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Abstract: Band-limited signals of finite energy, i.e., functions in , form the setting for much of signal processing. However, many signals of practical interest do not have finite energy. In this work we extend some results involving prolate spheroidal wave functions (PSWFs) to such signals. We consider those functions whose Fourier transforms have compact support in the sense of generalized functions and obtain convergence results for their PSWF expansions. [Copyright &y& Elsevier]

The theory of chromatic derivatives leads to chromatic series which replace Taylor's series for bandlimited functions. For such functions, these series have a global convergence property not shared by Taylor's series. In this work the theory is extended to bandlimited functions of slow growth. This includes many signals of practical importance such as polynomials, periodic functions and almost periodic functions. This extension also enables us to get improved local convergence results for chromatic series. [ABSTRACT FROM AUTHOR]

Chromatic series are series expansions in which the coefficients are linear combinations of derivatives of a function. They were introduced by Ignjatovic (Introduction to Signal Processing Based on Differential Operators, Tech. Rep. 1, Kromos Technology, available at http://www.kromos.com, February 2001.) as a replacement for Taylor's series and are based on orthogonal polynomials. However, real data usually involves the values of a function and not its derivatives which are needed in both theories. In this article, we replace the derivatives by discrete values in the calculations of the coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we construct infinite-band filterbanks for perfect reconstruction (PR) using Hermite polynomials and Hermite functions. The analysis filters are linear combinations of derivative operators based on these polynomials-the so-called chromatic derivative filters. Together with the synthesis filter- banks, they give PR for a large class of signals that may have infinite bandwidth. Several other related filterbanks are discussed as well. The error in reconstruction for finite-channel chromatic derivative filterbanks is calculated. Examples are given to demon- state the use of these chromatic filterbanks. [ABSTRACT FROM AUTHOR]

Slepian functions (Prolate Spheroidal Wave Functions) are obtained by maximizing the energy of a σ-bandlimiled function (nonnalized with total energy I) on a prescribed ititen'al [ -τ τ]. The solution to this problem leads to an eigenvalue problem λf(t) = ∫τ-τ{sinσ(t - x)/π(t - x)}f(x)dx, whose solutions, in turn, form an orthogonal sequence {ϕn}. This sequence is a basis of the Paley-Wiener space Bσ of σ-bandlimited functions. For σ = π, integer translates of the Slepian functions of order 0, {ϕ0(t - n)} form a Riesz basis of the same space. Furthermore, by using ϕ0, as a scaling function we can construct a wavelet theory based on them. Two methods of density estimations thus naturally arise; one based on the orthogonal system {ϕn} and the other on the scaling functions {ϕ0(t - n)}. The former gives more rapid convergence, while the latter avoids Gibbs phenomenon, is locally positive, and allows the use of thresholding methods. Both approaches exhibit a strong localization property. [ABSTRACT FROM AUTHOR]

Presents a method for recovering a computed tomography image from its sampled projections. Recovery of the image in the form of values of the Radon transform; Use of the properties of the raised-cosine wavelet in recovering the image; Rate of convergence of the resulting algorithm to the image density function.

Gibbs phenomenon occurs for most approximations based on standard orthogonal expansions, as well as for those based on integral operators. It also occurs in interpolations and other types of approximations. We consider a general approach to approximation based on delta sequences in an attempt to better understand the concept. [ABSTRACT FROM AUTHOR]

This paper is devoted to the discussion of a “hybrid” sampling series, a series of translates of a nonnegative summability function used in place of an orthogonal scaling function. The coefficients in the series are taken to be sampled values of the function to be approximated. This enables one to avoid the integration which arises in the other series. The approximations based on this hybrid series have certain desirable convergence properties: they are locally uniformly convergent for locally continuous functions, they have quadratic uniform convergence rate for functions in certain Sobolev spaces, they are locally bounded when the function is locally bounded and therefore, in particular, Gibbs'' phenomenon is avoided. Numerical experiments are given to illustrate the theoretical results and to compare these approximations with the scaling function approximations. [Copyright &y& Elsevier]

Abstract: Gibbs’ phenomenon occurs for most orthogonal wavelet expansions in one dimension. It also exists in higher dimensions but fundamental concepts must be redefined. This is done for both separable and non-separable wavelet expansions in severable variables. [Copyright &y& Elsevier]