Theory of the Optimum Interpolation Approximation in a Shift-Invariant Wavelet and Scaling Subspace

Yuichi KIDA; Takuro KIDA

doi:10.1093/ietfec/e90-a.9.1885

Theory of the Optimum Interpolation Approximation in a Shift-Invariant Wavelet and Scaling Subspace

Yuichi KIDA, Takuro KIDA

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In the main part of this paper, we present a systematic discussion for the optimum interpolation approximation in a shift-invariant wavelet and/or scaling subspace. In this paper, we suppose that signals are expressed as linear combinations of a large number of base functions having unknown coefficients. Under this assumption, we consider a problem of approximating these linear combinations of higher degree by using a smaller number of sample values. Hence, error of approximation happens in most cases. The presented approximation minimizes various worst-case measures of approximation error at the same time among all the linear and the nonlinear approximations under the same conditions. The presented approximation is quite flexible in choosing the sampling interval. The presented approximation uses a finite number of sample values and satisfies two conditions for the optimum approximation presented in this paper. The optimum approximation presented in this paper uses sample values of signal directly. Hence, the presented result is independent from the so-called initial problem in wavelet theory.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E90-A No.9 pp.1885-1903

Publication Date: 2007/09/01

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Online ISSN: 1745-1337

DOI: 10.1093/ietfec/e90-a.9.1885

Type of Manuscript: PAPER

Category: Digital Signal Processing

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Yuichi KIDA, Takuro KIDA, "Theory of the Optimum Interpolation Approximation in a Shift-Invariant Wavelet and Scaling Subspace" in IEICE TRANSACTIONS on Fundamentals, vol. E90-A, no. 9, pp. 1885-1903, September 2007, doi: 10.1093/ietfec/e90-a.9.1885.
Abstract: In the main part of this paper, we present a systematic discussion for the optimum interpolation approximation in a shift-invariant wavelet and/or scaling subspace. In this paper, we suppose that signals are expressed as linear combinations of a large number of base functions having unknown coefficients. Under this assumption, we consider a problem of approximating these linear combinations of higher degree by using a smaller number of sample values. Hence, error of approximation happens in most cases. The presented approximation minimizes various worst-case measures of approximation error at the same time among all the linear and the nonlinear approximations under the same conditions. The presented approximation is quite flexible in choosing the sampling interval. The presented approximation uses a finite number of sample values and satisfies two conditions for the optimum approximation presented in this paper. The optimum approximation presented in this paper uses sample values of signal directly. Hence, the presented result is independent from the so-called initial problem in wavelet theory.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1093/ietfec/e90-a.9.1885/_p

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@ARTICLE{e90-a_9_1885,
author={Yuichi KIDA, Takuro KIDA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Theory of the Optimum Interpolation Approximation in a Shift-Invariant Wavelet and Scaling Subspace},
year={2007},
volume={E90-A},
number={9},
pages={1885-1903},
abstract={In the main part of this paper, we present a systematic discussion for the optimum interpolation approximation in a shift-invariant wavelet and/or scaling subspace. In this paper, we suppose that signals are expressed as linear combinations of a large number of base functions having unknown coefficients. Under this assumption, we consider a problem of approximating these linear combinations of higher degree by using a smaller number of sample values. Hence, error of approximation happens in most cases. The presented approximation minimizes various worst-case measures of approximation error at the same time among all the linear and the nonlinear approximations under the same conditions. The presented approximation is quite flexible in choosing the sampling interval. The presented approximation uses a finite number of sample values and satisfies two conditions for the optimum approximation presented in this paper. The optimum approximation presented in this paper uses sample values of signal directly. Hence, the presented result is independent from the so-called initial problem in wavelet theory.},
keywords={},
doi={10.1093/ietfec/e90-a.9.1885},
ISSN={1745-1337},
month={September},}

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TY - JOUR
TI - Theory of the Optimum Interpolation Approximation in a Shift-Invariant Wavelet and Scaling Subspace
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1885
EP - 1903
AU - Yuichi KIDA
AU - Takuro KIDA
PY - 2007
DO - 10.1093/ietfec/e90-a.9.1885
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E90-A
IS - 9
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - September 2007
AB - In the main part of this paper, we present a systematic discussion for the optimum interpolation approximation in a shift-invariant wavelet and/or scaling subspace. In this paper, we suppose that signals are expressed as linear combinations of a large number of base functions having unknown coefficients. Under this assumption, we consider a problem of approximating these linear combinations of higher degree by using a smaller number of sample values. Hence, error of approximation happens in most cases. The presented approximation minimizes various worst-case measures of approximation error at the same time among all the linear and the nonlinear approximations under the same conditions. The presented approximation is quite flexible in choosing the sampling interval. The presented approximation uses a finite number of sample values and satisfies two conditions for the optimum approximation presented in this paper. The optimum approximation presented in this paper uses sample values of signal directly. Hence, the presented result is independent from the so-called initial problem in wavelet theory.
ER -