By the expansions (11) and (14), using Parseval equality
, we obtain that
and applying the Parseval equality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we obtain
Using the last theorem and Parseval equality, we obtain
By the Parseval equality
we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
[K.sub.d] ([psi], [chi]) can be evaluated either by direct integration or using convolution formula and Parseval equality for harmonic analysis on [S.sub.d-1].
By Parseval equality for harmonic analysis on [S.sub.d-1] we obtain the following expression:
Based on Parseval equality
(4), instead of (3) it is possible to write the following:
Since the spectrum is simple and discrete, then Parseval equality
(6) yields for any sequence of values F([[lambda].sub.n]) such that
Then the Fourier coefficients of the difference [f.sub.1] - f are zero and applying the Parseval equality
(3.9) to the function [f.sub.1] - f we get that [f.sub.1] - f = 0, so that the sum of series (4.1) is equal to f(t).
and that f (b) = 0 in case b is left-scattered, we have the Parseval equality (see )
We will show that the Parseval equality for problem (1.1), (1.2) can be obtained from (3.4) by letting b [right arrow] [infinity].
Now we apply Parseval equality (3.4) to the function (assuming b > [t.sub.0])