying signals. In this paper when f; f 2 L1(R)\C(R), the RHS of (PSF) is a well-defined we generalize the PSF for functions living in weighted Sobolev periodic function in L1([0; 1]) whose (possibly divergent) spaces that do not impose any decay on the functions. The Fourier series is the LHS. If f and f^ additionally satisfy only requirement is that the signal to be sampled and its weak ^ −1−" derivatives up to order 1=2 + " for arbitrarily small " > 0, grow jf(x)j+jf(x)j ≤ C(1+jxj) ; 8x 2 R, for some C; " > 0, slower than a polynomial in the L2 sense. The generalized PSF then it can be shown [6], [7] that (PSF) holds pointwise with will be interpreted in the language of distributions. both sides converging absolutely. It is also known [8], [9] ^ that (PSF) holds pointwise when f; f 2 L1(R) \C(R) and I. INTRODUCTION f^ has bounded total variation. Other versions of the PSF on L1(R) can also be found in [9]–[11]. The main limitation of A widely used result in engineering is that sampling in the above mathematical forms of the PSF is that they do not time corresponds to periodization in frequency and vice versa. apply to functions that are non-decreasing, such as our above This beautiful relation is captured in the celebrated Poisson example of a ramp signal or any realization of a stationary summation formula stochastic process, not to mention the non-stationary ones like X X the Brownian motion which may even grow at infinity [12]. f(k)e−2πikξ = f^(ξ + k): (PSF) Intuitively, however, there does not seem to be any compelling k2 k2 Z Z reason why a continuous signal cannot be sampled even if it In other words, the discrete-time Fourier transform of the sam- is not decaying. pled sequence is equal to the periodization of the continuous- In this paper we attempt to bridge the gap between the time Fourier transform of the analog signal. This identity is engineering and mathematics statements of the PSF. We will the key behind many results in sampling theory including show that (PSF) holds in the sense of distribution theory [13] the classical Shannon’s sampling theorem [1] for bandlimited for a broad class of continuous signals included in a weighted signals. What appears to be missing, however, is a proof of s Sobolev space L2;1=w(R), whose weak derivatives up to order f (PSF) for a general function beyond the classical hypotheses, s are in the weighted-L2 space L2;1=w(R). Here, the decaying which will be detailed later. In engineering texts, see for ex- weight 1=w allows growing signals. In particular, when the ample [2]–[5], the common “proof” of (PSF) for an arbitrary weight 1=w decays polynomially and the order of smoothness function f is based on the following observations: (1) the s is above the threshold 1=2, both sides of (PSF) are well- sampled sequence is obtained by multiplying the analog signal defined periodic distributions and they are equal when acting with a Dirac comb; (2) multiplication maps to convolution on periodic test functions. In light of our new result, it is in Fourier-domain; and (3) the Fourier transform of a Dirac easy to see that (PSF) holds for the ramp function, and more comb is also a Dirac comb. The problem with this argument generally, for any piecewise polynomials that are continuous. is that the multiplication of a Dirac comb with a general For brevity, most of the proofs in this paper are omitted or function is not necessarily a tempered distribution, so that the just sketched. Detailed proofs can be found in our extensive convolution theorem may not be applicable. Take, for example, papers [14], [15] that are currently in preparation.1 x+jxj the popular ramp signal f(x) = 2 . Since f(x) is not The rest of the paper is organized as follows. Section II differentiable at 0, the multiplication between f and the Dirac introduces some notations and definitions that will be used comb P δ(· − k) is prohibited. It means that the above k2Z throughout the paper. Section III discusses the sampling in argument for (PSF) does not even work for this seemingly weighted Sobolev spaces. The generalized PSF is stated in well-behaved signal. 978-1-4673-7353-1/15/$31.00 c 2015 IEEE 1The manuscripts are available from the authors upon request. 978-1-4673-7353-1/15/$31.00 ©2015 IEEE 1 2015 International Conference on Sampling Theory and Applications (SampTA) p Section IV with a sketched proof. Finally, some concluding w(0)=Cw ≥ 1=Cw, for all x. This means that every submul- remarks are made in Section V. tiplicative weighting function is lower-bounded. A prototypical example of submultiplicative weights is w(x) = (1+jxj2)α/2, II. NOTATION AND DEFINITION for some α ≥ 0. For this weight, we can choose Cw to be α/2 Throughout the paper, we deal with complex-valued func- Cα = 2 . tions and sequences. Following the convention in signal pro- Definition 2 (Weighted L and ` spaces). For p ≥ 1 and cessing, we use parentheses for functions and square brackets p p a weighting function w, a function f is in L ( ) if fw is for sequences to specify their values at a particular point. Espe- p;w R in Lp(R); a sequence c is in `p;w(Z) if fc[k]w(k)gk2Z is in cially, for a function f, f[·] denotes the sequence ff(k)gk2 . Z `p(Z). The corresponding weighted norms are defined as For p ≥ 1, the spaces Lp(R) and `p(Z) include functions and sequences, respectively, whose p-norms are finite. The scalar kfk := kfwk ; kck := kc w[·]k : Lp;w(R) Lp(R) `p;w(Z) `p(Z) product between two functions in L2(R) is defined as Z Note that if w is submultiplicative then Lp;w(R) ⊂ Lp(R), hf; gi := f(x)g(x)dx; for f; g 2 L ( ) 2 R and `p;w(Z) ⊂ `p(Z) because w is lower-bounded. R As usual, the notation h·; ·i is also used for the action of a Definition 3 (Weighted hybrid norm spaces). For p; q ≥ 1, the distribution on a test function. Sometimes, it is useful to write hybrid norm space Wp;q(R) includes all functions f : R ! C the complex sinusoid e−2πikξ as e−2πih·;ki to imply that it is a whose following norm is finite function in variable ξ. We also adopt standard notations used 0 !q=p 11=q in Schwartz’s distribution theory [13]. We use the symbols ^ Z 1 X kfk := jf(x + k)jp dx ; and _ to denote the forward and inverse Fourier transforms, Wp;q (R) @ A 0 respectively, of a tempered distribution f 2 S0(R), i.e. k2Z D E ^ with usual adjustments when p or q is infinity. The weighted f;^ ' = h(f) ;'i := hf; '^i ; 8' 2 S(R); hybrid norm space Wp;q;w(R) w.r.t. a weighting function w is _ f;ˇ ' = h(f) ;'i := hf; 'ˇi ; 8' 2 S(R) defined according to the following weighted norm where kfkW ( ) := kfwkW ( ): Z p;q;w R p;q R '^(ξ) = (')^(ξ) := '(x)e−2πiξxdx; These hybrid (mixed) norm spaces were mentioned in [9]. ZR _ 2πiξx They are similar to Wiener amalgam spaces Wfp;q(R) [16]– 'ˇ(x) = (') (x) := '(ξ)e dξ: [18], where the discrete and continuous norms are mixed in R reverse order, i.e. The set of continuous functions is denoted by C(R), whereas 1 1=p Cc (R) = D(R) denotes the space of bump functions that 1 p=q! X Z are infinitely differentiable and compactly supported. For an kfk := jf(x + k)jq dx : Wfp;q ( ) 1 R 0 interval T of R, C (T) denotes the space of T-periodic test k2Z functions that are infinitely differentiable. Here, a function These amalgam spaces play a central role in many important ' is T-periodic if ' = '(· + jTjk); 8k 2 Z. The constants throughout the paper are denoted by C with some subscripts results of the non-uniform sampling theory in shift-invariant denoting the dependence of the constants on those parameters. (spline-like) spaces [19]–[23]. Deeper results regarding Wiener amalgam spaces and their generalizations can be found For example, Cx;y;z is a constant that depends only on x; y and z.
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