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Related Experiment Videos

On localization for double Fourier series.

C Goffman1, D Waterman

  • 1Department of Mathematics, Purdue University, West Lafayette, Indiana 47906.

Proceedings of the National Academy of Sciences of the United States of America
|February 1, 1978
PubMed
Summary

Researchers explored Fourier series localization principles for multivariable functions. They introduced generalized bounded variation (LambdaBV) and harmonic bounded variation (HBV) to demonstrate that localization holds for rectangular partial sums only when LambdaBV equals HBV.

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Area of Science:

  • Mathematical Analysis
  • Harmonic Analysis
  • Fourier Series Theory

Background:

  • Localization theorems for Fourier series are well-established for single-variable functions.
  • Extending these theorems to multivariable functions, even with rectangular partial sums, presents significant challenges.
  • Classical methods are insufficient for understanding localization in higher dimensions.

Purpose of the Study:

  • To investigate the conditions under which localization principles hold for Fourier series of multivariable functions.
  • To introduce and analyze a new concept of generalized bounded variation (LambdaBV) for functions of several variables.
  • To determine the relationship between LambdaBV, harmonic bounded variation (HBV), and the validity of localization for rectangular and square partial sums.

Main Methods:

Related Experiment Videos

  • Introduction of the Lambda-bounded variation (LambdaBV) concept for functions on R(1) based on a sequence {lambda(n)}.
  • Definition of harmonic bounded variation (HBV) as a special case of LambdaBV (lambda(n)=n).
  • Extension of the LambdaBV definition to functions of several variables and analysis of its properties in the context of Fourier series partial sums.

Main Results:

  • The localization principle for rectangular partial sums of two-variable Fourier series holds if and only if the function space is LambdaBV = HBV.
  • If LambdaBV is not a subset of HBV, the localization principle fails for LambdaBV functions, even for square partial sums.
  • The study establishes a precise condition for the localization property in multivariable Fourier analysis.

Conclusions:

  • The concept of generalized bounded variation (LambdaBV) is crucial for understanding localization in multivariable Fourier series.
  • The equivalence of LambdaBV and HBV is a necessary and sufficient condition for the localization principle with rectangular partial sums.
  • This work provides new insights into the behavior of Fourier series for functions of several variables, highlighting the limitations of classical approaches.