A reconstruction formula for band limited functions in L2(Rd)L_2(R^d)L2​(Rd) (original) (raw)

Complete iterative reconstruction algorithms for irregularly sampled data in spline-like spaces

1997 IEEE International Conference on Acoustics, Speech, and Signal Processing

We prove that the exact reconstruction of a function s from its samples s(x i) on any "sufficiently dense" sampling set {x i } i∈Λ can be obtained, as long as s is known to belong to a large class of spline-like spaces in L p (R n). Moreover, the reconstruction can be implemented using fast algorithms. Since a limiting case is the space of bandlimited functions, our result generalizes the classical Shannon-Whittaker sampling theorem on regular sampling and the Paley-Wiener theorem on non-uniform sampling.

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 + .

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 (

Modified B-splines for the sampling of bandlimited functions

IEEE Transactions on Signal Processing, 1999

It is shown that there is an optimal finite linear combination of B-splines, denominated modified B-splines, such that a pertinent low frequency condition called M −flatness is satisfied. A profound relationship of the modified B-splines with the Beta distribution implies an asymptotic sampling theorem with exact reconstruction requiring only small oversampling.

Approximating a bandlimited function in terms of its samples

Computers & Mathematics with Applications, 2000

In order to reconstruct a bandlimited signal f from its sampled values, it is a standard practice to construct a step function from the samples, and then to smooth this step function by restricting its Fourier transform. The resulting function is used as an approximation for f. The aim of this paper is to generalize this process in order to decrease the approximation error.

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.

Approximation of bandlimited functions

Applied and Computational Harmonic Analysis, 2006

Many signals encountered in science and engineering are approximated well by bandlimited functions. We provide suitable error bounds for the approximation of bandlimited functions by linear combinations of certain special functions-the prolate spheroidal wave functions of order 0. The coefficients in the approximating linear combinations are given explicitly via appropriate quadrature formulae.

Deconvolution of band limited functions on non-compact symmetric spaces

Houston Journal of Mathematics, 2011

It is shown that a band limited function on a non-compact symmetric space can be reconstructed in a stable way from some countable sets of values of its convolution with certain distributions of compact support. A reconstruction method in terms of frames is given which is a generalization of the classical result of Duffin-Schaeffer about exponential frames on intervals. The second reconstruction method is given in terms of polyharmonic average splines.

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.

Sampling the Flow of a Bandlimited Function

The Journal of Geometric Analysis

We analyze the problem of reconstruction of a bandlimited function f from the space–time samples of its states f_t=\phi _t*fft=ϕt∗fresultingfromtheconvolutionwithakernelf t = ϕ t ∗ f resulting from the convolution with a kernelft=ϕt∗fresultingfromtheconvolutionwithakernel\phi _tϕt.Itiswell−knownthat,innaturalphenomena,uniformspace–timesamplesoffarenotsufficienttoreconstructfinastableway.Toenablestablereconstruction,aspace–timesamplingwithperiodicnonuniformlyspacedsamplesmustbeusedaswasshownbyLuandVetterli.Weshowthatthestabilityofreconstruction,asmeasuredbyaconditionnumber,controlsthemaximalgapbetweenthespacialsamples.Weprovideaquantitativestatementofthisresult.Inaddition,insteadofirregularspace–timesamples,weshowthatuniformdynamicalsamplesatsub−Nyquistspatialrateallowonetostablyreconstructthefunctionϕ t . It is well-known that, in natural phenomena, uniform space–time samples of f are not sufficient to reconstruct f in a stable way. To enable stable reconstruction, a space–time sampling with periodic nonuniformly spaced samples must be used as was shown by Lu and Vetterli. We show that the stability of reconstruction, as measured by a condition number, controls the maximal gap between the spacial samples. We provide a quantitative statement of this result. In addition, instead of irregular space–time samples, we show that uniform dynamical samples at sub-Nyquist spatial rate allow one to stably reconstruct the functionϕt.Itiswell−knownthat,innaturalphenomena,uniformspace–timesamplesoffarenotsufficienttoreconstructfinastableway.Toenablestablereconstruction,aspace–timesamplingwithperiodicnonuniformlyspacedsamplesmustbeusedaswasshownbyLuandVetterli.Weshowthatthestabilityofreconstruction,asmeasuredbyaconditionnumber,controlsthemaximalgapbetweenthespacialsamples.Weprovideaquantitativestatementofthisresult.Inaddition,insteadofirregularspace–timesamples,weshowthatuniformdynamicalsamplesatsub−Nyquistspatialrateallowonetostablyreconstructthefunction\widehat{f}$$ f ^ away from certain, explicitly described blind spots. We also consider several classes of finite dimensional subsets of bandlimited functions in w...