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The Gibbs Phenomenon in Higher Dimensions

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Book cover Harmonic Analysis and Applications

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

The concept of star discontinuity is defined for functions of several variables. A star discontinuity in dimension one is simply a jump discontinuity. It is then shown that in arbitrary dimensions the Gibbs phenomenon for square convergence occurs for periodic functions satisfying appropriate hypotheses at star discontinuities.

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References

  1. M. T. Cheng, The Gibbs phenomenon and Bochner’s summation method. I., Duke Math J., 17 (1950), pp. 83–90; The Gibbs phenomenon and Bochner’s summation method. II., Duke Math J., 17 (1950), pp. 477–490.

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  2. L. Colzani and M. Vignati, The Gibbs phenomenon for multiple Fourier integrals, J. Approx. Theory, 80 (1995), pp. 119–131.

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  3. B. I. Golubov, On the Gibbs phenomenon for Riesz spherical means of multiple Fourier series and Fourier integrals, Anal. Math., 1 (1975), pp. 31–53.

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  4. G. Helmberg, A corner point Gibbs phenomenon for Fourier series in two dimensions, J. Approx. Theory, 100 (1999), pp. 1–43.

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  5. G. Helmberg, Localization of a corner point Gibbs phenomenon for Fourier series in two dimensions, J. Fourier Anal. Appl., 8 (2002), pp. 29–41.

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  6. H. Weyl, Die Gibbs’sche Erscheinung in der Theorie der Kugelfunktionen, Rend. Circ. Mat. Palermo, 29 (1910), pp. 308–323.

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  7. H. Weyl, Uber die Gibbs’sche Erscheinung und verwandte Konvergenzphanomene, Rend. Circ. Mat. Palermo, 30 (1910), pp. 377–407.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Georgetown University, Washington, DC, 20057, USA

    George Benke

Authors
  1. George Benke
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Editor information

Editors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Christopher Heil

Additional information

Dedicated to John Benedetto—mentor, scholar, friend.

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© 2006 Birkhäuser Boston

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Benke, G. (2006). The Gibbs Phenomenon in Higher Dimensions. In: Heil, C. (eds) Harmonic Analysis and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4504-7_1

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