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182
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 207 (22 self)
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This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
Eigenfunctions, transfer matrices, and absolutely continuous spectrum of onedimensional Schrödinger operators
, 1999
"... ..."
Modulation Spaces on Locally Compact Abelian Groups
 University of Vienna
, 1983
"... this paper, which is now better accessible, in the same way as to the original report.) Hans G. Feichtinger 1 ..."
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Cited by 58 (2 self)
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this paper, which is now better accessible, in the same way as to the original report.) Hans G. Feichtinger 1
Wavelet theory demystified
 IEEE Trans. Signal Process
, 2003
"... Abstract—In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a Bspline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to red ..."
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Cited by 45 (22 self)
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Abstract—In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a Bspline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory—including some new extensions for fractional orders—in a selfcontained, accessible fashion. In particular, we prove that the Bspline component is entirely responsible for five key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multiscale differentiation property, and smoothness (regularity) of the basis functions. We also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions. This allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal. Additional results include a new peeling theory of smoothness, leading to the extended notion of wavelet differentiability in thesense and a sharper theorem stating that smoothness implies order. Index Terms—Approximation order, Besov spaces, Hölder smoothness, multiscale differentiation, splines, vanishing moments, wavelets. I.
Spectrum of dynamical systems arising from Delone sets
 American Math. Soc.: Providence RI
, 1997
"... We investigate spectral properties of the translation action on the orbit closure of a Delone set. In particular, sufficient conditions for pure discrete spectrum are given, based on the notion of almost periodicity. Connections with diffraction spectrum are discussed. 1 Introduction A set ae R d ..."
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Cited by 37 (2 self)
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We investigate spectral properties of the translation action on the orbit closure of a Delone set. In particular, sufficient conditions for pure discrete spectrum are given, based on the notion of almost periodicity. Connections with diffraction spectrum are discussed. 1 Introduction A set ae R d is a Delone set if there exist positive constants R and r such that every ball of radius R intersects and every ball of radius r contains at most one point of . The collection of all such sets with fixed R and r can be equipped with a metric to form a compact space. The group R d acts on this space by translations. We study the spectral properties of this action restricted to some invariant subsets. We begin with a description of eigenvalues (with continuous eigenfunctions) assuming that the restricted action is minimal. Then we consider dynamics with respect to an ergodic invariant measure and obtain sufficient conditions for the system to have pure discrete spectrum. These conditions a...
LogPeriodogram Regression Of Time Series With Long Range Dependence
 ANNALS OF STATISTICS
, 1999
"... This paper discusses the use of fractional exponential models (Robinson (1990), Beran (1994)) to model the spectral density f(x) of a covariance stationary process when f(x) may be decomposed as f(x) = x \Gamma2d f (x), where f (x) is bounded and bounded away from zero. A form of logperiodogram ..."
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Cited by 33 (0 self)
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This paper discusses the use of fractional exponential models (Robinson (1990), Beran (1994)) to model the spectral density f(x) of a covariance stationary process when f(x) may be decomposed as f(x) = x \Gamma2d f (x), where f (x) is bounded and bounded away from zero. A form of logperiodogram regression technique is presented both in the parametric context (i.e. f (x) is a finite order exponential model in the sense of Bloomfield (1973)) and the semiparametric context (f (x) is regarded as a nuisance parameter). Assuming gaussianity and additional conditions on the regularity of f (x) which seem mild, asymptotic normality of the parameter estimates in the parametric and the semiparametric context is established. As a byproduct, some improvements over the results presented by Robinson (1994) have been obtained for the large sample distribution of logperiodogram ordinates for Gaussian processes.
Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum
 PHYS. ACTA
, 1997
"... We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential q of a onedimensional Schrödinger operator H = − d2 dx2 + q determine the potential completely. Included are theorems for finite intervals and for the whole l ..."
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Cited by 33 (6 self)
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We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential q of a onedimensional Schrödinger operator H = − d2 dx2 + q determine the potential completely. Included are theorems for finite intervals and for the whole line. In particular, we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of H on a finite interval and knowledge of q over a corresponding fraction of the interval. The methods employed rest on Weyl mfunction techniques and densities of zeros of a class of entire functions.