Signal Recovery from Time and Frequency Samples

PREPRINT 1 Signal Reco v ery from T ime and Frequenc y Samples Mert Kayaalp and Oleg Szehr Abstract —W e analyze signal reco very when samples are taken concomitantly from a signal and its F ourier transf orm. This two-sided sampling framework extends classical one-sided re- construction and is particularly useful when measurements in either domain alone are insufficient because of sensing, storage, or bandwidth constraints. W e formulate the r esulting recovery problem in finite-dimensional spaces and repr oducing k ernel Hilbert spaces, and illustrate the infinite-dimensional setting in a Fourier -symmetric Sobolev space. Numerical experiments with sinc- and Hermite-based schemes indicate that, under a fixed sampling budget, two-sided sampling often yields better conditioned systems than one-sided appr oaches. A simplified spectrum-monitoring example further demonstrates impr oved reconstruction when limited time samples are supplemented with frequency-domain information. Index T erms —signal recov ery , two-sided sampling, analog-to- digital con version, non-bandlimited signals, spectrum monitoring I . I N T RO D U C T I O N S AMPLING and reconstruction are central to analog-to- digital con version, interpolation, and resampling in signal processing. The classical Whittaker-Shannon-Nyquist theorem [1] guarantees unique and stable recovery of bandlimited signals from sufficiently dense time-domain samples [2], [3]. Howe v er , this framew ork - and most of its extensions - rely on “one-sided” measurements, using either time or frequenc y samples alone. Recent mathematical advances have established a theory for function interpolation from the values of the function and its Fourier transform [4]–[7]. This work adopts an engineering perspectiv e, and adds applications and implementable two- sided sampling schemes that leverage simultaneous time- and frequency-domain samples for practical signal recovery . In practice, two-sided sampling is most useful when neither domain alone pro vides sufficient resolution for unique recov- ery , e.g. due to sensing, storage, or bandwidth constraints. Applications include spectrum monitoring, radar , wireless communication, and MRI (see Sec. I-A). Here, limited time- domain data can be supplemented with frequency-domain measurements - either direct Fourier samples, coarse spectral summaries, or a few targeted frequenc y observations - yielding additional information for reconstruction. This moti vates the central question of our work: How can samples fr om both time and fr equency domains be levera ged for practical signal recovery? W e provide a mathematical framework for reconstruction (see Secs. III and IV) and numerical e vidence that reconstruc- The authors are affiliated with the Dalle Molle Institute for Artificial Intelligence (IDSIA) - SUPSI/USI, V ia la Santa 1, 6962 Lugano-V iganello, Switzerland. Emails: { mert.kayaalp, oleg.szehr } @idsia.ch. tions with two-sided data impro ves identifiability (see Secs. V and VI). A. Motivation Joint use of time and frequency-domain samples can im- prov e e xisting sampling architectures and help ov ercome im- portant practical limitations. Spectrum monitoring & memory limitations: In many ap- plications the full set of time-domain samples cannot be stored due to memory constraints. This situation is common in spectrum monitoring, where storing all digitized time samples can be prohibiti ve. Thus systems retain only limited buf fered time samples after conv ersion to spectral information [8]–[10]. W e provide a simplified numerical example for this in Sec. VI, where supplementing the buf fer with frequency-domain in- formation improves the reconstruction of the underlying time signal compared to using stored time samples only . MRI & Prior information: In some applications, measure- ments are acquired in one domain, while samples in the other domain are a v ailable as a prior information. For example, in magnetic r esonance imaging (MRI) , the acquired measure- ments are inherently obtained in the Fourier domain ( k -space). At the same time, in some settings, a subset of the spatial- domain samples of the target image may be kno wn in adv ance (e.g., kno wn background pixels, prior anatomical structure, fixed from trusted auxiliary/reference data) [11]–[13]. Such spatial information can be combined with the acquired k -space data to improve the reconstruction. Communications & pr e-designed signal supports: In many applications, the signals’ supports follo w a regular structure. For instance, in r adar or communication systems , transmitted signals often lie on a pre-designed time-frequency grid. In multicarrier systems such as orthogonal frequency-di vision multiplexing ( OFDM ) [14], the signals are supported on specific grids defined by global communication standards such as 4G L TE and 5G NR [15]. Since these grids are known, receiv ers can directly target the corresponding frequencies and collect a small number of frequency samples there through a parallel analog front-end path, which can improv e channel estimation. This can become ev en more valuable in scenarios where frequency supports may shift slightly due to Doppler effects in high-mobility settings or due to har dwar e imperfec- tions [16]. B. Backgr ound & Related Literatur e The reconstruction of a function from its v alues on a discrete set is a fundamental problem in analysis. In classical sampling theory , the Whittaker–Shannon-Nyquist theorem asserts that band-limited functions are uniquely determined by samples taken at a rate beyond the Nyquist threshold. Beyond the PREPRINT 2 band-limited setting, this role is assumed by the notion of a uniqueness set : a set Λ ⊂ R such that any function in a given space X vanishing on Λ must vanish identically . The characterization of uniqueness sets is closely tied to the geometry of X and to quantitative density conditions on Λ [3]. More recently , in the context of Fourier interpolation and crystalline measures (see [4], [6], [7], [17] and related works), the notion of a uniqueness pair - a pair (Λ , M ) consisting of a spatial set Λ and a frequency set M - has been introduced. Definition 1 (Uniqueness pair , e.g. [6]) . Let X be a space of functions on R . A pair of sets (Λ , M ) , Λ , M ⊂ R is a uniqueness pair for X if for f ∈ X the conditions f | Λ = 0 and ˆ f | M = 0 imply that f = 0 . By linearity , (Λ , M ) is a uniqueness pair for X if prescrib- ing f | Λ and ˆ f | M determines f ∈ X uniquely . Such uniqueness typically requires the sampling densities of Λ and M to exceed appropriate critical thresholds. T o date, contributions in this area ha ve been purely math- ematical in nature. The pioneering work or Radchenko- V iazovska [4] introduces mixed Fourier interpolation via mod- ular forms. Subsequent studies [6], [17] determine density conditions for such schemes. They show , in particular , that the construction of [4] lies at a critical threshold consistent with the 2 W T /π -barrier of Landau-Pollak-Slepian [18]–[20], see [5], and also the extremal density condition of [6]. The work [7] extends the uniqueness theory to transforms beyond the Fourier setting. These mathematical works stop short of offering implementable reconstruction 1 algorithms or recipes accessible to signal processing practice. From an applied perspective, man y systems possess samples in both the time (or spatial) and frequency domains. Y et standard reconstruction pipelines typically treat these data separately , and ne glect the intrinsic coupling between a func- tion and its Fourier transform. Related in spirit are Papoulis- Gerchberg-type iterativ e schemes [21]–[24], which enforce function space prior constraints such as bandlimitedness. They can be interpreted as alternating projections between measure- ment consistenc y in one domain and structural constraints in the transform domain [25], [26]. In contrast, we develop a framew ork for reconstruction from concomitant spatial and Fourier samples that directly exploits the joint information contained in both domains. This two-sided approach has several consequences. First, it enables unique recov ery beyond the bandlimited setting. Sec- ond, when hardware constraints preclude sampling or storing at the Nyquist rate, additional Fourier measurements can sig- nificantly enhance reconstruction quality . Third, our numerical experiments indicate that, in finite-dimensional settings, the resulting two-sided linear systems are often better conditioned than their one-sided counterparts under equal sample budgets. C. Notation W e adopt the unitary con v ention for the Fourier transform, denoting time domain variables by t and frequency domain 1 The proposed interpolation formulas require sampling densities that in- crease with time and frequency making a direct translation into practical recovery schemes difficult. variables by ω , ˆ f ( ω ) = F [ f ]( ω ) = 1 √ 2 π Z R f ( t ) e − iω t d t, with f ( t ) = F ∗ [ ˆ f ( t )] for all suitable f . W e write X for a vector space of functions on R . In our exposition, we will focus on two guiding examples of function spaces. The first is the classical Pale y-W iener space of bandlimited functions, P W Ω ( R ) = n f ∈ L 2 ( R ) : ˆ f ( ω ) = 0 for | ω | > Ω o . The second example is the Fourier-symmetric Sobolev space (aka. modulation space), which is defined as [27] (see also [6], [7], [28]) H = n f ∈ L 2 ( R ) : Z R t 2 | f ( t ) | 2 dt + Z R ω 2 | ˆ f ( ω ) | 2 dω < ∞ o , where the first term represents the time-domain regularity and the second term reflects the frequency-domain regularity . Due to the symmetry of these terms, the space is in variant under the Fourier transform and therefore constitutes a natural framew ork for the study of uniqueness pairs. Furthermore, it admits an orthogonal basis of Hermite functions, which are eigenfunctions of the F ourier transform, see belo w . W e will treat these spaces as spanned by orthogonal bases or as Reproducing Kernel Hilbert Spaces (RKHS). Recall that an RKHS X of functions on a set X is defined by the requirement that the point ev aluation functional f 7→ f ( x ) is continuous. By the Riesz representation theorem, this is equiv alent to the existence of a unique function K x ∈ X with the reproducing property f ( x ) = ⟨ f , K x ⟩ X , and it induces a positiv e semidefinite reproducing kernel K : X × X → C , K ( x, y ) := ⟨ K y , K x ⟩ X , see [29] and [30] for applications to sampling. I I . C O N T R I B U T I O N S • F ormalization of two-sided time-frequency sampling. W e formulate reconstruction from measurements consisting of samples of both f in time and samples of ˆ f in frequency domains. W e describe basis sampling with expansions of f in suitable reconstruction families { Φ n } . For reconstruction, this yields a stacked linear system whose blocks enforce constraints in both domains as a coefficient vector solving problem. • T wo-sided sampling in finite-dimensional spaces. W e in- vestigate two-sided sampling in function spaces spanned by a finite family of N + 1 basis functions. In this finite- dimensional setting, unique signal recovery is equiv alent to the e xistence of a unique solution in a ( N + 1) × ( N + 1) linear system whose entries consist of ev aluations of the basis functions and their Fourier transforms at the prescribed time and frequency samples. Representativ e bases arising in signal processing include shifted sinc functions, Hermite functions, and prolate spheroidal wav e functions. W e demonstrate the resulting reco very schemes through numerical experiments in this framework. • T wo-sided sampling in r epr oducing kernel Hilbert spaces. W e introduce reconstruction from two-sided samples PREPRINT 3 within an RKHS framew ork and deriv e an explicit kernel- based representation for mixed constraints of f and ˆ f on sampling sets Λ and M , respecti vely . • T wo-sided sampling in F ourier-symmetric Sobolev space. As an illustrativ e example in an infinite-dimensional setting, we consider sampling in the Fourier-symmetric Sobolev space H , which has been identified as a suitable object for the study of two-sided interpolation in the mathematical literature. The space H naturally arises in contexts such as frequency modulation [27], and, unlike the Paley-W iener space, it allows for the study of settings that go beyond the band-limitation constraint. • Numerical comparison between one- and two-sided sam- pling: W e perform numerical experiments to compare reconstruction from one-sided and two-sided samples un- der a fixed sampling budget. Specifically , we analyze the condition numbers of the resulting finite-dimensional lin- ear systems arising in sinc- and Hermite-based sampling schemes. Our results show that two-sided systems typi- cally exhibit smaller condition numbers, which indicates reduced sensiti vity to noise, improv ed numerical stability of the recovery algorithms, and enhanced robustness of reconstruction. • Spectrum monitoring: As a practical case study , we ev aluate a simplified spectrum monitoring application in Sec. VI, where the system is constrained in the number of time-domain samples it can store. W e demonstrate that reconstruction performance impro ves when the a vailable time-domain samples are supplemented with measure- ments from monitored frequenc y bins. I I I . R E C O N S T R U CT I O N F R O M T W O - S I D E D S A M P L E S The classical Whittaker-Shannon-Nyquist sampling theorem asserts that if a function f is bandlimited to | ω | ≤ π /T (i.e. f ∈ P W π /T ) then it admits an expansion by taking samples at a sampling step of T , f ( t ) = X n ∈ Z f ( nT ) sinc  t − nT T  , sinc( x ) = sin( π x ) π x . This formula can be vie wed simultaneously as a prototypical instance of sampling in an orthonormal basis (the “basis- sampling” picture) and as sampling by point ev aluations in a reproducing kernel Hilbert space (the “RKHS” picture) [30]. In the former picture f is expanded as f ( t ) = X n ∈ Z α n Φ n ( t ) , (1) with Φ n ( t ) = r 1 T sinc  t − nT T  . Direct computation rev eals that Φ n are orthonormal ⟨ Φ n , Φ m ⟩ = Z ∞ −∞ Φ n ( t ) Φ m ( t ) dt = δ nm , with α n = ⟨ f , Φ n ⟩ = √ T f ( nT ) . In the RKHS picture f is expanded in reproducing kernels ev aluated at sampling points f ( t ) = X n ∈ Z α n K t n ( t ) (2) with K ( x, y ) = K y ( x ) = sin( π /T ( x − y )) π ( x − y ) = 1 T sinc  x − y T  and t n = nT . The expansion formulas (1) and (2) con- stitute fundamental components of sampling theory and find widespread application beyond the classical bandlimited framew ork. It is worth noting that in such settings the corre- sponding kernels need not be orthogonal. If data or hardware constraints preclude sampling at a sufficiently high rate, aliasing occurs and unique signal re- construction is no longer possible. In such cases, it is natural to attempt for incorporating additional information, such as measurements obtained in the Fourier domain. W e begin by illustrating reconstruction from two-sided sampling in a finite- dimensional setting. T o this end, we introduce a sample budget (which roughly corresponds to a maximal sampling frequency) and study signal recovery subject to this constraint in both basis sampling and RKHS sampling scenarios. A. F inite basis sampling with two-sided samples Assuming the signal lies in a finite v ector space X N = span { Φ 0 , ...., Φ N } , basis sampling represents a signal by expanding it in a given family of reconstruction functions { Φ n } N n =0 , with coefficients { α n } N n =0 , cf (1). The coefficients { α n } N n =0 serve as the discrete signal representations for digital processing. W e may also expand the Fourier transform in the cor- responding Fourier basis. In particular, taking the Fourier transform (and assuming suf ficient regularity) of (1) yields ˆ f ( ω ) = N X n =0 α n ˆ Φ n ( ω ) . If K samples can be obtained from f ( t ) and L samples can be obtained from ˆ f ( ω ) this yields K linear equations that constrain { α n } N n =0 through time-domain information and L linear equations for the frequenc y-domain. Combining the tw o, one can attempt to reconstruct f by solving a stacked linear system. Specifically , writing  c for the measurements obtained from f ( t ) and  ˆ c for the measurements obtained from ˆ f ( ω ) this yields the block system  c  ˆ c ! = (Φ ij )  ˆ Φ ij  !   α  . (3) Here we write (Φ ij ) for the matrix with entries Φ ij = Φ j ( t i ) , 0 ≤ j ≤ N , 0 ≤ i ≤ K − 1 and  ˆ Φ ij  for the matrix with entries ˆ Φ ij = ˆ Φ j ( ω i ) , 0 ≤ j ≤ N , 0 ≤ i ≤ L − 1 and  α for the coefficient vector of { α n } N n =0 . The solution set may be empty (no solution, i.e., inconsistency), a singleton (exactly one solution), or non-unique (infinitely many solutions), depending on the system’ s rank conditions and consistency . Assuming f ∈ X N and exact measurements guarantees the existence of a solution, but uniqueness necessitates K + L ≥ N + 1 and is not guaranteed in general. Systems of the form (3) can be treated uniformly via the Moore–Penrose pseudoinv erse A † . For PREPRINT 4 Ax = b , the pseudoin verse yields x ⋆ = A † b , which is a least- squares solution when the system is overdetermined (minimiz- ing ∥ Ax − b ∥ 2 ); when multiple least-squares solutions exist, it selects the minimum-norm one. In the underdetermined case, it returns the minimum-norm solution among all exact solutions when b ∈ range( A ) , and otherwise the minimum-norm least- squares solution. The pseudoin verse arises most transparently from the singular value decomposition: if A = U Σ V † (with singular values σ i on the diagonal of Σ ), then A † = V Σ † U † , Σ † = diag  σ − 1 i | σ i > 0  , with zeros left in place of any vanishing singular values. Thus the singular values determine both existence/uniqueness through rank( A ) and numerical behavior: small σ i correspond to weakly constrained directions and lead to ill-conditioned behavior in the sense of amplification in A † b . W e study conditioning experimentally in Sec. V. The rigorous study of uniqueness quickly leads into deep mathematics. A polynomial of degree N is uniquely deter- mined by its values at N + 1 distinct nodes. Consequently , if X N is (essentially) a polynomial space, any non-degenerate measurements yield a (one-sided) uniqueness set. Y et the corresponding two-sided linear system may admit multiple solutions. Thus, e ven in finite dimension and in the polynomial setting, a uniqueness set for pointwise interpolation does not automatically induce a unique solution for the stacked system. W e illustrate this point with the following example. 1) The finite Hermite function space: The Hermite func- tions are defined by the relation φ n ( x ) = 1 √ n !  1 √ 2 ( x − ∂ x )  n φ 0 ( x ) , n ≥ 1 , with φ 0 = π − 1 / 4 e − x 2 / 2 . It is obvious that e very Hermite function has the form φ n ( x ) = 1 π 1 / 4 √ 2 n n ! H n ( x ) e − x 2 / 2 with a degree- n polynomial H n . These polynomials are the Hermite polynomials. W e consider the ( N + 1) -dimensional Hermite function space H N = span { φ 0 , . . . , φ N } , where any f ∈ H N admits the unique e xpansion f ( t ) = N X n =0 α n φ n ( t ) . The Hermite functions are eigenfunctions of the Fourier trans- form F [ φ n ]( w ) = ( − i ) n φ n ( ω ) , n ≥ 0 . Therefore for f ∈ H N , it holds that ˆ f ( ω ) = N X n =0 α n ( − i ) n φ n ( ω ) . a) One-sided sampling: Pure time or frequency sampling at N + 1 distinct nodes allo ws for unique reconstruction. If measurements are taken from either time or frequency domain, then the system of equations (3) only contains Φ i,j = φ j ( t i ) or ˆ Φ i,j = ( − i ) j φ j ( ω i ) . Making use of the Hermite expansion, any f ∈ H N can be written in the form f ( x ) = e − x 2 / 2 p ( x ) with a polynomial p of degree N . Prescribing N + 1 zeros f ( t j ) = 0 at distinct points in time implies that the degree N polynomial p has N + 1 distinct zeros p ( t j ) = 0 . Thus p = 0 and f = 0 . In other words any distinct points { t 0 < t 1 < .... < t N } ⊂ R constitute a uniqueness set for H N . Along the lines of Proposition 1, a unique signal interpolates N + 1 distinct sampling points. The same reasoning applies also for one-sided frequency domain measurements. b) T wo-sided sampling: The situation is more delicate in the case of two-sided sampling. Choosing K distinct measure- ments from the time domain and L distinct measurements from the frequency domain with K + L = N + 1 is not guaranteed to yield a unique solution. W e illustrate this with a simple example: Example 1. Consider the space H 2 , where N + 1 = 3 measur ements ar e taken in total of whic h K = 1 measur ements ar e taken fr om the time domain and L = 2 fr om the fr equency domain: t 0 = 0 , ω 0 = 1 , ω 1 = − 1 . In this space, consider the function: f = 1 √ 2 φ 0 + φ 2 ∈ H 2 with ˆ φ 0 = φ 0 and ˆ φ 2 = ( − i ) 2 φ 2 = − φ 2 , so that ˆ f = 1 √ 2 φ 0 − φ 2 ∈ H 2 . W e have f ( t 0 = 0) = 1 √ 2 φ 0 (0) + φ 2 (0) = 1 √ 2 ( π − 1 / 4 − π − 1 / 4 ) = 0 . Since φ 0 and φ 2 ar e even and φ 0 ( ± 1) = π − 1 / 4 e − 1 / 2 , φ 2 ( ± 1) = 1 √ 2 π − 1 / 4 e − 1 / 2 we find ˆ f ( ω 0 / 1 = ± 1) = 1 √ 2 φ 0 ( ± 1) − φ 2 ( ± 1) = 0 . Thus, even though f ≡ 0 , it satisfies f ( t 0 ) = 0 , ˆ f ( ω 0 ) = 0 , ˆ f ( ω 1 ) = 0 . In other words Λ = { 0 } , M = {− 1 , 1 } is not a uniqueness pair for H 2 . Continuing in the space H 2 , more broadly , if A denotes the matrix of the 3 × 3 stacked linear system for one measurement in the time and two measurements in the frequency domain (cf. (3)), a straightforward computation re veals det( A ) = C · ( ω 0 − ω 1 )  t 2 0 − ω 0 ω 1 − 1 + i t 0 ( ω 0 + ω 1 )  , for some C = C ( t 0 , ω 0 , ω 1 )  = 0 . Thus, if ω 0  = ω 1 then det( A ) = 0 ⇐ ⇒ t 2 0 − ω 0 ω 1 − 1 + i t 0 ( ω 0 + ω 1 ) = 0 . For real t 0 , ω 0 , ω 1 , this complex equation is equi v alent to the two real equations t 2 0 − ω 0 ω 1 − 1 = 0 , t 0 ( ω 0 + ω 1 ) = 0 . PREPRINT 5 The system is singular , det( A ) = 0 iff both equations hold simultaneously . For t 0  = 0 we must ha ve ω 0 = − ω 1 , and hence t 2 0 + ω 2 0 = 1 . Therefore real singular configurations e xist only for | t 0 | < 1 , in which case there are exactly two ordered solutions, ( ω 0 , ω 1 ) =  q 1 − t 2 0 , − q 1 − t 2 0  ( ω 0 , ω 1 ) =  − q 1 − t 2 0 , q 1 − t 2 0  For t 0 = 0 , these conditions imply ω 0 ω 1 = − 1 . Fig. 1 illustrates this case. It shows a heat map of log ( σ min ( A ) σ max ( A ) ) as the frequency sampling points ω 0 and ω 1 vary . Configurations with ratios σ min /σ max below 1 . 85 ∗ 10 − 5 are considered nu- merically singular and are represented by contours. Fig. 1. Plot of log ( σ min ( A ) /σ max ( A )) for the stacked 3 × 3 system in H 2 , with t 0 = 0 fixed, as functions of ω 0 and ω 1 . Numerically singular configurations (ratios belo w 1 . 85 × 10 − 5 ) occur along the diagonal ω 0 = ω 1 (corresponding to repeated measurements) and the hyperbola ω 0 ω 1 = − 1 . From Fig. 1, it is e vident that matrix in v ersion is possible for almost all points in the sampling space, which therefore yield unique signal reconstruction. As expected from our analysis, reconstruction fails along the diagonal, ω 0 = ω 1 , and also along two symmetric contours ω 0 ω 1 = − 1 passing through the counterexample points (1 , − 1) and ( − 1 , 1) from Example 1. More generally , the zero set of a non-zero polynomial in m variables has Lebesgue measure zero in R m , which implies that a matrix sampled at random (according to a distrib ution that is absolutely continuous wrt. Lebesgue measure) is in- vertible with probability one. This alone does, howe ver , not imply that square matrices composed from entries Φ j ( t i ) and ˆ Φ j ( ω i ) are in vertible for almost all finite configurations of ( t i ) K − 1 i =0 , ( ω i ) L − 1 i =0 . A standard condition is that the mapping  ( t i ) K − 1 i =0 , ( ω i ) L − 1 i =0  7→ det( A ) is real-analytic. In this case the zero set has Lebesgue measure zero. This is true for the Hermite functions, but providing precise conditions for uniqueness pairs for H N is a difficult task. B. F inite RKHS sampling with two-sided samples If X is a finite-dimensional RKHS, dim( X ) = N + 1 < ∞ , one may choose a basis { Φ 0 , . . . , Φ N } and identify X = span { Φ i } equipped with an inner product specified by a symmetric positive definite Gram matrix G ∈ R ( N +1) × ( N +1) , i.e., for f = P i a i Φ i and g = P i b i Φ i , set ⟨ f , g ⟩ X = a ⊤ G b . The reproducing kernel then admits the explicit representation K ( x, y ) =  Φ( x ) ⊤ G − 1  Φ( y ) , where  Φ( x ) = (Φ 0 ( x ) , . . . , Φ N ( x )) ⊤ ; equiv alently , ev ery finite-dimensional RKHS corresponds to a feature map into R N +1 with a weighted Euclidean inner product, and the reproducing property follows from the simple computation ⟨ f , K x ⟩ X = a ⊤ G ((  Φ( x ) ⊤ G − 1 ) 0 , ..., (  Φ( x ) ⊤ G − 1 ) N ) = X i a i Φ i ( x ) = f ( x ) . One-sided sampling is governed by Eq. (2). T o incorporate two-sided samples we will need an appropriate mathematical framew ork, which is provided in Def. 2 and the subsequent discussion below . For now we prefer to keep the exposition heuristic - assuming the following quantities are all well- defined. As the kernel depends on two variables, we specify , which variable the Fourier transform applies to, follo wing the con vention: Fourier transform applied to first variable of K ( t, · ) : F  K s ( t )  = c K s ( ω ) (wide hat) , Fourier transform applied to second variable of K ( · , s ) : F  K s ( t )  = L ξ ( t ) . Suppose now that K samples can be obtained from f ( t ) and L samples can be obtained from ˆ f ( ω ) . Consider the two-sided expansion formula, which we will justify formally using a two-sided representer theorem, Thm. 1, belo w: f ( t ) = K − 1 X i =0 α i K t i ( t ) + L − 1 X i =0 β i L ω i ( t ) . (4) Applying F to this equation yields ˆ f ( ω ) = K − 1 X i =0 α i d K t i ( ω ) + L − 1 X i =0 β i d L ω i ( ω ) . Suppose we observe measurements  c from f ( t ) and  ˆ c from ˆ f ( ω ) , then the two equations yield a stacked system of equations:  c  ˆ c ! = ( K t i ( t j )) ( L ω i ( t j )) ( d K t i ( ω j )) ( d L ω i ( ω j )) !  (  α ) (  β )  . (5) Here we write ( K t i ( t j )) for the K × K matrix with entries K t i ( t j ) , 0 ≤ i, j ≤ K − 1 , ( L ω i ( t j )) for the K × L matrix with entries L ω i ( t j ) , 0 ≤ i ≤ L − 1 , 0 ≤ j ≤ K − 1 , ( d K t i ( ω j )) for the L × K matrix with entries d K t i ( ω j ) , 0 ≤ i ≤ K − 1 and 0 ≤ j ≤ L − 1 , finally , ( d L ω i ( ω j )) for the L × L matrix with entries d L ω i ( ω j ) , 0 ≤ i, j ≤ L − 1 . As outlined in the case of basis sampling, this system might ha ve an empty , a singleton or an infinite solution set, but it can be treated uniformly by the Moore-Penrose pseudoinv erse. PREPRINT 6 I V . R E C O N S T R U C T I O N I N T H E I N FI N I T E - D I M E N S I O NA L C A S E In signal sampling, natural sampling spaces like P W T , H hav e infinite dimension. This corresponds to a regime N → ∞ , where the number of measurements (cf. eq. (3) and (5)) “grows simultaneously with the dimension of the space”. A. Basis sampling with two-sided samples As the size of the system of equations (cf. (3), (5)) increases, the smallest singular value may decay tow ard zero, degrading conditioning and potentially leading to loss of injectivity in the limit. Thus, ev en if each finite- N problem admits a unique solution, the large- N system may become non-unique. The notion of a uniqueness pair ensures stability in this respect. Proposition 1. If (Λ , M ) is a uniqueness pair for a (poten- tially infinite-dimensional) function space X and the countable system of constraints admits a solution, then the solution is unique. Pr oof. Assume  α (1) ,  α (2) are two solutions, then  α (1) −  α (2) lies in the kernel of the linear system. The function f repre- sented by the coefficients { α (1) n − α (2) n } thus satisfies f | Λ = 0 and ˆ f | M = 0 , which is a contradiction unless f = 0 . The Paley-W iener model treats signals as e xactly bandlim- ited, so that the signals’ complexity is determined by the bandwidth. A more flexible model, developed by Landau, Pol- lak, and Slepian, considers functions that are simultaneously well concentrated in frequency on [ − W, W ] and in time on [ − T , T ] . The associated time-frequency concentration operator quantifies how compatible these two localization requirements are. The spectrum of this operator has a sharp plunge [18]– [20]: approximately 2 T W /π eigen v alues are close to 1 , while the rest are close to 0 , up to a narrow transition region. Thus, the space of signals that can be both time- and band- concentrated behaves, to first order, like a finite-dimensional space of dimension 2 T W /π . Equiv alently , this is the number of orthogonal modes that carry essentially all of the energy of such signals. This motiv ates a finite-dimensional analysis of two-sided reconstruction in Sec. III. In particular , any recon- struction based on K time samples and L frequency samples must satisfy the necessary condition K + L ≥ 2 T W /π . The prolate spheroidal wav e functions, being the eigenfunctions of the time-frequency concentration operator , pro vide the natural basis for the class of signals that are simultaneously concen- trated in time and frequency . When the dimension becomes large, these functions approximate Hermite functions and the respectiv e model spaces “con ver ge” to the Fourier -symmetric Sobolev space H . 1) The F ourier-symmetric Sobolev space: Using integration by parts and Plancherel’ s identity , the norm on H can be written in the form || f || 2 H = 2 Z R f H f d x, where H = 1 2  x 2 − ∂ 2 ∂ x 2  . Thus the natural scalar product for H is ⟨ f , g ⟩ = 2 R R f H g d x . H is the classical Hamiltonian operator of the quantum harmonic oscillator . It is well-known to be a self-adjoint and positiv e operator , whose eigenv ectors are the Hermite functions H φ n = E n φ n , E n = n + 1 2 . Thus in the Hermite basis H is the weighted l 2 -space with norm || f || H = ∞ X n =0 E n | α | 2 , f = ∞ X n =0 α n φ n . The Fourier -in v ariance of H manifests itself in the operator identity H = ˆ H in the sense that H = 1 2  x 2 − ∂ 2 ∂ x 2  , ˆ H = 1 2  k 2 − ∂ 2 ∂ k 2  and the fact that Hermite functions are eigenfunctions of the Fourier transform. H constitutes the natural infinite- dimensional extension of the finite Hermite function spaces considered in Sec. III-A. Uniqueness pairs for reconstruction in two-sided sampling in H have been identified recently in terms of a density condition (Λ , M ) . The pair (Λ = { λ j } j , M = { µ j } j ) is, respectively , called supercritical ( < ) or subcritical ( > ) if simultaneously the follo wing conditions hold 2 lim sup {| λ j | ( λ j +1 − λ j ) } ≶ π , lim sup {| µ j | ( µ j +1 − µ j ) } ≶ π . Proposition 2 (Uniqueness pairs in H [6]) . 1) If (Λ , M ) is supercritical then it is a uniqueness pair for H . 2) If (Λ , M ) is subcritical then it is a non-uniqueness pair for H . B. RKHS sampling with two-sided samples W e begin pro viding the formal mathematical framew ork for two-sided RKHS sampling. Definition 2 (T wo-sided RKHS) . Let X be a Hilbert space of functions f : X → C , where X is a domain equipped with the F ourier transform. W e call X a two-sided repr oducing kernel Hilbert space if, writing X ′ for the F ourier dual domain, for every t ∈ X the point evaluation functional in the time domain l t : X → C , f 7→ f ( t ) and for every ω ∈ X ′ the point evaluation functional in the F ourier domain ˆ l ω : X → C , f 7→ ˆ f ( ω ) ar e linear and continuous. W e will focus on X = R , but the definition is valid for other domains, e.g. X = R d . The Fourier functional is of the form ˆ l ω ( f ) = ˆ f ( ω ) = F [ f ]( ω ) . By the Riesz representation theorem, the functionals l t , ˆ l ω admit representations of the form l t ( f ) = ⟨ f , K t ⟩ X and ˆ l ω ( f ) = ⟨ f , L ω ⟩ X for certain 2 Our threshold differs from [6] since we use the unitary conv ention for the Fourier transforms. PREPRINT 7 representers K t , L ω ∈ X . Choosing f = K t in the last formula thus yields the pointwise identity c K t ( ω ) = ⟨ K t , L ω ⟩ X = L ω ( t ) . In case that X = R and K t is Fourier inte grable then under the unitary conv ention L ω ( t ) = c K t ( ω ) = 1 √ 2 π Z R K ( t, s ) e iω s d s. T o formally justify the reduction of two-sided sampling to the finite system of equations (5), we need a representer theorem, see [31], [32]. Theorem 1 (T w o-sided representer theorem) . Let ( X , ⟨· , ·⟩ X ) be a two-sided RKHS. Fix data points t 0 , ..., t K − 1 and ω 0 , ..., ω L − 1 . Let Ψ : C K + L → R ∪ {∞} be any function and let Ω : [0 , ∞ ) → R ∪ {∞} be strictly incr easing. Consider the optimization pr oblem min f ∈X ( Ψ  f ( t 0 ) , ..., f ( t K − 1 ) , ˆ f ( ω 0 ) , ..., ˆ f ( ω L − 1 )  +Ω( || f || X ) ) . If a minimizer f ∗ ∈ X e xists, then it is of the form f ∗ ( t ) = K − 1 X i =0 α i K t i ( t ) + L − 1 X i =0 β i L ω i ( t ) , wher e K t ar e Riesz r epr esenters of time domain point evalua- tion functionals and L ω ar e r epr esenters of frequency domain point evaluation functionals. Mor eover , L ω ( t ) = c K t ( ω ) . In the Euclidean case X = R , if K t is F ourier inte grable, then L ω ( t ) = 1 √ 2 π Z R K ( t, s ) e iω s d s. This justifies the reduction to the finite system of equa- tions (4). Pr oof. The proof is a consequence of a wider principle that representer theorems hold for continuous linear function- als [33] that even extends to Banach spaces [34], [35]. W e use the formulation of [33], which asserts that if a minimizer exists for the problem min ( Ψ  L 0 [ f ] , ..., L N − 1 [ f ])  + Ω( || f || X ) ) , where L 0 , ..., L N − 1 are continuous linear functionals, then the minimizer is of the form f ∗ = P N − 1 i =0 α i R i with R i the representer of L i . By assumption both time and Fourier domain point e valuation functionals are continuous, and we computed the explicit form of L ω ( t ) above. Remark 1. The statement r emains valid for a countable family of functionals. In this case f ∗ ∈ span { R n , n ∈ N } and for suitable sequences of coef ficients K X i =0 α ( K ) i K t i + L X i =0 β ( L ) i L ω i → f ∗ as K , L → ∞ . W e illustrate this representer theorem with the familiar example H . 1) The F ourier-symmetric Sobolev space as a two-sided RKHS: The Fourier functionals on H are continuous. W e have that | ˆ f ( ω ) | ≤ 1 √ 2 π Z R | f ( t ) | p 1 + t 2 d t √ 1 + t 2 ≤  π 2 Z R | f ( t ) | 2 (1 + t 2 ) d t  1 / 2 , where the first inequality uses the definition of the Fourier transform and second inequality is Cauchy-Schwarz. The right-hand side is bounded since f ∈ H . Making use of the in verse Fourier transform, the same argument sho ws that the point ev aluation functionals are continuous. The underlying vector space H admits sev eral equiv alent Hilbert norms that yield RKHS structures. A canonical choice that e xtends the finite dimensional discussion and preserves the symmetry with respect to the Fourier transform is to interpret the positive operator H as an infinite Gram matrix: Consider the scalar product defined for the Hermite basis expansions f = P ∞ n =0 α n φ n , g = P ∞ n =0 β n φ n by ⟨ f , g ⟩ H = ∞ X n =0 α n ¯ β n E n . The symmetric and positi ve function K ( x, y ) = ∞ X n =0 E − 1 n φ n ( x ) φ n ( y ) induces a reproducing kernel. Using the completeness of Hermite functions and the eigenv alue equation of H , direct computation rev eals that ⟨ f , K y ⟩ H = ∞ X n =0 α n E n E − 1 n φ n ( y ) = f ( y ) . This yields an explicit RKHS reconstruction scheme for two- sided sampling. The presented kernel representation together with the representer theorem 1 yield the linear system (5), which can be solved by standard tools such as the pseu- doin verse. In the finite-dimensional case the kernel can be truncated to a finite sum. There are two paths to ensure uniqueness of reconstruction. First, uniqueness is guaranteed if Ψ is con ve x and the regu- larizer Ω( || f || X ) is strictly con vex in f . In this case the entire optimization functional is strictly conv ex and the minimizer is unique if it exists [33]. The most important example is T ikhonov regularization, where the functional Ψ + λ || f || 2 X is strictly con ve x if Ψ is conv ex. Second, there is the injectivity of the sampling operator itself. This is ensured if the two- sided samples form a uniqueness pair . In this situation unique reconstruction holds ev en in the absence of a regularizer . Naturally , finding precise conditions is much harder in this situation. In this situation the formula of Rem. 1 takes the form of a unique representer . This leads to reconstruction formulas structurally analogous to those of Radchenko-V iazovska [4]. PREPRINT 8 Notice, howe ver , that their reconstruction nodes Λ = M = { √ 2 π n } n ∈ Z ≥ 0 are located at the edge of the regions of Prop. 2 since lim n →∞ √ 2 π n ( p 2 π ( n + 1) − √ 2 π n ) = π. No immediate conclusion about these nodes is possible here. Naturally , this path to the study of unique reconstruction is much harder . V . N U M E R I C A L E X P E R I M E N T S A. Comparison under a fixed sample b udget In this section, we consider N -dimensional Hermite func- tion space H N − 1 = span { φ 0 , . . . , φ N − 1 } , and compare one- and two-sided sampling under the same total sample budget D = K + L = N . W e measure reconstruction stability by the condition number σ max ( A ) /σ min ( A ) of the matrix A . Smaller values indicate a better conditioned in v ersion problem and lower sensiti vity to perturbations. Fig. 2. Condition numbers for one-sided and two-sided sampling with fixed total budget D = N . T ime samples are taken on [1 , 2] and frequency samples on [ − 1 , 0] . In both cases, the samples are equispaced. In our first experiment, for pure time-domain sampling, we take K = D time samples and no frequency samples ( L = 0 ). For each D , the sampling points are chosen on an equispaced grid ov er the interval [1 . 0 , 2 . 0] with spacing of 1 /D . For two- sided sampling, we retain the same total b udget of D samples but divide it between the time and frequency samples by K =  D 2  , L = D − K =  D 2  . The sampling points are then chosen on equispaced grids in time over [1 . 0 , 2 . 0] and in frequency over [ − 1 . 0 , 0] . As we can see in Fig. 2, the condition number increases with the dimension N , which indicates that the reconstruction problem becomes progressi vely more challenging in both cases ev en though the number of av ailable samples D also increases ( D = N ). Nev ertheless, two-sided reconstruction yields typi- cally smaller condition numbers than one-sided sampling and is therefore more stable numerically . Fig. 3. Condition numbers when both time and frequency samples lie in [ − 1 , 1] . Equispaced two-sided sampling is singular for some N , whereas random sampling avoids these failures and remains better conditioned. This behavior does not persist for all configurations. In Fig. 3, where both the time and frequenc y samples are taken on equispaced grids in [ − 1 , 1] , the resulting mixed sampling con- figuration becomes non-inv ertible for some values of N = D . This is consistent with the uniqueness-pair discussion above: certain aligned choices of time and frequency nodes can fail to determine the signal uniquely , leading to a singular sampling matrix. By contrast, in the same figure, we can see that if the sampling points are chosen randomly (uniformly) rather than equispaced within the same [ − 1 , 1] interv al, two- sided sampling once again outperforms one-sided sampling consistently . This suggests that some degree of misalignment between the time and frequency sampling sets is useful. In particular , when the relev ant uniqueness sets are not known a priori, randomization may serve as an ef fectiv e practical strategy . Related forms of misalignment are also known to play a fundamental role in compressed sensing [36]. Fig. 4. Condition numbers are unchanged when part of the time-domain data is post-processed by the DFT and treated as frequency-domain data. Finally , we consider a different experiment. W e begin with D = N time-domain samples and apply the discrete Fourier transform to a subset of the measurement vector , treating the resulting coef ficients as surrogate frequency-domain data. As expected, this leav es the condition number unchanged, cf. Fig. 4: in vertible post-processing of e xisting measurements cannot increase the information content. PREPRINT 9 B. Bandlimited signal space W e now turn to the standard sinc interpolation setting. As in the previous section, we in vestigate the condition number and compare the classical one-sided time-domain sampling with two-sided sampling under a fixed sampling budget D = N . More precisely , consider the finite-dimensional space P W N − 1 π = span { s 0 , . . . , s N − 1 } , where s n ( t ) = sinc( t − n ) , n = 0 , 1 , . . . , N − 1 , with T = 1 . Under the unitary Fourier con vention, we ha ve d sinc( ω ) = 1 √ 2 π 1 | ω |≤ π , c s n ( ω ) = 1 √ 2 π e − iω n 1 | ω |≤ π . In other words, P W N − 1 π is bandlimited in the frequency domain and integer shifts in time translate into phase shifts in frequency . For frequency-domain sampling, we consider the interval [ − 3 , 3] , since sampling outside the range [ − π , π ] carries no additional information. In the time-domain, as opposed to the previous section, we consider sampling intervals that increase with increasing sample size D , i.e., [  − D 2 + 1  , · · · ,  D 2  ] . In both settings, the sampling locations are chosen uniformly at random from the corresponding interv als, as in Fig 3. Fig. 5. Condition numbers in a finite-dimensional bandlimited space spanned by integer-shifted sinc functions. The time-sampling interval increases with the dimension of the function space, while the frequency-sampling interv al is fixed at [ − 3 , 3] . Sampling points are chosen uniformly at random. Fig. 5 shows that, as in the Hermite-generated case, the reconstruction problem becomes increasingly ill-conditioned as N grows. Moreov er , the two-sided sampling scheme consis- tently yields smaller condition numbers than the corresponding one-sided scheme in the bandlimited setting as well. V I . S P E C T R U M M O N I T O R I N G A P P L I C AT I O N In spectrum monitoring, the goal is to observe and analyze the radio-frequency en vironment in order to detect signals of interest, identify interference, and ensure efficient spec- trum usage [8], [9]. This is critical for applications such as telecommunication networks and defence systems. In such settings, the receiver digitizes a time-domain signal and then applies a discrete Fourier transform to conv ert those samples into spectra. Ho wev er , acquiring and storing Nyquist-rate time samples over very large bandwidths is often prohibitively expensi ve. In practice, this leads to memory bottlenecks due to limited RAM depth or insufficient storage throughput. Nev ertheless, reconstructing the time series afterward is still required for many tasks, such as detecting intermittent interferers and hunting for anomalies. T ypically , systems main- tain a rolling circular buf fer and overwrite old data until a trigger ev ent occurs. This practice is also reflected in standards for spectrum management [10]. As we demonstrate next, the reconstruction can be improv ed if the time samples in the buf fer are supplemented with the frequency samples in the spectra in a joint reconstruction. Fig. 6. T ime-domain signal reconstruction after a trigger in a spectrum- monitoring system: ground truth, reconstruction from time-domain samples only , reconstruction from time samples plus 2 DFT bins, and reconstruction from time samples plus 4 DFT bins. T o that end, we consider a spectrum monitoring scenario in which the system must reconstruct the time-domain signal after a trigger event. Specifically , we examine whether the last Z = 1024 samples of the input signal can be recovered when, due to storage constraints, only Z/ 2 samples can be retained. In this setting, the system stores every other time sample. At the same time, it maintains a long recursi ve sliding DFT with DFT length equal to Z . For the reconstruction task, we assume that the signal is a 4 -tone signal with SNR = 16 corrupted by additiv e white Gaussian noise. In Fig. 6, we compare the reconstruction performance for three cases: ( a ) using only the stored Z / 2 time samples, PREPRINT 10 ( b ) augmenting them with the two maximum-magnitude DFT bins, and ( c ) augmenting them with the four maximum- magnitude DFT bins. In all cases, reconstruction is performed in the least-squares sense. The results show that incorporating the strongest complex frequency-bin samples improves the reconstruction compared with using time samples alone, and that using 4 frequency bins yields a further improv ement ov er using 2 . Fig. 6 also reports the normalized MSE values, a v- eraged over 10 independent experiments. The error decreases from 0 . 62 when only time samples are used, to 0 . 37 when 2 frequency samples are added, and further to 0 . 25 when 4 frequency samples are included. V I I . C O N C L U S I O N It is common in signal processing to treat time and frequency-domain information as if they are independent di- mensions, with their relationship typically entering the discus- sion only through uncertainty principles. This article presents a complementary perspectiv e by examining joint reconstruction from two-sided samples in both domains. W e analyzed this problem in finite-dimensional model spaces and related it to infinite-dimensional settings through RKHS and uniqueness results. These connections show that two-sided sampling is not only a computational paradigm in finite dimensions but also part of a broader functional-analytic framework. While our article focused on several representative applications, we believ e that the main message is broader . W e e xpect that this two-sided viewpoint will be useful in other reconstruction problems where measurements in dual domains are naturally av ailable and should be e xploited jointly . A C K N O W L E D G M E N T The authors are grateful to Andrii Bondarenko, V isa K oivunen, and Kristian Seip for valuable discussions. This work was supported by the Swiss National Science Founda- tion, SNF grant No. CRSK-2 229036. R E F E R E N C E S [1] E. T . Whittaker , “On the functions which are represented by the expansions of the interpolation-theory , ” Pr oceedings of the Royal Society of Edinbur gh , vol. 35, pp. 181–194, 1915. [2] H. Landau, “Sampling, data transmission, and the Nyquist rate, ” Pro- ceedings of the IEEE , vol. 55, no. 10, pp. 1701–1706, 1967. [3] K. Seip, Interpolation and sampling in spaces of analytic functions . American Mathematical Society , 2004, vol. 33. [4] D. Radchenko and M. V iazo vska, “Fourier interpolation on the real line, ” Publications math ´ ematiques de l’IH ´ ES , vol. 129, no. 1, pp. 51–81, 2019. [5] A. Kuliko v , “Fourier interpolation and time-frequency localization, ” Journal of F ourier Analysis and Applications , vol. 27, no. 3, 2021. [6] A. Kulikov , F . Nazarov , and M. Sodin, “Fourier uniqueness and non- uniqueness pairs, ” , 2023. [7] O. Szehr , “Spectral criteria for unique signal recov ery from two-sided sampling, ” , 2025. [8] E. Axell, G. Leus, E. G. Larsson, and H. V . Poor, “Spectrum sensing for cognitiv e radio : State-of-the-art and recent advances, ” IEEE Signal Pr ocessing Magazine , vol. 29, no. 3, pp. 101–116, 2012. [9] J. Lunden, V . Koi vunen, and H. V . Poor , “Spectrum exploration and ex- ploitation for cognitiv e radio: Recent advances, ” IEEE Signal Pr ocessing Magazine , vol. 32, no. 3, pp. 123–140, 2015. [10] ITU-R, “Data format definition for exchanging stored I/Q data for the purpose of spectrum monitoring, ” International T elecommunication Union, Geneva, Switzerland, T ech. Rep. SM.2117-0, Sep. 2018. [Online]. A v ailable: https://www .itu.int/rec/R-REC-SM.2117 [11] Z.-P . Liang and P . C. Lauterbur , “ An efficient method for dynamic magnetic resonance imaging, ” IEEE T ransactions on Medical Imaging , vol. 13, no. 4, pp. 677–686, 1994. [12] Z.-P . Liang and P . Lauterbur , “Constrained imaging: overcoming the limitations of the F ourier series, ” IEEE Engineering in Medicine and Biology Magazine , vol. 15, no. 5, pp. 126–132, 1996. [13] L. W eizman, Y . C. Eldar , and D. Ben Bashat, “Reference-based MRI, ” Medical Physics , vol. 43, no. 10, pp. 5357–5369, 2016. [14] A. R. Bahai, B. R. Saltzberg, and M. Ergen, Multi-carrier digital communications: Theory and applications of OFDM . Springer , 2004. [15] 3GPP, “NR; Physical channels and modulation, ” 3GPP, T ech. Rep. TS 38.211, 2024. [16] T . S. Rappaport, W ir eless communications: Principles and practice . Pearson Education, 2010. [17] J. P . Ramos and M. Sousa, “Fourier uniqueness pairs of powers of integers, ” J . Eur . Math. Soc.(JEMS) , vol. 24, pp. 4327–4351, 2022. [18] D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty - I, ” Bell System T echnical Journal , vol. 40, no. 1, p. 43–63, 1961. [19] ——, “Prolate spheroidal wav e functions, Fourier analysis and uncer- tainty - II, ” Bell System T ec hnical J ournal , v ol. 40, no. 1, p. 65–84, 1961. [20] H. J. Landau and H. O. Pollak, “Prolate spheroidal wav e functions, Fourier analysis and uncertainty-iii: The dimension of the space of es- sentially time- and band-limited signals, ” Bell System T ec hnical Journal , vol. 41, no. 4, p. 1295–1336, 1962. [21] R. Gerchberg, “Super-resolution through error energy reduction, ” Optica Acta: International Journal of Optics , vol. 21, no. 9, pp. 709–720, 1974. [22] A. Papoulis, “ A new algorithm in spectral analysis and band-limited extrapolation, ” IEEE T ransactions on Circuits and Systems , vol. 22, no. 9, pp. 735–742, 1975. [23] A. N. Hirani and T . T otsuka, “Combining frequency and spatial domain information for fast interactiv e image noise remov al, ” in Pr oc. Annual Confer ence on Computer Gr aphics and Interactive T echniques , 1996, pp. 269–276. [24] Y . Matsuki, M. T . Eddy , and J. Herzfeld, “Spectroscopy by integration of frequency and time domain information for fast acquisition of high- resolution dark spectra, ” Journal of the American Chemical Society , vol. 131, no. 13, pp. 4648–4656, 2009. [25] P . J. S. Ferreira, “Interpolation and the discrete Papoulis-Gerchber g algorithm, ” IEEE T ransactions on Signal Processing , vol. 42, no. 10, pp. 2596–2606, 2002. [26] M. Marques, A. Neves, J. S. Marques, and J. Sanches, “The Papoulis- Gerchberg algorithm with unkno wn signal bandwidth, ” in International Confer ence Image Analysis and Recognition , 2006, pp. 436–445. [27] M. Ehler and K. Gr ¨ ochenig, “ An abstract approach to Marcinkiewicz- Zygmund inequalities for approximation and quadrature in modulation spaces, ” Mathematics of Computation , vol. 93, no. 350, pp. 2885–2919, 2024. [28] D. Zelent, “Time frequency localization in the Fourier symmetric Sobolev space, ” , 2025. [29] N. Aronszajn, “Theory of reproducing kernels, ” Tr ansactions of the American Mathematical Society , vol. 68, no. 3, p. 337–404, 1950. [30] M. Unser, “Sampling-50 years after Shannon, ” Pr oceedings of the IEEE , vol. 88, no. 4, pp. 569–587, 2000. [31] G. Whaba, Support vector mac hines, r epr oducing kernel Hilbert spaces and the randomized GA CV . Advances in K ernel Methods, MIT Press, 1999. [32] A. Christmann and I. Steinwart, Support V ector Machines , 2nd ed. Information Science and Statistics, Springer, 2008. [33] F . Dinuzzo and B. Sch ¨ olkopf, “The representer theorem for Hilbert spaces: a necessary and suf ficient condition, ” in Advances in Neural Information Pr ocessing Systems , F . Pereira, C. Burges, L. Bottou, and K. W einber ger , Eds., vol. 25. Curran Associates, Inc., 2012. [34] H. Zhang and Y . Xu, “Reproducing kernel Banach spaces for machine learning, ” Journal of Machine Learning Resear ch , vol. 10, pp. 2741– 2775, 2009. [35] O. Szehr and R. Zarouf, “Interpolation without commutants, ” Journal of Operator Theory , vol. 84, no. 1, p. 239–256, 2020. [36] E. J. Cand ` es, J. Romberg, and T . T ao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency informa- tion, ” IEEE T ransactions on Information Theory , vol. 52, no. 2, pp. 489–509, 2006.