Signal and System Approximation from General Measurements

In this paper we analyze the behavior of system approximation processes for stable linear time-invariant (LTI) systems and signals in the Paley-Wiener space PW_\pi^1. We consider approximation processes, where the input signal is not directly used to generate the system output, but instead a sequence of numbers is used that is generated from the input signal by measurement functionals. We consider classical sampling which corresponds to a pointwise evaluation of the signal, as well as several more general measurement functionals. We show that a stable system approximation is not possible for pointwise sampling, because there exist signals and systems such that the approximation process diverges. This remains true even with oversampling. However, if more general measurement functionals are considered, a stable approximation is possible if oversampling is used. Further, we show that without oversampling we have divergence for a large class of practically relevant measurement procedures.Comment: This paper will be published as part of the book "New Perspectives on Approximation and Sampling Theory - Festschrift in honor of Paul Butzer's 85th birthday" in the Applied and Numerical Harmonic Analysis Series, Birkhauser (Springer-Verlag). Parts of this work have been presented at the IEEE International Conference on Acoustics, Speech, and Signal Processing 2014 (ICASSP 2014

Исследование и совершенствование компоновки бурильной колонны для повышения эффективности бурения геологоразведочных скважин: автореферат диссертации на соискание ученой степени кандидата технических наук: спец. 25.00.14

AbstractThe approximate sampling theorem with its associated aliasing error is due to J.L. Brown (1957). This theorem includes the classical Whittaker–Kotel’nikov–Shannon theorem as a special case. The converse is established in the present paper, that is, the classical sampling theorem for f∈Bπwp, 1⩽p<∞, w>0, implies the approximate sampling theorem. Consequently, both sampling theorems are fully equivalent in the uniform norm.Turning now to Lp(R)-space, it is shown that the classical sampling theorem for f∈Bπwp, 1<p<∞ (here p=1 must be excluded), implies the Lp(R)-approximate sampling theorem with convergence in the Lp(R)-norm, provided that f is locally Riemann integrable and belongs to a certain class Λp. Basic in the proof is an intricate result on the representation of the integral ∫R|f(u)|pdu as the limit of an infinite Riemann sum of |f|p for a general family of partitions of R; it is related to results of O. Shisha et al. (1973–1978) on simply integrable functions and functions of bounded coarse variation on R. These theorems give the missing link between two groups of major equivalent theorems; this will lead to the solution of a conjecture raised a dozen years ago

Characterization of best algebraic approximation by weighted moduli of continuity

11 refs.SIGLEDEGerman

Linaer prediction by samples from the past

SIGLETIB Hannover: RN 2414(379) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Approximating a bandlimited function in terms of its samples

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 smoothen this step function by restricting its Fourier transform. The resulting function is used as an approximation for f. The aim ot this paper is to generalize this process in order to decrease the approximation error. (orig.)Available from TIB Hannover: RN 2414(482) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

De la Vallee Poussin's paper of 1908 on interpolation and sampling theory, and its influence

SIGLETIB Hannover: RN 2414(361) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Approximation by generalized sampling series

SIGLETIB: RN 2414 (294) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

The sampling theorem and its unique role in various branches of mathematics

SIGLECopy held by FIZ Karlsruhe; available from UB/TIB Hannover / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman