| 释义 |
Bessel's InequalityIf  is piecewise Continuous and has a general Fourier Series
 | (1) |
 , it must be true that
 | (2) |
 | (3) |
 | (4) |
 | (5) |
 | (6) |
 are not Complete. If they areComplete, then the inequality (2) becomes an equality, so (6) becomes anequality and is known as Parseval's Theorem. If has a simple Fourier Series expansion withCoefficients  ,  ,  ,  and  , ...,  , then
 | (7) |
 | (8) |
 in a superposition of Eigenfunctions of  ,  . Then
 | (9) |
 | (10) |
 and
 | (11) |
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 526-527, 1985.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1102, 1980.
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