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13
Diffusion polynomial frames on metric measure spaces. submitted
, 2006
"... We construct a multiscale tight frame based on an arbitrary orthonormal basis for the L 2 space of an arbitrary sigma finite measure space. The approximation properties of the resulting multiscale are studied in the context of Besov approximation spaces, which are characterized both in terms of suit ..."
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Cited by 23 (6 self)
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We construct a multiscale tight frame based on an arbitrary orthonormal basis for the L 2 space of an arbitrary sigma finite measure space. The approximation properties of the resulting multiscale are studied in the context of Besov approximation spaces, which are characterized both in terms of suitable K–functionals and the frame transforms. The only major condition required is the uniform boundedness of a summabilility operator. We give sufficient conditions for this to hold in the context of a very general class of metric measure spaces. The theory is illustrated using the approximation of characteristic functions of caps on a dumbell manifold, and applied to the problem of recognition of hand–written digits. Our methods outperforms comparable methods for semi–supervised learning.
Polynomial operators and local smoothness classes on the unit interval
 Journal of Approximation Theory
"... We prove the existence of quadrature formulas exact for integrating high degree polynomials with respect to Jacobi weights based on scattered data on the unit interval. We also obtain a characterization of local Besov spaces using the coefficients of a tight frame expansion. 1 ..."
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Cited by 20 (11 self)
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We prove the existence of quadrature formulas exact for integrating high degree polynomials with respect to Jacobi weights based on scattered data on the unit interval. We also obtain a characterization of local Besov spaces using the coefficients of a tight frame expansion. 1
Parameter estimation for exponential sums by approximate Prony method
, 2009
"... The recovery of signal parameters from noisy sampled data is a fundamental problem in digital signal processing. In this paper, we consider the following spectral analysis problem: Let f be a real–valued sum of complex exponentials. Determine all parameters of f, i.e., all different frequencies, all ..."
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Cited by 13 (2 self)
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The recovery of signal parameters from noisy sampled data is a fundamental problem in digital signal processing. In this paper, we consider the following spectral analysis problem: Let f be a real–valued sum of complex exponentials. Determine all parameters of f, i.e., all different frequencies, all coefficients, and the number of exponentials from finitely many equispaced sampled data of f. This is a nonlinear inverse problem. In this paper, we present new results on an approximate Prony method (APM) which is based on [1]. In contrast to [1], we apply matrix perturbation theory such that we can describe the properties and the numerical behavior of the APM in detail. The first part of APM estimates the frequencies and the second part solves an overdetermined linear Vandermonde–type system in a stable way. We compare the first part of APM also with the known ESPRIT method. The second part is related to the nonequispaced fast Fourier transform (NFFT). Numerical experiments show the performance of our method.
Localized linear polynomial operators and quadrature formulas on the sphere
 SIAM J. Numer. Anal
"... the sphere ..."
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Polynomial operators for spectral approximation of piecewise analytic functions
, 2009
"... functions ..."
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Polynomial frames: a fast tour
 In Approximation theory XI: Gatlinburg 2004, Mod. Methods Math. Nashboro
, 2005
"... Abstract. We present a unifying theme in an abstract setting for some of the recent work on polynomial frames on the circle, the unit interval, the real line, and the Euclidean sphere. In particular, we describe a construction of a tight frame in the abstract setting, so that certain Besov approxima ..."
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Cited by 5 (2 self)
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Abstract. We present a unifying theme in an abstract setting for some of the recent work on polynomial frames on the circle, the unit interval, the real line, and the Euclidean sphere. In particular, we describe a construction of a tight frame in the abstract setting, so that certain Besov approximation spaces can be characterized using the absolute values of the frame coefficients. We discuss the localization properties of the frames in the context of trigonometric, Jacobi, and spherical polynomials, and discuss some applications. Wavelet analysis is perhaps one of the fastest growing areas of analysis. Some of the applications include image processing, signal processing, time series analysis, probability density estimation, neural networks, data compression, etc. Traditionally [4, 6, 39], wavelets are defined using the notion
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, 811
"... Localized linear polynomial operators and quadrature formulas on the sphere ..."
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Localized linear polynomial operators and quadrature formulas on the sphere
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"... On a filter for exponentially localized kernels based on Jacobi polynomials ..."
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On a filter for exponentially localized kernels based on Jacobi polynomials
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"... On a filter for exponentially localized kernels based on Jacobi polynomials ..."
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On a filter for exponentially localized kernels based on Jacobi polynomials