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Shannon’s sampling theorem is one of the most important results of modern signal theory. It describes the reconstruction of any band-limited signal from a finite number of its samples. On the other hand, although less well known, there is the discrete sampling theorem, proved by Cooley while he was working on the development of an algorithm to speed up the calculations of the discrete Fourier transform. Cooley showed that a sampled signal can be resampled by selecting a smaller number of samples, which reduces computational cost. Then it is possible to reconstruct the original sampled signal using a reverse process. In principle, the two theorems are not related. However, in this paper we will show that in the context of Non-Standard Mathematical Analysis (NSA) and Hyperreal Numerical System ∗R, the two theorems are equivalent. The difference between them becomes a matter of scale. With the scale changes that the hyperreal number system allows, the discrete variables and functions become continuous, and Shannon’s sampling theorem emerges from the discrete sampling theorem. Resumen El teorema del muestreo de Shannon es uno de los resultados más importantes de la moderna teoría de señales. Este describe la reconstrucción de toda señal de banda limitada desde un número finito de sus muestras. Por otra parte, aunque menos conocido, se tiene el teorema del muestreo discreto, demostrado por Cooley mientras trabajaba en la elaboración de un algoritmo para acelerar los cálculos de la transformada discreta de Fourier. Cooley demostró que una señal muestreada se puede volver a muestrearla mediante la selección de un número menor de muestras, lo cual reduce el costo computacional. Luego, es posible reconstruir la señal muestreada original mediante un proceso inverso. En principio, los dos teoremas no están relacionados. Sin embargo, en este artíclo demostraremos que, en el contexto del Análisis Matemático No Estándar (ANS) y el Sistema Numérico Hiperreal ∗R, los dos teoremas son equivalentes. La diferencia entre ellos se vuelve un asunto de escala. Con los cambios de escala que permite realizar el sistema numérico hiperreal, las variables y funciones discretas se vuelven continuas, y el teorema del muestreo de Shannon emerge del teorema del muestreo discreto.

We consider the problem of phase retrieval from magnitudes of short-time Fourier transform (STFT) measurements. It is well-known that signals are uniquely determined (up to global phase) by their STFT magnitude when the underlying window has an ambiguity function that is nowhere vanishing. It is less clear, however, what can be said in terms of unique phase-retrievability when the ambiguity function of the underlying window vanishes on some of the time-frequency plane. In this short note, we demonstrate that by considering signals in Paley-Wiener spaces, it is possible to prove new uniqueness results for STFT phase retrieval. Among those, we establish a first uniqueness theorem for STFT phase retrieval from magnitude-only samples in a real-valued setting., Comment: 13 pages, 5 figures

A continuous-time white Gaussian channel can be formulated using a white Gaussian noise, and a conventional way for examining such a channel is the sampling approach based on the Shannon–Nyquist sampling theorem, where the original continuous-time channel is converted to an equivalent discrete-time channel, to which a great variety of established tools and methodology can be applied. However, one of the key issues of this scheme is that continuous-time feedback and memory cannot be incorporated into the channel model. It turns out that this issue can be circumvented by considering the Brownian motion formulation of a continuous-time white Gaussian channel. Nevertheless, as opposed to the white Gaussian noise formulation, a link that establishes the information-theoretic connection between a continuous-time channel under the Brownian motion formulation and its discrete-time counterparts has long been missing. This paper is to fill this gap by establishing causality-preserving connections between continuous-time Gaussian feedback/memory channels and their associated discrete-time versions in the forms of sampling and approximation theorems, which we believe will play important roles in the long run for further developing continuous-time information theory. As an immediate application of the approximation theorem, we propose the so-called approximation approach to examine continuous-time white Gaussian channels in the point-to-point or multi-user setting. It turns out that the approximation approach, complemented by relevant tools from stochastic calculus, can enhance our understanding of continuous-time Gaussian channels in terms of giving alternative and strengthened interpretation to some long-held folklore, recovering “long-known” results from new perspectives, and rigorously establishing new results predicted by the intuition that the approximation approach carries. More specifically, using the approximation approach complemented by relevant tools from stochastic calculus, we first derive the capacity regions of continuous-time white Gaussian multiple access channels and broadcast channels, and we then analyze how feedback affects their capacity regions: feedback will increase the capacity regions of some continuous-time white Gaussian broadcast channels and interference channels, while it will not increase capacity regions of continuous-time white Gaussian multiple access channels.

We survey some aspects of the classical prediction theory for stationary processes, in discrete time in Section 1, turning in Section 2 to continuous time, with particular reference to reproducing-kernel Hilbert spaces and the sampling theorem. We discuss the discrete-continuous theories of ARMA-CARMA, GARCH-COGARCH, and OPUC-COPUC in Section 3. We compare the various models treated in Section 4 by how well they model volatility, in particular volatility clustering. We discuss the infinite-dimensional case in Section 5, and turn briefly to applications in Section 6.

Computer science education for kids has received considerable attention. However, we can hardly find educational cases for kids in information science, compared to computational science, such as programming. Aiming to develop classes and their materials about information science for kids, the author has tried to develop lectures to deliver core ideas of the Fourier transform, which is an important topic of information science, using the notion of average. However, these lectures did not include the sampling theorem, which plays an important role when we decide the sampling rate when digitizing analog data, and thus it is essential in information science. One of the challenges to incorporate it to lessons for kids is that it is necessary to explain notions of the wave and its frequency while lectures about the Fourier transform itself do not require these notions since the author explains the main idea of it using average. To incorporate the sampling theorem into our lectures, the author tried to explain waves and frequencies by timing. In this paper, the author shares experiences extending already developed lectures so that they include the sampling theorem. One important feature of these lessons is that we have used many hands-on materials. For the sampling theorem, we also use an equipment, called "water pearl", which makes us confuse that droplets of dropping water seem to be climbing up when we see it through flashing light. This is useful when we learn basic ideas of the sampling theorem because this phenomenon happens when two frequencies of dropping and flashing are appropriately selected. In this paper, the author also explain how we use this equipment to explain the sampling theorem.

The state of the art in sampling theory now contains several theorems for signals that are non-bandlimited. For signals on the sphere however, most theorems still require the assumptions of bandlimitedness. In this work we show that a particular class of non-bandlimited signals, which have a finite rate of innovation, can be exactly recovered using a finite number of samples. We consider a sampling scheme where K weighted Diracs are convolved with a kernel on the rotation group. We prove that if the sampling kernel has a bandlimit L=2k then (2k - 1) (4k - 1) + 1 equiangular samples are sufficient for exact reconstruction. If the samples are uniformly distributed on the sphere, we argue that the signal can be accurately reconstructed using 4K2 samples and validate our claim through numerical simulations. To further reduce the number of samples required, we design an optimal sampling kernel that achieves accurate reconstruction of the signal using only 3K samples, the number of parameters of the weighted Diracs. In addition to weighted Diracs, we show that our results can be extended to sample Diracs integrated along the azimuth. Finally, we consider kernels with antipodal symmetry which are common in applications such as diffusion magnetic resonance imaging. ASTAR (Agency for Sci., Tech. and Research, S’pore) Accepted version

We obtain a $q$-linear analogue of Gegenbauer's expansion of the plane wave. It is expanded in terms of the little $q$-Gegenbauer polynomials and the \textit{third} Jackson $q$-Bessel function. The result is obtained by using a method based on bilinear biorthogonal expansions., Comment: 12 pages, to appear in Adv. in Appl. Math. arXiv admin note: text overlap with arXiv:0909.0067

We study an extension of the classical Paley-Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structure includes both sampling expansions and Fourier-Neumann type series as special cases, and it also provides a bilinear expansion for the Dunkl kernel (in the rank 1 case) which is a Dunkl analogue of Gegenbauer's expansion of the plane wave and the corresponding sampling expansions. In fact, we show how to derive sampling and Fourier-Neumann type expansions from the results related to the bilinear expansion for the Dunkl kernel., 16 pages

We introduce a sampling theory for cyclic lattice invariant spaces in the context of locally compact Abelian groups. The key element of the theory is a new periodization condition which, for L a lattice in a group G, characterizes the L cyclic invariant subspaces V ⊆ L 2 ( G ) which allow sampling. When L is contained in a larger lattice L ′ , we characterize the collection of L -invariant subspaces of L 2 ( G ) which are also L ′ -invariant. We use this to show that, if V is cyclic with respect to both L and L ′ , then V admits sampling with respect to L if and only if V admits sampling with respect to L ′ .

We study some problems related to the effect of bounded, additive sample noise in the bandlimited interpolation given by the Whittaker–Shannon–Kotelnikov (WSK) sampling formula. We establish a generalized form of the WSK series that allows us to consider the bandlimited interpolation of any bounded sequence at the zeros of a sine-type function. The main result of the paper is that if the samples in this series consist of independent, uniformly distributed random variables, then the resulting bandlimited interpolation almost surely has a bounded global average. In this context, we also explore the related notion of a bandlimited function with bounded mean oscillation. We prove some properties of such functions, and in particular, we show that they are either bounded or have unbounded samples at any positive sampling rate. We also discuss a few concrete examples of functions that demonstrate these properties.

A perfect reconstruction of functions in a reproducing kernel Hilbert space from a given set of sampling points is discussed. A necessary and sufficient condition for the corresponding reproducing kernel and the given set of sampling points to perfectly recover the functions is obtained in this paper. The key idea of our work is adopting the reproducing kernel Hilbert space corresponding to the Gramian matrix of the kernel and the given set of sampling points as the range space of a sampling operator and considering the orthogonal projector, defined via the range space, onto the closed linear subspace spanned by the kernel functions corresponding to the given sampling points. We also give an error analysis of a reconstructed function by incomplete sampling points.

A ladder shaped filter of two real parameters a1,a2∈(−1,1) is introduced in this note. The impulse response of the corresponding Linear Time Invariant (LTI) system is a generalized Sinc function of two parameters. Consequently a generalized Shannon-type sampling theorem is established for a class of non-bandlimited signals with special spectrum properties associated with a ladder shaped filter of two parameters. Finally, a mathematical characterization for the class of non-bandlimited signals satisfying the generalized sampling theorem is offered. These signals are restrictions to the real line of certain analytic functions in stripped domains symmetric about the real axis in the complex plane. For these signals, their spectra in higher frequency bands are measured by the spectrum of their base bands.

Chapter 7 We discuss the relationships between linear canonical transform (LCT) domains, fractional Fourier transform (FRT) domains, and the space-frequency plane. In particular, we show that LCT domains correspond to scaled fractional Fourier domains and thus to scaled oblique axes in the space-frequency plane. This allows LCT domains to be labeled and monotonically ordered by the corresponding fractional order parameter and provides a more transparent view of the evolution of light through an optical system modeled by LCTs. We then study the number of degrees of freedom of optical systems and signals based on these concepts. We first discuss the bicanonical width product (BWP), which is the number of degrees of freedom of LCT-limited signals. The BWP generalizes the space-bandwidth product and often provides a tighter measure of the actual number of degrees of freedom of signals. We illustrate the usefulness of the notion of BWP in two applications: efficient signal representation and efficient system simulation. In the first application we provide a sub-Nyquist sampling approach to represent and reconstruct signals with arbitrary space-frequency support. In the second application we provide a fast discrete LCT (DLCT) computation method which can accurately compute a (continuous) LCT with the minimum number of samples given by the BWP. Finally, we focus on the degrees of freedom of first-order optical systems with multiple apertures. We show how to explicitly quantify the degrees of freedom of such systems, state conditions for lossless transfer through the system and analyze the effects of lossy transfer.

This work is concerned with approximation of a signal from local averages. It improves a result of Butzer and Lei [P.L. Butzer, J. Lei, Approximation of signals using measured sampled values and error analysis, Commun. Appl. Anal. 4 (2000) 245–255].

We present the sampling theorem with sampling functions of general form for entire functions satisfying one of the growth conditions 1+|y| f(z) ⩽A 1+|z| N 1 exp τ|x|+σ|y| , 1+|x| f(z) ⩽A 1+|z| N 1 exp τ|x|+σ|y| for some A>0, τ,σ⩾0, N 1 ∈ N ∪{0} and any z=x+iy∈ C . It will be shown that many well-known sampling theorems included in SIAM J. Math. Anal. 19 (1988) 1198–1203 and Inform. Control 8 (1965) 143–158 can be interpreted as special cases of this sampling theorem. As examples, we provide sampling representations for entire functions which are bounded, of polynomial growth, or of exponential growth on R . We also provide sampling representations involving derivatives of entire functions and nonuniform sampling representations. Taking the set of sampling points in which a finite number of points are arbitrarily distributed, we obtain a sampling representation.

It is well known that exponential Riesz bases are stable. The celebrated theorem by Kadec shows that 1/4 is a stability bound for the exponential basis on L2(−π;π). In this paper we prove that α/π (where α is the Lamb-Oseen constant) is a stability bound for the sinc basis on L2(−π;π). The difference between the two values α/π−1/4, is ≈ 0.15, therefore the stability bound for the sinc basis on L2(−π;π) is greater than Kadec's stability bound (i.e. 1/4).

We show that every function in a spline subspace is uniquely determined and can be reconstructed by its local averages near certain points. Regular and irregular average sampling theorems for spline subspaces are obtained.

The sampling theory says that a function may be determined by its sampled values at some certain points provided the function satisfies some certain conditions. In this paper, we consider the convergence of the sampling series for bounded functions with the sampling function from some multiresolution analysis. We also present a convergence theorem on the sampling series with irregular sampling points and give the approximation error.

Stable sampling formula using basic hypergeometric series for reconstructing analytic functions from exponentially spaced samples is considered. Criterion for selecting regularizing parameter and error estimates are obtained.

The harmonizable Piranashvili-type stochastic processes are approximated by finite time shifted average sampling sums. Explicit truncation error upper bounds are established. Various corollaries and special cases are discussed., Comment: 16 pages. This is an Author's Accepted Manuscript of an article published in the Communications in Statistics - Theory and Methods, Vol. 40, Issue 19-20, 2011, 3587-3598. [copyright Taylor \& Francis], available online at: http://www.tandfonline.com/ [DOI: 10.1080/03610926.2011.581180]

We consider the problem of reconstructing a wide sense stationary band-limited process from its local averages taken either at the Nyquist rate or above. As a result we obtain a sufficient condition under which average sampling expansions hold in mean square and for almost all sample functions. Truncation and aliasing errors of the expansion are also discussed.

Meaningful upper bounds for the Fourier transform of polynomial exponential functions are often hard to come by. Regarding Fourier transforms of rational exponential functions, which are of importance, for example in Campbell's sampling theorem, the purpose of finding significant upper bounds is an even more demanding exercise. In this article, we propose a new approach in order to obtain significant upper bounds for Fourier transforms of general exponential functions. The technique is shown to allow further generalization in order to deal with Fourier-like integrals and rational exponential integrals.

We discuss a novel sampling theorem on the sphere developed by McEwen & Wiaux recently through an association between the sphere and the torus. To represent a band-limited signal exactly, this new sampling theorem requires less than half the number of samples of other equiangular sampling theorems on the sphere, such as the canonical Driscoll & Healy sampling theorem. A reduction in the number of samples required to represent a band-limited signal on the sphere has important implications for compressive sensing, both in terms of the dimensionality and sparsity of signals. We illustrate the impact of this property with an inpainting problem on the sphere, where we show superior reconstruction performance when adopting the new sampling theorem., 9 pages, 2 figures, Proceedings of Wavelets and Sparsity XIV, SPIE Optics and Photonics 2011

We define an analogue of the Paley-Wiener space in the context of the Askey-Wilson function transform, compute explicitly its reproducing kernel and prove that the growth of functions in this space of entire functions is of order two and type ln q−1, providing a Paley-Wiener Theorem for the Askey-Wilson transform. Up to a change of scale, this growth is related to the refined concepts of exponential order and growth proposed by J. P. Ramis. The Paley-Wiener theorem is proved by combining a sampling theorem with a result on interpolation of entire functions due to M. E. H. Ismail and D. Stanton. CMUC/FCT; FCT post-doctoral grant SFRH/BPD/26078/2005, POCI 2010; FSE

Sampling moiré effects are well known in signal processing. They occur when a continuous periodic signal g ( x ) is sampled using a sampling frequency f s that does not respect the Nyquist condition, and the signal-frequency f folds over and gives a new, false low frequency in the sampled signal. However, some visible beating artefacts may also occur in the sampled signal when g ( x ) is sampled using a sampling frequency f s which fully respects the Nyquist condition. We call these phenomena sub-Nyquist artefacts . Although these beating effects have already been reported in the literature, their detailed mathematical behaviour is not widely known. In this paper, we study the behaviour of these phenomena and compare it with analogous results from the moiré theory. We show that both sampling moirés and sub-Nyquist artefacts obey the same basic mathematical rules, in spite of the differences between them. This leads us to a unified approach that explains all of these phenomena and puts them under the same roof. In particular, it turns out that all of these phenomena occur when the signal-frequency f and the sampling frequency f s satisfy f ≈( m / n ) f s with integer m , n , where m / n is a reduced integer ratio; cases with n =1 correspond to true sampling moiré effects.

The well known Poisson Summation Formula is analysed from the perspective of the coherent state systems associated with the Heisenberg--Weyl group. In particular, it is shown that the Poisson summation formula may be viewed abstractly as a relation between two sets of bases (Zak bases) arising as simultaneous eigenvectors of two commuting unitary operators in which geometric phase plays a key role. The Zak bases are shown to be interpretable as generalised coherent state systems of the Heisenberg--Weyl group and this, in turn, prompts analysis of the sampling theorem (an important and useful consequence of the Poisson Summation Formula) and its extension from a coherent state point of view leading to interesting results on properties of von Neumann and finer lattices based on standard and generalised coherent state systems., 20 pages, Latex

Sampling theorems are one of the basic tools in information theory. The signal function f whose band–region is contained in a certain interval can be reconstructed from their values f ðxkÞ at the sampling points fxkg. We obtain analogues of this theorem for the cases of the Fourier–Jacobi series, the complex sphere S 1 c and the complex semisimple Lie groups. And as an application of these formulae, we show a version of the sampling theorem for the Radon transform on the complex hyperbolic space.

A lecture note introducing the sampling theorem as an interpolation method is presented. The relationship between piecewise linear approximation and the sampling theorem is highlighted by the use of triangular pulses instead of sampling functions. Furthermore, a comparison of the linear interpolation with a series on a nonorthogonal basis composed of equally spaced triangular pulses is provided. The interpolation uses the sample values of the function whereas the series coefficients are obtained by minimizing the quadratic error between the original function and the series. Uma aula sobre o teorema da amostragem como um método de interpolação é apresentada neste trabalho. A relação entre a interpolação linear e o teorema da amostragem é realçada pelo uso de pulsos triangulares aproximando funções sampling. Além disso, é feita uma comparação entre a interpolação linear e a série em uma base não ortogonal composta de pulsos triangulares igualmente espaçados. A interpolação usa os valores das amostras da função e os coeficientes da série são obtidos minimizando o erro quadrático entre a função original e a série.

This work investigates some spectral characteristics of the errors of optimal linear predictors for weakly stationary random fields. More specifically, for errors of optimal linear predictors, results here explicitly bound the fraction of the variance attributable to some set of frequencies. Such a bound is first obtained for random fields on $\mathbb{R}^d$ observed on the infinite lattice $\deltaJ$ for all J on the d-dimensional integer lattice. If the spectral density exists, then the faster the spectral density tends to 0 at high frequencies, the more quickly this bound tends to 0 as $\delta \downarrow 0$. Under certain conditions on the spectral density, a similar result is given for processes on $\mathbb{R}$ where both observations and predictands are confined to a finite interval and observations may not be evenly spaced. These results provide a powerful tool for studying a problem the author has previously addressed using different methods: the properties of linear predictors calculated under an incorrect spectral density. Specifically, this work gives a number of new rates of convergence to optimality for predictors based on an incorrect spectral density when the ratio of the incorrect to the correct spectral density tends to 1 at high frequencies.

It has been known since long that an oversampled function can be represented by a generalized sinc series which converges much faster than the usual one. We show that the speed of convergence is exponential and give numerical approximations to the kernel function involved in the generalized series. The practical consequence is that a bandlimited function which is oversampled can be evaluated at any point with accuracy O( L e− L ) using O( L ) operations.

Consider a regular self-adjoint eigenvalue problem of n th order, and the associated (Whittaker–Shannon)–Kramer sampling series, the samples being taken at the eigenvalues, and the sinc-function being replaced by the generalized Fourier coefficients of the kernel (arising from the eigenvalue problem) with respect to the complete set of (orthogonal) eigenfunctions. The major aim of this paper is to show that each such sampling expansion can be written as a Lagrange interpolation series, provided the kernel satisfies suitable conditions. Two concrete fourth-order eigenvalue problems are examined, a regular one as an application of the general result, and a singular one. The problem whether the two resulting sampling series are really new (or can be reduced to already known results) is in part open.

The notion of sampling for second-order q -variate processes is defined. It is shown that if the components of a q -variate process (not necessarily stationary) admits a sampling theorem with some sample spacing, then the process itself admits a sampling theorem with the same sample spacing. A sampling theorem for q -variate stationary processes, under a periodicity condition on the range of the spectral measure of the process, is proved in the spirit of Lloy's work. This sampling theorem is used to show that if a q -variate stationary process admits a sampling theorem, then each of its components will admit a sampling theorem too.

This paper continues the work started in [1]; a second continuous Jacobi transform is defined for suitable functionsf(x). Properties of the transform are studied. In particular, the first continuous Jacobi transform in [1] and the second continuous Jacobi transform are shown to be inverse to each other. The paper concludes with an extension of Campbell's sampling theorem [2].

In this paper, we propose an algorithm for the stable and efficient computation of Fourier expansions of square integrable functions on the unit sphere S ⊂ R 3, as well as for the evaluation of these Fourier expansions at special knots. The heart of the algorithm is an efficient realization of discrete Legendre function transforms based on a modified and stabilized version of the Driscoll-Healy algorithm.

In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon's R(D) theory in the case of Gaussian stationary processes, which says that transforming into a Fourier basis followed by block coding gives an optimal lossy compression technique; practical developments like transform-based image compression have been inspired by this result. In this paper we also discuss connections perhaps less familiar to the information theory community, growing out of the field of harmonic analysis. Recent harmonic analysis constructions, such as wavelet transforms and Gabor transforms, are essentially optimal transforms for transform coding in certain settings. Some of these transforms are under consideration for future compression standards. We discuss some of the lessons of harmonic analysis in this century. Typically, the problems and achievements of this field have involved goals that were not obviously related to practical data compression, and have used a language not immediately accessible to outsiders. Nevertheless, through an extensive generalization of what Shannon called the "sampling theorem", harmonic analysis has succeeded in developing new forms of functional representation which turn out to have significant data compression interpretations. We explain why harmonic analysis has interacted with data compression, and we describe some interesting recent ideas in the field that may affect data compression in the future.