Read e-book online An Introduction to Nonharmonic Fourier Series (Pure and PDF By Robert M. Young

ISBN-10: 0080874096

ISBN-13: 9780080874098

ISBN-10: 0127728503

ISBN-13: 9780127728506

An creation to Non-Harmonic Fourier sequence, Revised variation is an replace of a well-known and hugely revered vintage textbook.Throughout the e-book, fabric has additionally been additional on contemporary advancements, together with balance thought, the body radius, and functions to sign research and the keep an eye on of partial differential equations.

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Extra resources for An Introduction to Nonharmonic Fourier Series (Pure and Applied Mathematics (Academic Pr))

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1 however, it is sufficient to know that a sequence of vectors, although not necessarily a basis, does nevertheless have this approximation property. Such sequences will be called complete. Definition. A sequence of vectors {xl, x2, x 3 , . } in a normed vector space X is said to be completet fi its linear span is dense in X , that is, if for each vector x and each E > 0 there is a finite linear combination c 1x, + ’ . cnx, such that + IIX - (C’XI + . . + c,xn)II < E. It is a direct and important consequence of the Hahn-Banach theorem that the completeness of a sequence of vectors {x,} in X is equivalent to the following condition : if p E X * (the topological dual of X ) and if p ( x , ) = 0 ( n = 1,2,3,.

A) Show that (x,} has a biorthogonal sequence if and only if it is minimal. (b) Show that a biorthogonal sequence for [xln} is uniquely determined if and only if { x,} is complete in X . (c) Show that { x l l } is a basis for X if and only if it possesses a bif,(x)x, converges to x orthogonal sequence (1,)such that for every x in X . 3. (Karlin) Let {x,} be a sequence of elements of a Banach space X and let be biorthogonal to {x,,}. Prove that if { f , } is a basis for X * , then {xll) is a basis for X .

This proves that X and Y are isomorphic. 24 Bases in Banach Spaces Suppose now that x Then for every n, = cp==l cnxnis a fixed but [Ch. 1 arbitrary element of X. This proves that each f , is continuous and that IIf,II 5 2IIT-'Il/llx,II. Choosing M = 2 1 T - 11, we have IIxn II . I f n II 5 M for every n. The remaining inequality is trivial: Definition. Let {x,} be a basis for a Banach space X and let i f , } be the associated sequence of coeflcient functionals. For n = 1,2,3,. . the nth partial sum operator S, is the linear operator on X defined by Theorem 3 implies that each partial sum operator is bounded, and a closer examination of the proof reveals further that As an application of this fact, we establish a simple and useful criterion for determining when a complete sequence is a basis.