Fourier Series in Several Variables with Applications to Partial Differential Equations: 1st Edition (Hardback) book cover

Fourier Series in Several Variables with Applications to Partial Differential Equations

1st Edition

By Victor Shapiro

Chapman and Hall/CRC

352 pages

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Hardback: 9781439854273
pub: 2011-03-28
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Fourier Series in Several Variables with Applications to Partial Differential Equations illustrates the value of Fourier series methods in solving difficult nonlinear partial differential equations (PDEs). Using these methods, the author presents results for stationary Navier-Stokes equations, nonlinear reaction-diffusion systems, and quasilinear elliptic PDEs and resonance theory. He also establishes the connection between multiple Fourier series and number theory.

The book first presents four summability methods used in studying multiple Fourier series: iterated Fejer, Bochner-Riesz, Abel, and Gauss-Weierstrass. It then covers conjugate multiple Fourier series, the analogue of Cantor’s uniqueness theorem in two dimensions, surface spherical harmonics, and Schoenberg’s theorem. After describing five theorems on periodic solutions of nonlinear PDEs, the text concludes with solutions of stationary Navier-Stokes equations.

Discussing many results and studies from the literature, this book demonstrates the robust power of Fourier analysis in solving seemingly impenetrable nonlinear problems.


I found the book interesting … Certainly, all readers and even the experts will find something new and relevant in it.

—Atanas G. Stefanov, Mathematical Reviews, Issue 2012a

Table of Contents

Summability of Multiple Fourier Series


Iterated Fejer Summability of Fourier Series

Bochner-Riesz Summability of Fourier Series

Abel Summability of Fourier Series

Gauss-Weierstrass Summability of Fourier Series

Further Results and Comments

Conjugate Multiple Fourier Series


Abel Summability of Conjugate Series

Spherical Convergence of Conjugate Series

The Cα-Condition

An Application of the Cα-Condition

An Application of the Lp–Condition

Further Results and Comments

Uniqueness of Multiple Trigonometric Series

Uniqueness for Abel Summability

Uniqueness for Circular Convergence

Uniqueness, Number Theory, and Fractals

Further Results and Comments

Positive Definite Functions

Positive Definite Functions on SN-l

Positive Definite Functions on TN

Positive Definite Functions on SN1-l × TN

Further Results and Comments

Nonlinear Partial Differential Equations

Reaction-Diffusion Equations on the N-Torus

Quasilinear Ellipticity on the N-Torus

Further Results and Comments

The Stationary Navier-Stokes Equations

Distribution Solutions

Classical Solutions

Further Results and Comments

Appendix A: Integrals and Identities

Appendix B: Real Analysis

Appendix C: Harmonic and Subharmonic Functions



About the Author

Victor L. Shapiro is a Distinguished Professor Emeritus in the Department of Mathematics at the University of California, Riverside, where he has taught for 46 years. He earned his Ph.D. from the University of Chicago and completed postdoctoral work at the Institute for Advanced Study, where he was an NSF fellow.

About the Series

Chapman & Hall/CRC Applied Mathematics & Nonlinear Science

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Differential Equations
MATHEMATICS / Functional Analysis