Download e-book for iPad: Fourier Transforms. by Salomon Bochner, Komaravolu Chandrasekharan

By Salomon Bochner, Komaravolu Chandrasekharan

ISBN-10: 0691095787

ISBN-13: 9780691095783

The description for this ebook, Fourier Transforms. (AM-19), might be forthcoming.

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Are orthonormal bases of 0. If {ek) and Indeed, let f= ® L{f1, . and L( g1, . f = k,1 . . 4) cJ, j and thus >0. >) is a pre-Hilbert space. The completion of this pre-Hilbert space (cf. 3) will be denoted by H1® H2 and called Therefore (H1 <. , the (complete) tensor product of the Hubert spaces H1 and H2. 5) k 1 where (ek) and {efl are orthonormal systems in L(f1, .. , f,j and and L( g1, are contained in , g,,}, respectively; the elements . . L(f1, . and L( g1, . EXAMPLE 1. Let Pi and .

If T1 and T2 are subspaces of a pre-Hilbert space with T1 I T2, then we have T1 n T2 = (0). In this case we call the direct sum T1 -I- T2 an orthogonal sum and we denote it by T1 T2. 3. (a) Let H be a pre-Hilbert space, and let T1 and T2 be orthogonal subspaces. If T1 T2 is closed, then T1 and T2 are closed. (b) If H is a Hubert space and T1, T2 are closed orthogonal subspaces, then T1 T2 is closed. (c) If H is a Hilbert space and T and T1 are closed subspaces such that such that T2 C T, T1 C T, then there exists exactly one closed subspace T2 J..

Such that g± L(e1, .. 0= thus E L(e1,.. + g =— = l,j . =L(f1,.. , then have i= + 1, .. , n, necessarily has the form Consequently, 10 bn+1 1=1 > 0, therefore +d +11 —1 = 1. Consequently, n = j=i 1=1 E L(f1, construction, we have From the formula for c L(f1,. L(e1, . , i)' therefore we also have L(f1, . By . . L(e1,.. it follows that hence . , . . , C L(e1, . . , E , D 1) the set F= nE with x" is a linearly independent system.

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Fourier Transforms. by Salomon Bochner, Komaravolu Chandrasekharan


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