Multidimensional of gradually series band limited functions and frames (original) (raw)

Sampling and recovery of multidimensional bandlimited functions via frames

Journal of Mathematical Analysis and Applications, 2010

In this paper, we investigate frames for L2[−π, π] d consisting of exponential functions in connection to oversampling and nonuniform sampling of bandlimited functions. We derive a multidimensional nonuniform oversampling formula for bandlimited functions with a fairly general frequency domain. The stability of said formula under various perturbations in the sampled data is investigated, and a computationally managable simplification of the main oversampling theorem is given. Also, a generalization of Kadec's 1/4 Theorem to higher dimensions is considered. Finally, the developed techniques are used to approximate biorthogonal functions of particular exponential Riesz bases for L2[−π, π], and a well known theorem of Levinson is recovered as a corollary.

Irregular sampling theorems and series expansions of band-limited functions

Journal of Mathematical Analysis and Applications, 1992

We present a new approach to the problem of irregular sampling of band-limited functions that is based on the approximation and factorization of convolution operators. A special case of our main result is the following theorem: If 52 G R" is compact, gE Li(R") a band-limited function with d(t) # 0 on Q and (

Sampling and recovery of bandlimited functions and applications to signal processing

Advanced Courses of Mathematical Analysis IV - Proceedings of the Fourth International School – In Memory of Professor Antonio Aizpuru Tomás, 2011

Bandlimited functions, i.e square integrable functions on R d , d ∈ N, whose Fourier transforms have bounded support, are widely used to represent signals. One problem which arises, is to find stable recovery formulae, based on evaluations of these functions at given sample points. We start with the case of equally distributed sampling points and present a method of Daubechies and DeVore to approximate bandlimited functions by quantized data. In the case that the sampling points are not equally distributed this method will fail. We are suggesting to provide a solution to this problem in the case of scattered sample points by first approximating bandlimited functions using linear combinations of shifted Gaussians. In order to be able to do so we prove the following interpolation result. Let (x j : j ∈ Z) ⊂ R be a Rieszbasis sequence. For λ > 0 and f ∈ P W , the space of square-integrable functions on R, whose Fourier transforms vanish outside of [−1, 1], there is a unique sequence (a j) ∈ 2 (Z), so that the function I λ (f)(x) := a j e −λ x−xj 2 2 , x ∈ R is continuous, square integrable, and satisfies the interpolatory conditions I λ (f)(x k) = f (x k), for all k ∈ Z. It is shown that I λ (f) converges to f in L 2 (R d) and uniformly on R, as λ → 0 + .

The exponential sampling theorem of signal analysis

1998

The Shannon sampling theory of signal analysis, the so-called WKSsampling theorem, which can be established by methods of Fourier analysis, plays an essential role in many fields. The aim of this paper is to study the so-called exponential sampling theorem (ESF) of optical physics and engineering in which the samples are not equally spaced apart as in the Shannon case but exponentially spaced, using the Mellin transform methods. One chief aim of this paper is to study the reproducing kernel formula, not in its Fourier transform setting, but in that of Mellin, for Mellin-bandlimited functions as well as an approximate version for a less restrictive class, the MRKF, namely for functions which are only approximately Mellin-bandlimited. The rate of approximation for such signals will be measured in terms of the strong Mellin fractional differential operators recently studied by the authors. The final aim is to show that the exponential sampling theorem ESF is equivalent to the MRKF for Mellinbandlimited functions (signals) in the sense that each is a corollary of the other. Three graphical examples are given, as illustration of the theory.

Nonuniform sampling of bandlimited signals with polynomial growth on the real axis

IEEE Transactions on Information Theory, 1997

We derive a sampling expansion for bandlimited signals with polynomial growth on the real axis. The sampling expansion uses nonuniformly spaced sampling points. But unlike other known sampling expansions for such signals, ours converge uniformly to the signal on any compact set. An estimate of the truncation error of such a series is also obtained.

Riesz bases in related to sampling in shift-invariant spaces

2005

The Fourier duality is an elegant technique to obtain sampling formulas in Paley-Wiener spaces. In this paper it is proved that there exists an analogous of the Fourier duality technique in the setting of shift-invariant spaces. In fact, any shiftinvariant space V ϕ with a stable generator ϕ is the range space of a bounded oneto-one linear operator T between L 2 (0, 1) and L 2 (R). Thus, regular and irregular sampling formulas in V ϕ are obtained by transforming, via T , expansions in L 2 (0, 1) with respect to some appropriate Riesz bases.

A generalized sampling theory without band-limiting constraints

IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 1998

We consider the problem of the reconstruction of a continuous-time function f (x) ∈ H from the samples of the responses of m linear shift-invariant systems sampled at 1/m the reconstruction rate. We extend Papoulis' generalized sampling theory in two important respects. First, our class of admissible input signals (typ. H = L 2 ) is considerably larger than the subspace of bandlimited functions. Second, we use a more general specification of the reconstruction subspace V (ϕ), so that the output of the system can take the form of a bandlimited function, a spline, or a wavelet expansion. Since we have enlarged the class of admissible input functions, we have to give up Shannon and Papoulis' principle of an exact reconstruction. Instead, we seek an approximationf ∈ V (ϕ) that is consistent in the sense that it produces exactly the same measurements as the input of the system. This leads to a generalization of Papoulis' sampling theorem and a practical reconstruction algorithm that takes the form of a multivariate filter. In particular, we show that the corresponding system acts as a projector from H onto V (ϕ). We then propose two complementary polyphase and modulation domain interpretations of our solution. The polyphase representation leads to a simple understanding of our reconstruction algorithm in terms of a perfect reconstruction filterbank. The modulation analysis, on the other hand, is useful in providing the connection with Papoulis' earlier results for the bandlimited case. Finally, we illustrate the general applicability of our theory by presenting new examples of interlaced and derivative sampling using splines.