“Parseval's Theorem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Parseval's Theorem

释义

Parseval's Theorem

Let be a continuous function and and be Fourier Transform pairs so that

			(1)
			(2)

Then

where

is the Delta Function.

For finite Fourier Transform pairs and ,

(4)

If a function has a Fourier Series given by

(5)

then Bessel's Inequality becomes an equality known as Parseval's theorem. From (5),

Integrating

(8)

For a generalized Fourier Series with a Complete Basis

, an analogousrelationship holds. For a Complex Fourier Series,

(9)

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1101, 1979.

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