JFourierAnalAppl(2017)23:442–461 DOI 10.1007/s00041-016-9475-9

Generalized Poisson Summation Formulas for Continuous Functions of Polynomial Growth

Ha Q. Nguyen1 Michael Unser1 John Paul Ward·1,2 ·

Received: 28 July 2015 / Published online: 24 May 2016 ©SpringerScience+BusinessMediaNewYork2016

Abstract The Poisson summation formula (PSF) describes the equivalence between the sampling of an analog signal and the periodization of its frequency spectrum. In engineering textbooks, the PSF is usually stated formally without explicit conditions on the signal for the formula to hold. By contrast, in the mathematics literature, the PSF is commonly stated and proven in the pointwise sense for various types of L1 signals. This L1 assumption is, however, too restrictive for many signal-processing tasks that demand the sampling of possibly growing signals. In this paper, we present two gen- eralized versions of the PSF for d-dimensional signals of polynomial growth. In the first generalization, we show that the PSF holds in the space of tempered distributions for every continuous and polynomially growing signal. In the second generalization, the PSF holds in a particular negative-order Sobolev space if we further require that d/2 ε derivatives of the signal are bounded by some polynomial in the L2 sense. +

Communicated by Eric Todd Quinto.

This work was presented in part at the International Conference on Sampling Theory and Applications, Washington, DC, USA, May 2015.

B Ha Q. Nguyen ha.nguyen@epfl.ch Michael Unser michael.unser@epfl.ch John Paul Ward [email protected]

1 Biomedical Imaging Group, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland 2 Present Address: University of Central Florida, Orlando, FL 32816, USA JFourierAnalAppl(2017)23:442–461 443

Keywords Poisson summation formula Sampling theory Polynomially growing functions Weighted Sobolev spaces Tempered· distributions· · ·

1 Introduction

Awidelyusedresultinsignalprocessingisthatsamplinginthespacedomaincor- responds to periodization in the frequency domain and vice versa. This relation is mathematically described by the Poisson summation formula

2πj k, f (k)e− ⟨ ·⟩ f ( ℓ), (PSF) = ˆ ·+ k d ℓ d !∈Z !∈Z where the left-hand side (LHS) is the discrete-space Fourier transform of the dis- cretized signal f (k) k d ,andtheright-handside(RHS)istheperiodizationofthe { } ∈Z continuous-space Fourier transform of the analog signal f (x) d .ThePSFlies x R at the heart of sampling theories including the classical Shannon’s{ } ∈ sampling theo- rem [25]becauseithelpsconnectthedigitalworldtotheanalogworldthroughthe Fourier transform. In the mathematics literature, the PSF is often stated the other way round with the periodization appearing in the space domain and the sampling occurring in the Fourier domain. Throughout this paper, however, we shall consistently refer to Equation (PSF) as the statement of the PSF. Various versions of the PSF have been proved in the pointwise sense when both the signal f and its Fourier transform f are in appropriate d d d ˆ subspaces of L1(R ) (R ),where (R ) is the space of continuous functions. Given that f, f L ( ∩dC) ( d ),theLHSofPSFcanbeinterpretedasthe(possiblyC ˆ 1 R R divergent) Fourier∈ series of∩ theC periodic function in the RHS. Note that, by Katznelson’s counterexample [15], it is not sufficient to require f, f to be continuous and in L . ˆ 1 One way to make PSF valid pointwise is to impose an appropriate decay on both the signal and its Fourier transform. For example, if f and f are continuous and satisfy ˆ the decay

d ε d f (x) f (x) C(1 x )− − , x R , | |+| ˆ |≤ +∥ ∥ ∀ ∈ for some C,ε>0, then it was shown [10,27]thatPSFholdspointwisewithabsolute convergence of both sides. More generally, Kahane and Lemarié-Rieusset [14]consid- ered signals f such that x a f L ( d ) and ξ b f L ( d ),witha, b 0, and p R ˆ q R gave a characterization of∥ parameters∥ ∈ a, b, p, q∥such∥ that∈ the PSF holds. Gröchenig≥ obtained in [11]asimilarresultinconnectionwiththeuncertaintyprinciple.Inthe 1-D case, when d 1, it is also known [4,36]thatPSFholdspointwisewhen = f, f L1(R) (R) and f has bounded total variation. Many other forms of the ˆ ∈ ∩ C ˆ PSF on L1(R) can also be found in the extensive work of Benedetto and Zimmer- mann [4]. The main limitation of all the above versions of the PSF is that they do not apply to functions that might be growing, such as realizations of a non-stationary stochastic process, the typical example being Brownian motion [31]. Surprisingly, the 444 J Fourier Anal Appl (2017) 23:442–461 existing forms of the PSF cannot even handle the simple (1-D) one-sided power signals xα (max(0, x))α,forα>0. +What:= appears to be missing is a rigorous proof of the PSF for a broad class of signals beyond the L1 hypothesis. In engineering texts, see for example [6,20,21], the PSF is often stated without explicitly specifying sufficient conditions on the signal f .ThecommonproofofthePSFthatispresentedtoengineersisbasedonsymbolic manipulations with the following steps: (1) The LHS of PSF is the Fourier transform of the multiplication between f and the Dirac comb δperiod d δ( k);(2) := k Z ·+ the RHS of PSF is the convolution of f with δ ;and(3)thetwosidesareequal∈ ˆ period " due to the convolution theorem and to the fact that δperiod is the Fourier transform of itself. The problem with this argument, in light of Schwartz’ distribution theory [24], is that the multiplication of a Dirac comb with an arbitrary function is not necessar- ily a tempered distribution,sothattheconvolutiontheoremmaynotbeapplicable. This can also result in erroneous conclusions such as the distributional version of Shannon’s sampling theorem in [34,Theorem7.6.1].Infact,variousgeneralizations of the sampling theorem for bandlimited functions of polynomial growth were rig- orously established in [7,12,16,22,32,33,35]. More recently, Fischer [8]fixedthe Dirac-comb-based argument in proving the PSF by requiring the signal f to be infi- nitely differentiable and bounded by a polynomial. This condition, although allowing the signal to grow, is unnecessarily restrictive. Intuitively, in order for the signal to be sampled, it only needs to be continuous. Our work is an attempt to bridge the gap between the engineering and mathematics statements of the PSF. We provide in this paper two generalizations of the PSF for (possibly) growing signals. In the first generalization, we prove that PSF holds in the d space ′(R ) of tempered distributions if the signal f is continuous and bounded by a polynomial.S This condition is minimal for the signal to be sampled and for its Fourier transform to make sense. In this case, the RHS of PSF should be interpreted as a Cesàro sum which converges conditionally, and the two sides are identical con- tinuous linear functionals on Schwartz’ space (Rd ) of infinitely differentiable and rapidly decaying functions. The second generalizationS of the PSF is for signals whose d/2 ε derivatives, for arbitrary ε>0, are bounded by a polynomial in the L2 + d/2 ε d sense. In particular, if f is included in the weighted Sobolev space L2, +2n (R ) with order of differentiability d/2 ε and with order of growth 2n,forn −N,thenPSF + 2n d ∈ holds in the corresponding negative-order Sobolev space L2− (R ).Inthisversion, both sums in PSF converge unconditionally; hence no need for summability methods. 2n d Moreover, it should be understood that the two sides of PSF are equal in L2− (R ) in the sense that their Fourier transforms are growing functions that are equal almost everywhere. Alternatively, by duality, the second generalized PSF can be interpreted as the equality of two continuous linear functionals on the positive-order Sobolev space 2n d L2 (R ). This paper is a development of the previous works [18,19] of the first two authors. AweakerversionofthegeneralizedPSFonweightedSobolevspaceswasbriefly introduced in [19]withoutdetailedproofs.Ageneraltheoryforsamplinggrowing signals was given in [18]. The outline of the rest of the paper is as follows: In Sect. 2, we provide some mathematical preliminaries, mainly for the benefit of readers with JFourierAnalAppl(2017)23:442–461 445 an engineering background. In Sect. 3,westateandprovethefirstgeneralizationof the PSF for polynomially growing functions. The second generalization of the PSF for functions in weighted Sobolev spaces is presented in Sect. 4. Finally, we provide a discussion in Sect. 5.

2 Preliminaries

2.1 Notation

Let d 1beafixeddimension.Allfunctionsinthispaperarecomplex-valued and ≥ multidimensional with variables in Rd .Let x denote the Euclidean norm of a vector ∥ ∥ x Rd .Fork Nd ,weusethefollowingstandardmulti-indexnotation: ∈ ∈

d d k k ki k ∂| | k ki , k k1 kd , x x ,∂f f. | |:= !:= !··· ! := i := k1 kd i 1 i 1 ∂x1 ...∂xd != #= The constants throughout the paper are denoted by C with subscripts denoting the dependence of the constants on some parameters. We denote by f the sampled [·] sequence of a function f on the integer multigrid; that means f k f (k), k Zd . d d [ ]:= ∀ ∈ For 1 p , L p(R ) and ℓp(Z ) denote the spaces of functions and sequences, respectively,≤ ≤∞ whose p-norms are finite. The scalar product of two functions f and g is defined by

f, g d f (x)g(x)dx, (1) ⟨ ⟩R := d $R whenever the integral exists. Complex conjugation is unconventionally dropped in the definition of a scalar product to preserve bilinearity. For a topological vector space V on the complex field C,itsdual space V ′ is defined to be the set of all continuous linear functionals f V C.Theactionofafunctional f V ′ on a vector ϕ V : → ∈ ∈ is denoted by f,ϕ (without the subscript Rd ). Background on topological vector spaces can be found⟨ ⟩ in, for example, [23]. d The forward and inverse Fourier transforms of a function f L1(R ) are defined by ∈

2πj ξ,x f (ξ) f (ξ) f (x)e− ⟨ ⟩dx, {F } := ˆ := d $R 1 2πj ξ,x − f (x) f (ξ)e ⟨ ⟩dξ, {F } := d $R

2 d respectively, where j 1and ξ, x i 1 ξi xi .NotethattheFouriertransform =− d ⟨ ⟩ := = is not an isomorphism on L1(R ).However,byPlancherel’stheoremandbythedensity d d d " of L1(R ) L2(R ) in L2(R ),theFouriertransformcanbeextendedtoanisometry ∩ 446 J Fourier Anal Appl (2017) 23:442–461

d on L2(R ),leadingto

d f L ( d ) f L ( d ), f L2(R ). ∥ ˆ∥ 2 R =∥ ∥ 2 R ∀ ∈

d More generally, for f, g L2(R ),Parseval’srelationtakestheform ∈

1 f, g d f , − g . R d ⟨ ⟩ = F F R % & Note that, this form of Parseval’s relation is unconventional1 due to the definition of scalar product in (1). The extension of the Fourier transform to the class of generalized functions called tempered distributions will be discussed in Sect. 2.3 with a brief review of the underlying theory. Afunction f is called polynomially growing (or of polynomial growth) if there exist C > 0andn N such that ∈

n d f (x) C(1 x ) , x R . | |≤ +∥ ∥ ∀ ∈ Despite its name, a polynomially growing function is not necessarily growing but instead bounded poinwise by a polynomial. When we want to control the L2-norm of a growing or decaying function, it is necessary to consider weighted-L2 spaces. In particular, we define

d 2 2 n L2,n(R ) f f (x) (1 x ) dx < , := : d | | +∥ ∥ ∞ ' $R ( where the parameter n Z specifies the order of growth (when n 0) or decay (when ∈ ≤ d n > 0) of the function. The weighted norm of a function f L2,n(R ) is given by ∈

1 2 2 2 n d f L2,n( ) f (x) (1 x ) dx . ∥ ∥ R := d | | +∥ ∥ )$R *

d Given an n N,allfunctionsinL2, n(R ) are said to be bounded by a polynomial ∈ − of degree n in the L2 sense. It is not difficult to see that every function of polynomial d growth must belong to L2, n(R ) for some n N. − ∈ Similarly, for n Z,thespaceofweighted-ℓ2 sequences is defined by ∈

d 2 2 n ℓ2,n(Z ) c c k (1 k ) < . := ⎧ : | [ ]| +∥ ∥ ∞⎫ k d ⎨ !∈Z ⎬ ⎩ ⎭ 1 , , , Parseval’s relation is more often seen in the literature as f g L2 f g with f g L2 f g. ⟨ ⟩ = ˆ ˆ L2 ⟨ ⟩ := ¯ % & 1 JFourierAnalAppl(2017)23:442–461 447

d The weighted norm of a sequence c ℓ2,n(Z ) is given by ∈ 1 2 2 2 n c d c k (1 k ) . ∥ ∥ℓ2,n (Z ) := ⎛ | [ ]| +∥ ∥ ⎞ k d !∈Z ⎝ ⎠ 2.2 Fejér Kernel

Let T denote the interval 0, 1 .Afunction f is 1-periodic if f f ( k),forallk d d [ ] = ·+ ∈ Z .LetL1(T ) be the space of all 1-periodic functions f such that d f (x) dx < . d T | | ∞ The Fourier coefficients of a function f L1(T ) are defined by ∈ 1

2πj k,x d f k f (x)e− ⟨ ⟩dx, k Z . ˆ[ ]:= d ∈ $T 2πj k,x When studying the convergence of the Fourier series d f k e to f (x), k Z ˆ ⟨ ⟩ it is often useful to consider the (separable) d-dimensional∈ Fejér[ kernel] " (1) (1) FN (x) F (x1) F (xd ), N 1, (2) := N ··· N ≥ (1) where the 1-dimensional Fejér kernel FN is defined as the Cesàro partial sum of the complex exponentials, i.e.,

k F(1)(x) 1 | | e2πjkx. (3) N := − N k

N 1 1 − F(1)(x) D(1)(x), N := N n n 0 != (1) n 2πjkx where Dn (x) k n e is the 1-dimensional Dirichlet kernel. Putting (2) and (3)together,wewrite:= =− " 2πj k,x FN (x) FN k e ⟨ ⟩, = ˆ [ ] k

d ki FN k 1 | | , k < N. (4) ˆ [ ]= − N ∥ ∥∞ i 1 ) * #= 448 J Fourier Anal Appl (2017) 23:442–461

It is well-known [15,36]thattheFejérkernelisapositivesummabilitykernelwith properties such as d Periodicity FN (x k) FN (x), k Z . • + = d ∀ ∈ Positivity FN (x) 0, x R . • ≥ ∀ ∈ Normalization d FN (x)dx 1, N 1. • T = ∀ ≥ d Asymptotic sampling limN d g( y x)FN (x)dx g( y), y ,forall 1 T R • continuous functions g. →∞ − = ∀ ∈ 1 These properties are needed in the proof of Theorem 1.

2.3 Distribution Theory

In this section, we briefly review some important concepts of distribution theory; the standard reference for the subject is Schwartz’ treatise [24], but more accessible texts are also available [13,29,34]. The theory of distributions generalizes the classical notion of functions to objects called distributions, characterized through their actions on well-behaved test functions. To be precise, let (Rd ) denote the space of test functions that are infinitely differentiable and compactlyD supported. Test functions are d d also known as bump functions,andthespace (R ) is sometimes denoted by c∞(R ). D d d C The space of distributions is defined as the dual space ′(R ) of (R ) that consists d D D d of all continuous linear functionals on (R ).Adistribution f ′(R ) is specified by the mapping D ∈ D

d f (R ) C : D → ϕ f,ϕ , -→ ⟨ ⟩ where the scalar product notation f,ϕ (without the subscript Rd )denotestheaction of the functional f on the test function⟨ ⟩ ϕ.Aregular type of distribution can be associated to a locally integrable function f ,whichisintegrableoneverycompact subset of its domain, by defining

f,ϕ f (x)ϕ(x)dx. ⟨ ⟩ := d $R By contrast, there are also irregular distributions that cannot be described by a scalar product with some function. The most famous example of irregular distributions is the Dirac pulse δ ϕ δ,ϕ ϕ(0),whichisoften(mis)treatedasanordinary function in engineering: -→ texts.⟨ ⟩ = For the sake of extending the Fourier transform, we are only interested in the space d d ′(R ) of tempered distributions.ThisisthethedualspaceofSchwartz’ space (R ) ofS infinitely differentiable and rapidly decreasing functions. Specifically, a functionS ϕ is included in (Rd ) if it is infinitely differentiable and if S k ℓ ϕ k,ℓ sup x ∂ ϕ(x) < ∥ ∥ = d ∞ x R ∈ 6 6 6 6 JFourierAnalAppl(2017)23:442–461 449 for every pair of multi-indices k, ℓ Nd .Suchfunctionϕ is called a Schwartz function ∈ d and the quantities ϕ k,ℓ are called the Schwartz seminorms of ϕ.Thespace (R ) equipped with this family∥ ∥ of seminorms becomes a Fréchet space (complete metrizableS locally convex space). In this space, a sequence of functions ϕn converges to a function ϕ if

d ϕn ϕ k,ℓ 0asn , k, ℓ N . ∥ − ∥ → →∞ ∀ ∈ d The continuity of a tempered distribution f ′(R ) means f,ϕn f,ϕ when- d ∈ S ⟨ ⟩ → ⟨ ⟩ ever ϕn ϕ in (R ). Since→ every testS function is a Schwartz function, a tempered distribution is indeed a distribution. The advantage of using the space (Rd ) is that the Fourier transform is an S isomorphism on (Rd ),whichisnottrueon (Rd ).Withthat,wecaneasilydefine S D d the Fourier transform of a tempered distribution f ′(R ) as another tempered distribution f f given by ∈ S ˆ = F

d f ,ϕ f ,ϕ f, ϕ , ϕ (R ). ˆ = ⟨F ⟩ := ⟨ F ⟩ ∀ ∈ S % & The above definition can be thought of as Parseval’s relation for tempered distributions. d d As the Fourier transform is isomorphic on (R ), it is also isomorphic on ′(R ). S d S The inverse Fourier transform operator on ′(R ) is then given by S

1 1 d − f ,ϕ f, − ϕ , ϕ (R ). F = F ∀ ∈ S % & % & ℓ For a multi-index ℓ Nd ,the(weak)partialderivative∂ f of a tempered distribution d ∈ f ′(R ) is also a tempered distribution defined by ∈ S

ℓ ℓ ℓ d ∂ f ,ϕ ( 1)| | f,∂ ϕ , ϕ (R ). (5) := − ∀ ∈ S % & % & The derivative defined above is called weak because it is also defined for (generalized) functions that are not differentiable in the ordinary sense, otherwise, the two notions are equivalent. It is important to mention here the differentiation property of the Fourier d d transform: for f ′(R ) and ℓ N ,onehas ∈ S ∈

ℓ ℓ ℓ ∂ f (2πj)| |( ) f . (6) F = · ˆ 7 8 d In order to define the two other important operations on ′(R ),namely,multipli- S d d cation and convolution, we need to consider the two spaces M (R ) and C′ (R ). d O O In particular, a function ϕ is in M (R ) if it is infinitely differentiable and not grow- O d ing faster than some polynomial, whereas a tempered distribution f is in C′ (R ) d d d O if f M (R ).Notethat,forallψ M (R ) and ϕ (R ),wehavethat ˆ ∈ O ∈ O ∈ S ψϕ (Rd ).Thus,wecandefinethemultiplicationandconvolutionwithatempered ∈ S d distribution f ′(R ) by ∈ S 450 J Fourier Anal Appl (2017) 23:442–461

d d f ψ,ϕ f,ψϕ , ψ M (R ),ϕ (R ) ⟨ · ⟩ := ⟨ ⟩ ∀ ∈ O ∈ S d d and f g,ϕ f, g( ) ϕ , g ′ (R ),ϕ (R ), ⟨ ∗ ⟩ := ⟨ −· ∗ ⟩ ∀ ∈ OC ∈ S respectively. Importantly, these definitions retain the classical duality between multi- d plication and convolution so that, for f ′(R ), ∈ S d ( f g) f g, g ′ (R ) F ∗ = F · F ∀ ∈ OC ; d ( f ψ) f ψ, ψ M (R ). F · = F ∗ F ∀ ∈ O d In the remainder of the paper, we also need the translation operator on ′(R ),which corresponds to S

d d d f ( x0),ϕ f,ϕ( x0) , for f ′(R ),ϕ (R ), x0 R . ⟨ ·+ ⟩ = ⟨ ·− ⟩ ∈ S ∈ S ∈

3PSFforPolynomiallyGrowingFunctions

We are now ready to state the first generalization of the PSF for continuous functions of polynomial growth. Note that the periodization of the Fourier transform of the function is interpreted, in this case, as a Cesàro sum. Theorem 1 (PSF for polynomially growing functions) If f is a continuous function of polynomial growth, then

2πj k, d f (k)e− ⟨ ·⟩ lim FN ℓ f ( ℓ) in ′(R ), (7) = N ˆ [ ] ˆ ·+ S k d →∞ ℓ

2πj k, 2πj k, f (k)e− ⟨ ·⟩,ϕ f (k) e− ⟨ ·⟩,ϕ f (k)ϕ(k) = Rd = ˆ :k d ; k d k d !∈Z !∈Z % & !∈Z g(k), (8) = k d !∈Z where g f ϕ.Thesumin(8)convergesabsolutelysinceϕ (Rd ) and since := ˆ ˆ ∈ S f (k)ϕ(k) C (1 k )n ϕ(k) < , (9) | ˆ |≤ +∥ ∥ |ˆ | ∞ k d k d !∈Z !∈Z where we have used the fact that f (x) C(1 x )n,forallx Rd and for some | |≤ +∥ ∥ ∈ n N. ∈ JFourierAnalAppl(2017)23:442–461 451

On the other hand, since f is bounded by a polynomial and ϕ is rapidly decaying, the action of the RHS of (7)onϕ can be written as ˆ

lim FN ℓ f ( ℓ),ϕ lim FN ℓ f ( ℓ),ϕ N ˆ [ ] ˆ ·+ = N ˆ [ ] ˆ ·+ : →∞ ℓ

lim FN ℓ f, ϕ( ℓ) = N ˆ [ ] ⟨ F{ ·− }⟩ →∞ ℓ

2πj ℓ,x lim FN ℓ f (x)ϕ(x)e− ⟨ ⟩dx = N ˆ [ ] d ˆ →∞ ℓ

2πj ℓ,x lim g(x) FN ℓ e− ⟨ ⟩dx = N d ˆ [ ] →∞ $R ℓ

lim g( x)FN (x)dx = N d − →∞ $R

lim g(k x)FN (x)dx. (10) = N d − →∞ k d $T !∈Z

Note that, for each k Zd ,theintegralin(10)isboundedby ∈

g(k x)FN (x)dx sup g(k x) FN (x) dx sup g(k x) , d − ≤ d | − |· d | | = d | − | 6$T 6 x T $T x T 6 6 ∈ ∈ 6 6 (11) 6 6 where we have used the continuity of g and the fact that FN (x) is a positive kernel that integrates to unity on Td .Asg f ϕ decays faster than any polynomial, it must = ˆ be that the sequence sup d g(k x) is absolutely summable. Therefore, x T k Zd we can invoke Lebesgue’s dominated-convergence∈ | − | ∈ theorem to exchange the limit and sum in (10)as < =

lim FN ℓ f ( ℓ),ϕ lim g(k x)FN (x)dx N ˆ [ ] ˆ ·+ = N d − : →∞ ℓ

4PSFonWeightedSobolevSpaces

4.1 Weighted Sobolev Spaces

When sampling in weighted-L2 spaces, we need to impose some order of differen- tiability on the signals by considering weighted Sobolev spaces. To that end, we first define the (unweighted) Sobolev spaces of integer orders k N as ∈

k d d ℓ d H (R ) f ′(R ) ∂ f L2(R ), ℓ k , (13) 2 := ∈ S : ∈ ∀| |≤ 7 8 where ∂ℓ f is the ℓ-th weak derivative of f defined in (5). In words, the Sobolev space d of order k consists of functions whose weak derivatives up to order k are all in L2(R ). k The Sobolev norm of a function f H (Rd ) is given by ∈ 2

ℓ f k d ∂ f . (14) H ( ) d ∥ ∥ 2 R := L2(R ) ℓ k > > |!|≤ > > > > Meanwhile, the Sobolev spaces of fractional orders s R are defined via the Fourier transform as ∈

s d d s d L (R ) f ′(R ) J f L2(R ) , (15) 2 := ∈ S : ∈ 7 8 where the operator J s is defined as

s 1 2 s J f − (1 ) 2 f . (16) := F +∥·∥ ˆ 7 8 s The Sobolev norm of an element f L (Rd ) is given by ∈ 2 s f Ls ( d ) J f d . (17) ∥ ∥ 2 R := L2(R ) > > We want to emphasize that definition (15)isindeedanextensionofdefinition(> > 13): k d one can show by using Fourier multipliers [9,10]that,fork N,thespacesH2 (R ) k d ∈ and L2(R ) are identical and their corresponding norms given in (14)and(17)are equivalent. We now extend (15)todefinetheweightedSobolevspaces(offractionalorders)as

s d d s d L (R ) f ′(R ) J f L2,n(R ) , s R, n Z. 2,n := ∈ S : ∈ ∈ ∈ 7 8 s The weighted Sobolev norm of an element f L (Rd ) is given by ∈ 2,n

s f Ls ( d ) J f d . ∥ ∥ 2,n R := L2,n(R ) > > > > JFourierAnalAppl(2017)23:442–461 453

It is also useful to introduce the Bessel potential kernels

(s) 1 2 s/2 β − (1 )− , s R. (18) := F +∥·∥ ∈ 7 8 Informally speaking, the operator J s accumulates in some sense all the derivatives up to order s.Therefore,s is referred to as the order of differentiability and an element s d of the space L2,n(R ) is said to have s derivatives that are bounded by a polynomial s d of degree n in the L2 sense. Since any f L (R ) can be written as ∈ 2,n

f (J s f ) β(s), (19) = ∗ we can say, in the language of signal processing, that a signal lives in the weighted Sobolev space of order s if it is the result of filtering a weighted-L2 signal with the Bessel potential kernel of order s. For a deeper understanding of Sobolev spaces, interested readers are referred to [2, 9,17]; excellent resources are also available online, see, for example, [28]. The Bessel potentials were discussed thoroughly in [1,3,26]. Results regarding the sampling of growing signals in weighted Sobolev spaces can be found in [18]. Proposition 1 asserts s d that elements of the weighted Sobolev space L2,n(R ) are polynomially growing functions when the order of differentiability s exceeds the threshold d/2.

s d Proposition 1 If f is a function in the weighted Sobolev space L2, n(R ) with s > d/2 and n N,thenf ispolynomiallygrowing. − ∈ s (s) s d s Proof We first write f (J f ) β .Since f L2, n(R ),itmustbethatJ f d = ∗ ∈ − (s) d ∈ L2, n(R ).Furthermore,from[18,Proposition7],wealsoknowthatβ L2,n(R ). It then− follows from [18,Proposition1]thattheconvolution(J s f ) β(s) is a∈ continuous function that is bounded pointwise by the polynomial C(1 ∗x )n.Thus, f is a +∥ ∥ polynomially growing function. !

4.2 PSF on Weighted Sobolev Spaces

s d By Proposition 1,theweightedSobolevspaceL2, n(R ),withs > d/2, n N,is embedded in the space of polynomially growing functions.− Therefore, as shown∈ in The- d s d orem 1,PSFautomaticallyholdsin ′(R ) for every function f L2, n(R ),where S ∈ − the periodization of f converges conditionally in the sense of the Cesàro summation. ˆ In this section, we show that PSF actually holds in a stronger sense for functions in aparticularweightedSobolevspace.Specifically,PSFistrueinthenegative-order 2n d s Sobolev space L2− (R ) for every f L2, 2n,wheres > d/2andn N.With ∈ − ∈ this stricter requirement on the function f ,theperiodizationof f converges uncon- ˆ ditionally, and so there is no need to use the Cesàro summation. The equality in PSF now means that the Fourier transforms of both sides are growing functions that are equal almost everywhere. The central result of this section is stated in Theorem 4;but, before that, we need several auxiliary results: Theorem 2,whichwaspresentedin[18], 454 J Fourier Anal Appl (2017) 23:442–461 makes sure that the sampling is bounded in weighted Sobolev spaces with sufficient order of differentiability. Theorem 3 shows that the periodization in the RHS of PSF is a well-defined distribution.

Theorem 2 If f is a continuous function that is included in the weighted Sobolev s d space L ( ) with s > d/2 and n ,thenthesampledsequence f (k) d is 2,n R Z k Z d ∈ { } ∈ an element of ℓ2,n(Z ) and we have that

f ℓ ( d ) Cn,s f Ls ( d ). (20) ∥ [·]∥ 2,n Z ≤ ∥ ∥ 2,n R

Proof See [18,Theorem3]foraproofforn 0. When n > 0, the same result is ≤ obtained by using duality. !

s d Theorem 3 Let f L2, 2n(R ) with s > d/2 and n N.Then,theseries ∈ − ∈ d k d f ( k),ϕ is absolutely convergent for every test function ϕ c∞(R ) ∈Z ˆ ·+ ∈ C and we have that " % &

f ( k),ϕ Cn,s f Ls ( d ) ϕ 2n d . (21) ˆ 2, 2n R L2 (R ) ·+ ≤ ∥ ∥ − ∥ ∥ k Zd 6% &6 !∈ 6 6 6 6 Proof See Appendix. !

Theorem 4 (PSF on weighted Sobolev spaces) If f is a continuous function in the s d weighted Sobolev space L2, 2n(R ) with s > d/2 and n N,then − ∈

2πj k, 2n d f (k)e− ⟨ ·⟩ f ( ℓ) in L− (R ). (22) = ˆ ·+ 2 k d ℓ d !∈Z !∈Z

2n d Proof We first recall the well-known result [5,Corollary6.2.8]thatL2− (R ) 2n d = L2 (R ) ′.Therefore,itsufficestoshowthatthetwosidesof(22) are identical in 2n d the dual space of L2 (R ). ? @ d Let us fix a test function ϕ ∞(R ).As f is polynomially growing (according ∈ Cc to Proposition 1)andϕ (Rd ),then,similarlytotheproofofTheorem1,theaction of the LHS of (22)onϕ∈isS well defined by

2πj k, 2πj k, f (k)e− ⟨ ·⟩,ϕ f (k) e− ⟨ ·⟩,ϕ g(k), (23) = Rd = :k d ; k d k d !∈Z !∈Z % & !∈Z JFourierAnalAppl(2017)23:442–461 455 where g f ϕ is a continuous rapidly decaying function. Furthermore, from Theo- rem 2 and:= by theˆ Cauchy–Schwarz inequality, we have the bound

2πj k, f (k)e− ⟨ ·⟩,ϕ f (k)ϕ(k) 6 6 ≤ ˆ 6:k d ;6 k d 6 !∈Z 6 !∈Z 6 6 6 6 f 6 d 6 ϕ d 6 6 ℓ2, 2n(Z ) ℓ2,2n(Z ) 6 6 ≤∥ [·]∥ − ∥ˆ[·]∥ Cn,s f s d ϕ s d L2, 2n (R ) L2,2n (R ) ≤ ∥ ∥ − ∥ˆ∥ Cn,s f s d ϕ 2n d . (24) L2, 2n (R ) L2 (R ) ≤ ∥ ∥ − ∥ ∥ On the other hand, the action of the RHS of (22)onϕ is now interpreted as

f ( ℓ),ϕ f ( ℓ),ϕ , (25) ˆ ·+ = ˆ ·+ :ℓ d ; ℓ d !∈Z !∈Z % & which is finite from Theorem 3. Let us consider the periodization h(x) d g(x k),whoseFouriercoeffi- k Z cients are given by := ∈ + "

2πj ℓ,x h ℓ g(x k)e− ⟨ ⟩dx ˆ[ ]= d + $T k d !∈Z 2πj ℓ,x k g(x k)e− ⟨ + ⟩dx = d + k d $T !∈Z 2πj ℓ,x g(x)e− ⟨ ⟩dx, = d $R where the exchange of integral and sum is justified by the dominated convergence d theorem and by the fact that g L1(R ).Withthesamemanipulationasintheproof of Theorem 1,weobtain ∈

h ℓ f ( ℓ),ϕ (26) ˆ[ ]= ˆ ·+ % & which, together with Theorem 3,establishesthattheFouriercoefficientsofh are absolutely summable. We can also show that h(x) is continuous at 0.Indeed,asg is rapidly decaying, for any ε>0, there exists N N such that ∈ ε g(x k) < , x < 1. | + | 3 ∀∥ ∥ k >N ∥ !∥

Moreover, since g is continuous, the finite sum k N g( k) is continuous. Thus, there exists δ>0suchthat ∥ ∥≤ ·+ " 456 J Fourier Anal Appl (2017) 23:442–461

ε (g(x k) g(k)) < , x <δ. 6 + − 6 3 ∀∥ ∥ 6 k N 6 6∥ !∥≤ 6 6 6 6 6 Hence, for every x6 < min(1,δ),wehavethat6 ∥ ∥

(g(x k) g(k)) (g(x k) g(k)) g(x k) 6 + − 6 ≤ 6 + − 6 + | + | 6k d 6 6 k N 6 k >N 6 !∈Z 6 6∥ !∥≤ 6 ∥ !∥ 6 6 6 6 6 6 6 g(k) <ε, 6 6 6 +6 | | 6 k >N ∥ !∥ which implies the continuity of h at 0.Therefore,theFourierseriesofh at x 0 converges to =

h(0) h ℓ . = ˆ[ ] ℓ d !∈Z Combining this with (23), (25), (26)yields

2πj k, d f (k)e− ⟨ ·⟩,ϕ f ( ℓ),ϕ , ϕ ∞(R ). (27) = ˆ ·+ ∀ ∈ Cc :k d ; :ℓ d ; !∈Z !∈Z 2n d The final step of the proof is to extend the validity of (27)toallϕ L2 (R ). From bounds (21)and(24)andbytheHahn-Banachtheorem[23,Theorem3.3],each∈ 2n d side of (22)canbeextendedtoacontinuouslinearfunctionalonL2 (R ).(Notethat, for linear operators between normed spaces, boundedness is equivalent to continuity.) d 2n d We also know that c∞(R ) is a dense subspace of L2 (R ),asaconsequenceofthe Meyers-Serrin theoremC [17,Theorem10.15].Hence,thecontinuousextensionson 2n d both sides must be identical on the whole space L2 (R ),thuscompletingtheproof. !

5Discussion

We have derived two generalizations of the Poisson summation formula, one for con- tinuous functions of polynomial growth and the other for functions in some weighted s d Sobolev space L2, 2n(R ) with s > d/2andn N.Inthefirstgeneralization,with the minimal assumption− on the continuity and the∈ polynomial growth of the function, d the PSF identity holds in the space of tempered distributions ′(R ),wheretheperi- odization on the RHS of the formula converges only conditionallyS in the sense of the Cesàro summation. The second generalization gives a stronger sense of equality when s d the function is restricted to the space L2, 2n(R ) which is embedded in the space of polynomially growing functions provided− that s > d/2. With this assumption, the 2n d PSF identity holds in the negative-order Sobolev space L2− (R ),whichisasubspace JFourierAnalAppl(2017)23:442–461 457

d of ′(R ).Furthermore,bothsidesoftheformulaconvergeunconditionallywithout usingS any summability methods. We conclude the paper by demonstrating the application of the two generalized PSFs to some simple growing signals. Let us apply the generalized PSFs to the one- sided power functions xα mentioned in the introduction section, for some particular values of α>0. Recall [+30]thatthe(distributional)Fouriertransformofxα is given by +

+(α 1) + , if α>0,α/ α (2πjξ)α 1 N x (ξ) + n ∈ F{ } = ⎧ n 1 j (n) + ! n 1 2 2π δ (ξ), if α n N, ⎨ (2πjξ) + + = ∈ A B ⎩ where + is Euler’s gamma function and δ(n) is the nth derivative of the Dirac pulse. Obviously, xα is continuous and polynomially growing for all α>0. In particular, when α 1/+2, the first generalized PSF yields =

2πjkξ ℓ √π/2 √k e− lim 1 | | in ′(R). (28) = N − N (2πj(ξ ℓ))3/2 S k →∞ ℓ

2πjkξ j 1 1 2 ke− δ′(ξ ℓ) in L− (R). (29) = 4π + − 4π 2 (ξ ℓ)2 2 k ℓ ℓ !∈N !∈Z !∈Z + Although the validity of formulas (28)and(29)areguaranteedbyTheorems1 and 4, respectively, their interpretations need further investigation.

Acknowledgements The authors thank Prof. Hans Georg Feichtinger for his interest in the topic and for helpful discussions, especially for bringing to our attention the references [34,35]. We also thank the anonymous reviewers for their constructive comments that greatly improved the paper. This research was funded by the Swiss National Science Foundation under Grant 200020-162343.

Appendix: Proof of Theorem 3

Before starting the proof of Theorem 3,weneedatechnicallemma.

d Lemma 1 For any r R and k N ,thereexistsaconstantCr,k depending on r and k such that ∈ ∈

k 2 r 2 r d ∂ (1 ξ ) Cr,k (1 ξ ) , ξ R . +∥ ∥ ≤ +∥ ∥ ∀ ∈ 6 6 6 6 6 6 458 J Fourier Anal Appl (2017) 23:442–461

Proof of Lemma 1 It suffices to show, by induction, that

k k 2 r | | 2 r n d ∂ (1 ξ ) Pn(ξ)(1 ξ ) − , k N , (30) +∥ ∥ = +∥ ∥ ∀ ∈ n 0 !=

ℓ 2 n d where each term Pn(ξ) satisfies ∂ Pn(ξ) Cr,ℓ(1 ξ ) , ℓ N . Obviously, | |≤ +∥ ∥ ∀ ∈ the claim is true for k 0 by choosing P0(ξ) 1. Suppose (30)istrueforsome d = = k N ,weshallshowthatitisalsotruefork′ (k1,...,ki 1,...,kd ) for arbitrary i.Indeed,fromtheinductionhypothesisandbyLeibniz’rule,wehavethat∈ = +

k k 2 r ∂ | | 2 r n ∂ ′ (1 ξ ) Pn(ξ)(1 ξ ) − +∥ ∥ = ∂ξ +∥ ∥ i n 0 != k | | ∂ 2 r n 2 r n 1 Pn(ξ)(1 ξ ) − 2(r n)ξi Pn(ξ)(1 ξ ) − − = ∂ξ +∥ ∥ + − +∥ ∥ n 0 ) i * != k′ | | 2 r n Qn(ξ)(1 ξ ) − , = +∥ ∥ n 0 != where

∂ P (ξ), n 0 ∂ξi n ∂ = Qn(ξ) Pn(ξ) 2(r n 1)ξi Pn 1(ξ), n 1,..., k := ⎧ ∂ξi + − + − = | | ⎨⎪2(r n 1)ξi Pn 1(ξ) n k 1. − + − =| |+ ⎪ ⎩ 2 From the induction hypothesis and from the simple inequality 2 ξi 1 ξ ,it | |≤ ℓ+∥ ∥ is easy to check that, for each n k′ , Qn(ξ) satisfies the property ∂ Qn(ξ) 2 n d ≤| | | |≤ Cr,ℓ(1 ξ ) , ℓ N .ThiscompletestheproofofLemma1. +∥ ∥ ∀ ∈ ! Proof of Theorem 3 Let w (1 2)n be the weighting function. Let B denote := +d∥·∥ the support of ϕ.Foreachk Z ,wedefineϕk ϕ( k) whose support is ∈ := ·− denoted by Bk B k.ItisnothardtoseethatthereexistsaconstantCs such 2 :=s/2 + s that (1 ξ )− Cs (1 k )− ,forallξ Bk.Also,letusputϕk,s +2∥ ∥s/2 ≤ +∥ ∥ ∈ := (1 ) ϕk.Withthesenotations,wenowperformthemanipulation +∥·∥ −

f ( k),ϕ f ,ϕk ˆ ·+ = ˆ k d k d !∈Z 6% &6 !∈Z 6% &6 6 6 6 6 2 s 2 s 6 6 6 (1 6 ) 2 f ,(1 )− 2 ϕk = +∥·∥ ˆ +∥·∥ k d !∈Z 6% &6 6 1 2 s 6 6 − (1 ) 2 f , ϕk,s 6 (31) = F +∥·∥ ˆ F k Zd 6% 7 8 &6 !∈ 6 < = 6 6 6 JFourierAnalAppl(2017)23:442–461 459

s J f , ϕk,s = F{ } k d !∈Z 6D E6 6 s 16 1 (J f )w− , − w ϕk,s , (32) = F F { F{ }} Rd k Zd 6% & 6 !∈ 6 < = 6 6 6 where (31)and(32)arebothduetoParseval’srelation.Asw(x) (1 x 2)n n 2ℓ = +∥ ∥ = ℓ n ℓ! x ,itfollowsfromthedifferentiationpropertyoftheFouriertransform mentioned| |≤ ! in (6)that "

1 n 1 2ℓ − w ϕk,s (ξ) ! ∂ ϕk,s(ξ). (33) F { F{ }} = ℓ (2πj)2 ℓ ℓ n | | |!|≤ !

1 This implies that the support of − w ϕk,s is the same as that of ϕk,s.Thus,we F {d F{ }} can replace the integration domain R in the RHS of (32)byBk.Ontheotherhand, by Leibniz’ rule, for every multi-index ℓ,wehavethat

ℓ ℓ 2 s/2 ∂ ϕk,s(ξ) ∂ (1 ξ )− ϕk(ξ) = +∥ ∥ A ℓ m 2B s/2 n ! ∂ (1 ξ )− ∂ ϕk(ξ). = m n +∥ ∥ m n ℓ !+ = ! ! From this and Lemma 1,weobtain

ℓ 2 s/2 ℓ n d ∂ ϕk,s(ξ) Cℓ,s (1 ξ )− ! ∂ ϕk(ξ) , ℓ N . ≤ +∥ ∥ m n ∀ ∈ 6 6 m n ℓ ! ! 6 6 !+ = 6 6 6 6 6 6 Plugging this inequality into (33)yields

1 2 s/2 ℓ d − w ϕk,s (ξ) Cn,s (1 ξ )− ∂ ϕk(ξ) , ξ R . (34) F { F{ }} ≤ +∥ ∥ ∀ ∈ 6 6 ℓ 2n 6 6 6 6 | !|≤ 6 6 6 6 6 6 We continue to estimate the RHS of (32)as

s 1 1 f ( k),ϕ (J f )w− , − w ϕk,s ˆ ·+ = F F F{ } Bk k d k d 6 6 !∈Z 6% &6 !∈Z 6% < = < =& 6 6 s 6 1 6 1 6 6 (J f6)w− 6 − w ϕk,s 6 (35) ≤ F L2(Bk) F F{ } L2(Bk) k Zd > > > > !∈ > < => > < => > > > > s 1 s ℓ (J f )w− Cn,s(1 k )− ∂ ϕk (36) ≤ F L2(Bk) +∥ ∥ > > k d > ℓ 2n > !∈Z > < => >| !|≤ 6 6>L2(Bk) > > > 6 6> > ℓ > s 1 > 6 6> s Cn,s ∂ ϕ (J f )w− > (1 k >)− (37) ≤ L2(B) F L2(Bk) +∥ ∥ ℓ 2n > > k Zd > > | !|≤ > > !∈ > < => > > > > 460 J Fourier Anal Appl (2017) 23:442–461

1/2 C ∂ℓϕ (1 k ) 2s n,s d − ≤ L2(R ) ⎛ +∥ ∥ ⎞ ℓ 2n > > k Zd | !|≤ > > !∈ > > ⎝ 1/2 ⎠ 2 s 1 (J f )w− (38) × ⎛ F L2(Bk)⎞ k Zd > > !∈ > < => ⎝ > > ⎠ 1/2 2 C ∂ℓϕ (J s f )w 1 (39) n,s d − ≤ L2(R ) ⎛ F L2(Bk)⎞ ℓ 2n k d | !|≤ > > !∈Z > < => > > ⎝ > > ⎠ C >∂ℓϕ> (J>s f )w 1 > (40) n,s d − d ≤ L2(R ) F L2(R ) ℓ 2n > > > > | !|≤ > > > < => C (J s f>)w >1 > ∂ℓϕ > (41) n,s − d d = L2(R ) L2(R ) > > ℓ 2n > > > > | !|≤ > > Cn,s > f s d> ϕ 2n d > > (42) L2, 2n(R ) L2 (R ) = ∥ ∥ − ∥ ∥ where (35)isaconsequenceoftheCauchy–Schwarzinequality;(36) follows from (34) 2 s/2 s and from the bound (1 ξ )− Cs (1 k )− over Bk;(37)wasobtainedbya change of variable and+∥ by∥ Minkowski’s≤ inequality;(+∥ ∥ 38)followsfromtheCauchy– Schwarz inequality for sequences; (39)isbecause2s > d;(40)isduetothe compactness of the support B;(41)followsfromParseval’srelation;and,finally,the desired bound (42)wasobtainedbythedefinitionsofSobolevnormsin(14)and(17) d d and by the equivalence between H 2n (R ) and L2n (R ). ∥·∥ 2 ∥·∥ 2 ⊓2

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