Fourier series, fourier transforms, and function spaces : a second course in analysis

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書誌事項

Fourier series, fourier transforms, and function spaces : a second course in analysis

Tim Hsu

（AMS/MAA textbooks, v. 59）

MAA Press, an imprint of the American Mathematical Society, c2020

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注記

Includes bibliographical references (p. 343-345) and index

内容説明・目次

内容説明

Fourier Series, Fourier Transforms, and Function Spaces is designed as a textbook for a second course or capstone course in analysis for advanced undergraduate or beginning graduate students. By assuming the existence and properties of the Lebesgue integral, this book makes it possible for students who have previously taken only one course in real analysis to learn Fourier analysis in terms of Hilbert spaces, allowing for both a deeper and more elegant approach. This approach also allows junior and senior undergraduates to study topics like PDEs, quantum mechanics, and signal processing in a rigorous manner. Students interested in statistics (time series), machine learning (kernel methods), mathematical physics (quantum mechanics), or electrical engineering (signal processing) will find this book useful. With 400 problems, many of which guide readers in developing key theoretical concepts themselves, this text can also be adapted to self-study or an inquiry-based approach. Finally, of course, this text can also serve as motivation and preparation for students going on to further study in analysis.

目次

Overture Complex functions of a real variable: Real and complex numbers Complex-valued calculus Series of functions Fourier series and Hilbert spaces: The idea of a function space Fourier series Hilbert spaces Convergence of Fourier series Operators and differential equations: PDEs and diagonalization Operators on Hilbert spaces Eigenbases and differential equations The Fourier transform and beyond: The Fourier transform Applications of the Fourier transform What's next? Rearrangements of series Linear algebra Bump functions Suggestions for problems Bibliography Index of selected notation Index.

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