A Basis Theory Primer: Expanded EditionThe classical subject of bases in Banach spaces has taken on a new life in the modern development of applied harmonic analysis. This textbook is a self-contained introduction to the abstract theory of bases and redundant frame expansions and their use in both applied and classical harmonic analysis. The four parts of the text take the reader from classical functional analysis and basis theory to modern time-frequency and wavelet theory. Extensive exercises complement the text and provide opportunities for learning-by-doing, making the text suitable for graduate-level courses. The self-contained presentation with clear proofs is accessible to graduate students, pure and applied mathematicians, and engineers interested in the mathematical underpinnings of applications. No other text develops the ties between classical basis theory and its modern uses in applied harmonic analysis. |
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1-periodic Banach space bases basis for H basis for LČ(R Bessel sequence biorthogonal system Cauchy closed span closed subspace Cnxn coefficient functionals compact compactly supported complete continuous function converges unconditionally Corollary define Definition denote dense equivalent Example Exercise exists finite finite-dimensional Fourier series Fourier transform frame bounds function ƒ Gabor frames Gabor systems given Haar system Hamel basis harmonic analysis Hence Hilbert space implies Inequality infinite inner product integrable isometric Lč(R Lč(T LČ(T Lebesgue Lemma Lp(T mapping nonzero normed space Notation operator norm orthonormal basis Parseval frame partial sums properties Prove PW(R range(C refinable function refinement equation Riesz basis satisfies scalars scaling function Schauder basis Show space H subset Suppose topological isomorphism translates trigonometric system unconditional basis unique vector space Vo(g wavelet wavelet set zero