Shapiro Fourier Series in Several Variables with Applications to Partial Differential Equations

1. Auflage 2011
ISBN: 978-1-4398-5428-0
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 352 Seiten

Reihe: Chapman & Hall/CRC Applied Mathematics & Nonlinear Science

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.

Zielgruppe

Graduate students and professionals in mathematics; applied mathematicians and physicists.

Autoren/Hrsg.

Shapiro, Victor

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

Summability of Multiple Fourier Series
Introduction
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
Introduction
Abel Summability of Conjugate Series
Spherical Convergence of Conjugate Series
The Ca-Condition
An Application of the Ca-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

Bibliography
Index

Über Autor(innen)

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.