Bessel inequality

In this relation we have obtained the counterpart to the Bessel inequality. For all possible expansions it selects in an invariant fashion the one with the smallest possible coefficient contributions. 

We hasten to add that we have introduced even more Bessel functions neither for completeness nor for the further aggrandizement of Friedrich Wilhelm Bessel (1784-1846), who, with a veritable zoo of functions to his credit, not to mention infinite series, a revered inequality, an interpolation scheme, and various other mathematical artifacts, needs no publicity as we shall see, (4.13) and (4.14) will save some labor, a sufficient reason for admitting more functions into our larder. 

As it follows from these inequalities, the argument of Bessel s function in the expression (21) is very small, and coth(fku/2kT) 1. The resulting formula can be deduced, if we take into account that AE/hat — p, where p is integer and Ip(x) = I-P(x) [14],... 

Let ( , be an inner product defined on a space X, and

[Pg.311]

If we recall the Fourier analysis, we understand that f = (/, are the Fourier coefficients. Note that Bessel s inequality (C.18) is reduced to an equality if we have... 

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