Harmonic Analysis: From Fourier to Wavelets
In the last 200 years, harmonic analysis has been one of the most influential bodies of mathematical ideas, having been exceptionally significant both in its theoretical implications and in its enormous range of applicability throughout mathematics, science, and engineering. In this book, the authors convey the remarkable beauty and applicability of the ideas that have grown from Fourier theory. They present for an advanced undergraduate and beginning graduate student audience the basics of harmonic analysis, from Fourier's study of the heat equation, and the decomposition of functions into sums of cosines and sines (frequency analysis), to dyadic harmonic analysis, and the decomposition of functions into a Haar basis (time localization). While concentrating on the Fourier and Haar cases, the book touches on aspects of the world that lies between these two different ways of decomposing functions: time-frequency analysis (wavelets). Both finite and continuous perspectives are presented, allowing for the introduction of discrete Fourier and Haar transforms and fast algorithms, such as the Fast Fourier Transform (FFT) and its wavelet analogues. The approach combines rigorous proof, inviting motivation, and numerous applications. Over 250 exercises are included in the text. Each chapter ends with ideas for projects in harmonic analysis that students can work on independently. This book is published in cooperation with IAS/Park City Mathematics Institute.
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Pointwise convergence of Fourier series
Meansquare convergence of Fourier series
A tour of discrete Fourier and Haar analysis
The Fourier transform in paradise
algorithm approximation bounded interval boundedness Cesàro Chapter coeﬁicients compactly supported complete complex numbers continuous functions converge uniformly convolution deﬁned Deﬁnition denote differentiable dilation equation Dirichlet kernel dyadic intervals example Exercise Fejér kernel ﬁlter filter coefficients finite form an orthonormal formula Fourier coefficients Fourier series Fourier side Fourier transform frequency func function f functions of moderate given Haar basis Haar functions Hilbert space Hilbert transform inner product integrable functions inverse Lebesgue integrable Lemma linear Lp norm Lp spaces Lp(I Lp(R Lp(T mathematician matrix moderate decrease multiplication operator orthogonal MRA orthogonal projection orthonormal basis Parseval’s Identity partial Fourier sums Plancherel’s Identity pointwise Poisson kernel project in Section proof properties prove real numbers Riemann integrable scaling function Schwartz functions sequence Show signal step functions subspace tempered distribution Theorem time–frequency tion trigonometric polynomials Verify wavelet transform