Paul L. Butzer

Sampling theory for not necessarily band-limited functions: a historical overview

Shannon’s sampling theorem is one of the most powerful results in signal analysis. The aim of this overview is to show that one of its roots is a basic paper of de la Vallee Poussin of 1908. The historical development of sampling theory from 1908 to the present, especially the matter dealing with not necessarily band-limited functions (which includes the duration-limited case actually studied in 1908), is sketched. Emphasis is put on the study of error estimates, as well as on the delicate point-wise behavior of sampling sums at discontinuity points of the signal to be reconstructed.

Mellin transform analysis and integration by parts for Hadamard-type fractional integrals

Abstract This paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of Hadamard, in the space Xpc of Lebesgue measurable functions f on R + =(0,∞) such that ∫ 0 ∞ u c f(u) p du u ess sup u>0 u c f(u) for c∈ R =(−∞,∞) , in particular in the space Lp(0,∞) (1⩽p⩽∞). Formulas for the Mellin transforms of the four Hadamard-type fractional integral operators are established as well as relations of fractional integration by parts for them.

Fractional calculus in the Mellin setting and Hadamard-type fractional integrals

Abstract The purpose of this paper and some to follow is to present a new approach to fractional integration and differentiation on the half-axis R + =(0,∞) in terms of Mellin analysis. The natural operator of fractional integration in this setting is not the classical Liouville fractional integral Iα0+f but J α 0+,c f (x):= 1 Γ(α) ∫ 0 x u x c log x u α−1 f(u) du u (x>0) for α>0, c∈ R . The Mellin transform of this operator is simply (c−s) −α M [f](s) , for s=c+it, c,t∈ R . The Mellin transform of the associated fractional differentiation operator D α 0+,c f is similar: (c−s) α M [f](s) . The operator D α 0+,c f may even be represented as a series in terms of xkf(k)(x), k∈ N 0 , the coefficients being certain generalized Stirling functions Sc(α,k) of second kind. It turns out that the new fractional integral J α 0+,c f and three further related ones are not the classical fractional integrals of Hadamard (J. Mat. Pure Appl. Ser. 4, 8 (1892) 101–186) but far reaching generalizations and modifications of these. These four new integral operators are first studied in detail in this paper. More specifically, conditions will be given for these four operators to be bounded in the space Xcp of Lebesgue measurable functions f on (0,∞), for c∈(−∞,∞), such that ∫ ∞ 0 |u c f(u)| p du/u for 1⩽p ess sup u>0 [u c |f(u)|] for p=∞, in particular in the space Lp(0,∞) for 1⩽p⩽∞. Connections of these operators with the Liouville fractional integration operators are discussed. The Mellin convolution product in the above spaces plays an important role.

Compositions of Hadamard-type fractional integration operators and the semigroup property

Abstract This paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of Hadamard in the space Xcp of functions f on R + =(0,∞) such that ∫ 0 ∞ u c f(u) p du u ess sup u>0 u c |f(u)| for c∈ R =(−∞,∞) , in particular in the space Lp(0,∞) (1⩽p⩽∞). The semigroup property and its generalizations are established for the generalized Hadamard-type fractional integration operators under consideration. Conditions are also given for the boundedness in Xcp of these operators; they involve Kummer confluent hypergeometric functions as kernels.

A direct approach to the mellin transform

The aim of this paper is to present an approach to the Mellin transform that is fully independent of Laplace or Fourier transform theory, in a systematic, unified form, containing the basic properties and major results under natural, minimal hypotheses upon the functions in questions. Cornerstones of the approach are two definitions of the transform, a local and global Mellin transform, the Mellin translation and convolution structure, in particular approximation-theoretical methods connected with the Mellin convolution singular integral enabling one to establish the Mellin inversion theory. Of special interest are the Mellin operators of differentiation and integration, more correctly of anti-differentiation, enabling one to establish the fundamental theorem of the differential and integral calculus in the Mellin frame. These two operators are different than those considered thus far and more general. They are of particular importance in solving differential and integral equations. As applications, the wave equation onℝ+ × ℝ+ and the heat equation in a semi-infinite rod are considered in detail. The paper is written in part from an historical, survey-type perspective.

On Lagrange interpolation and Kramer-type sampling theorems associated with Sturm-Liouville problems

This article is devoted to a connection between Kramer’s sampling theorem and sampling expansions generated by Lagrange interpolation. It is shown that any function that has a sampling expansion in the scope of Kramer’s theorem also has a Lagrange-type interpolation expansion provided that the kernel associated with Kramer’s theorem arises from a second-order Sturm–Liouville boundary-value problem. This new approach, which for a variety of regular and singular Sturm–Liouville problems leads to associated sampling theorems, recovers not only many known sampling expansions but also gives new ways to calculate the corresponding sampling functions. New sampling series are included.

Central factorial numbers; their main properties and some applications.

The purpose of this paper is to present a systematic treatment of central factorial numbers (cfn), including their main properties, as well as to employ them in a variety of applications. The cfn are related more closely to the Stirling numbers than to the other well-known numbers of Bernoulli, Euler, etc., and they are at least as important as Stirlings numbers, said to be “as important as Bernoullis, or even more so”.

On dyadic analysis based on the pointwise dyadic derivative

AbstractВ пРЕДыДУЩИх РАБОтАх АВтОРы В ОсНОВНОМ РАж ВИВАлИ ДВОИЧНыИ АНАлИж, ОсНО ВАННыИ НА пОНьтИИ сИльНОИ ДВ ОИЧНОИ пРОИжВОДНОИ Д ль ФУНкцИИ, ОпРЕДЕлЕННых НА ДИАД ИЧЕскОИ ГРУппЕ ИлИ НА [0,1) с пЕРИО ДОМ 1. цЕльУ НАстОьЩЕИ Р АБОты ьВльЕтсь пОстРОЕНИЕ ДВОИЧНОгО ДИФФЕРЕНцИАльНОгО И ИНтЕгРАльНОгО ИсЧИс лЕНИИ НА ОсНОВЕ БОлЕЕ слОжНОг О, НО жАтО И БОлЕЕ клАссИЧЕскОгО пОНьт Иь ДВОИЧНОИ пРОИжВОД НОИ В тОЧкЕ. ИсслЕДУУтсь тЕ пРОст РАНстВА ФУНкцИИ, Дль кОтОРых пРИМЕНИМ ДВОИЧНыИ АНАлИж, А тАк жЕ ОпРЕДЕльУтсь гРАНИц ы ЕгО пРИМЕНИМОстИ. тАк ОкАжАлОсь, ЧтО пРО стРАНстВОLp(0, l), 1≦∞, ьВльЕ тсь БОлЕЕ ЕстЕстВЕННыМ п РОстРАН стВОМ Дль пОстРОЕНИь ДВОИЧНОгО АНАлИжА, ЧЕ М клАссИЧЕскОЕ пРОстР АНстВОс[0,1]. НАпРИМЕР, ЕслИ пЕРВАь ДВОИЧНАь пРОИжВОДНАь пРИНАДл ЕжИтс[0,1], тОf=const. с ДРУгОИ стОРОНы, ЕслИfεс[0,1], тО ДВОИЧНыИ ИНтЕгРАл, пОстРОЕННы И Дльf, НЕ пРИНАДлЕжИтс[0,1]. Уст АНОВлЕНО тАкжЕ, ЧтО сИльНАь ДВО ИЧНАь пРОИжВОДНАь И Д ВОИЧНАь пРОИжВОДНАь В тОЧкЕ с ОВпАДАУт пОЧтИ ВсУДУ Дль ФУНкцИИ, пРИ НАДлЕжАЩИх ОпРЕДЕлЕ ННОМУ пОДклАссУLp[0, 1].пОлУЧЕННыЕ РЕжУльтА ты пРИМЕНьУтсь к пОЧл ЕННОМУ ДИФФЕРЕНцИРОВАНИУ И ИНтЕгРИРОВАНИУ РьДОВ пО сИстЕМЕ УОлш А, к ОцЕНкАМ ВЕлИЧИН кОЁФФИцИЕНтОВ ФУРьЕ-УОлшА, к ДОкАжАтЕльст ВУ АНАлОгА ОсНОВНОИ тЕО РЕМы О НАИлУЧшЕМ пРИБ лИжЕНИИ Дль пОлИНОМОВ пО сИстЕМЕ УОлшА, А тАкжЕ к РЕшЕНИ У ДВОИЧНОгО ВОлНОВОг О УРАВНЕНИь.

The Shannon sampling series and the reconstruction of signals in terms of linear, quadratic and cubic splines

The sine-kernel function of the sampling series is replaced by spline functions having compact support, all built up from the B-splines

Anniversary Volume on Approximation Theory and Functional Analysis

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