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Approximation of bivariate functions via smooth extensions.

Zhihua Zhang1

  • 1College of Global Change and Earth System Science, Beijing Normal University, Beijing 100875, China.

Thescientificworldjournal
|April 1, 2014
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Researchers developed a novel extension method for bivariate functions on complex domains, enabling accurate Fourier and wavelet approximations. This technique significantly reduces approximation errors for functions on irregular shapes.

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Area of Science:

  • Numerical Analysis
  • Approximation Theory
  • Harmonic Analysis

Background:

  • Approximating smooth bivariate functions on general domains with arbitrary shapes using Fourier or wavelet methods is challenging.
  • Existing techniques struggle with the geometric complexity of these domains, leading to potential inaccuracies.
  • The need for efficient and accurate approximation methods for such functions is critical in various scientific and engineering fields.

Purpose of the Study:

  • To develop a method for extending bivariate functions defined on general, arbitrarily shaped domains to smooth, periodic, or compactly supported functions in the entire space.
  • To enable accurate Fourier and wavelet approximations for these functions on their original domains.
  • To achieve fast decay of Fourier/wavelet coefficients and vanishing wavelet coefficients for improved approximation accuracy.

Main Methods:

  • Extension of bivariate functions from general domains to smooth, periodic functions on the whole space.
  • Extension of bivariate functions to smooth, compactly supported functions on the whole space.
  • Expansion of extended functions into Fourier series, periodic wavelet series, or wavelet series.
  • Utilizing polynomial-based extension tools and the moment theorem.

Main Results:

  • The proposed extension methods yield smooth functions with simple representations.
  • The resulting Fourier and wavelet coefficients exhibit rapid decay.
  • The moment theorem, applied to polynomial-based extensions, leads to vanishing wavelet coefficients.
  • Accurate Fourier and wavelet approximations with small errors are achieved for bivariate functions on general domains.

Conclusions:

  • The developed extension technique effectively overcomes the difficulties in approximating bivariate functions on complex domains.
  • The method provides a robust framework for applying Fourier and wavelet analysis to functions on arbitrary shapes.
  • The fast decay and vanishing properties of coefficients ensure high-accuracy approximations, beneficial for scientific computations.