By Robert M. Young

An creation to Non-Harmonic Fourier sequence, Revised variation is an replace of a well known and hugely revered vintage textbook.Throughout the publication, fabric has additionally been additional on fresh advancements, together with balance concept, the body radius, and purposes to sign research and the regulate of partial differential equations.

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**Example text**

While it is of course true that every result of the preceding section applies equally well to orthonormal bases, the added structure of a Hilbert space provides additional and stronger stability criteria. We begin by reformulating Theorem 10. Theorem 13. Let ( e n }he an orthonormal basis for a Hilbert space H and let { be "close" to {en}in the sense that ff1} I , 0 2 I < 1, and urbitrury scalars c l , . ,c, ( n = 1,2,3,. ). for H . 1 Application : The stability of the trigonometric system It follows from what has already been done that the trigonometric system { e i n f } ?

For fixed k and m > n > k we have by hypothesis As m ,n + r y j , this last expression tends to zero. Since x k # 0, it follows that for each k the sequence { c , ~ } ; = ~is a Cauchy sequence, and therefore the limit exists. It is now a routine matter to show that The details are left to the reader. We complete the proof by showing that this representation is unique. 26 Bases in Banach Spaces [Ch. 1 It suffices to show that if C k X k = 0 , then ck = 0 for every k . For fixed k and n 2 k we have once again by hypothesis Letting n + 00, we find 1 CkXk 11 = 0, and hence ck = 0 for every k .

Example 2. mis complete in Lp[-n, n] for 1 S p < 00. e. t The terminology of the subject is not uniform-the terms “closed”, “total”, and “l’undamental” are also used. Sec. 51 21 Complete Sequences Example 3. In C[a,h ] the sequence of powers { 1, t , t 2 , . } is complete by virtue of the Weierstrass theorem on polynomial approximation : the polynomials are dense in C [ a , b ] . It follows that if y is an element of C [ a , h ] for which all of the "moments" l (n = 0, 1,2,. ) t"g(t) dt are zero, then y must be identically zero.