Parseval–Gutzmer formula - Alchetron, the free social encyclopedia

In mathematics, the Parseval–Gutzmer formula states that, if ƒ is an analytic function on a closed disk of radius r with Taylor series

Proof

The Cauchy Integral Formula for coefficients states that for the above conditions:

a n = 1 2 π i ∫ γ f ( z ) z n + 1 d z

where γ is defined to be the circular path around 0 of radius r. We also have that, for x in the complex plane C,

x ¯ x = | x | 2

We can apply both of these facts to the problem. Using the second fact,

∫ 0 2 π | f ( r e i ϑ ) | 2 d ϑ = ∫ 0 2 π f ( r e i ϑ ) f ( r e i ϑ ) ¯ d ϑ

Now, using our Taylor Expansion on the conjugate,

= ∫ 0 2 π f ( r e i ϑ ) ∑ k = 0 ∞ a k ( r e i ϑ ) k ¯ d ϑ

Using the uniform convergence of the Taylor Series and the properties of integrals, we can rearrange this to be

= ∑ k = 0 ∞ ∫ 0 2 π f ( r e i ϑ ) a k ¯ ( r k ) ( e i ϑ ) k , d ϑ

With further rearrangement, we can set it up ready to use the Cauchy Integral Formula statement

= ∑ k = 0 ∞ ( 2 π a k ¯ r 2 k ) ( 1 2 π i ∫ 0 2 π f ( r e i ϑ ) ( r e i ϑ ) k + 1 r i e i ϑ ) d ϑ

Now, applying the Cauchy Integral Formula, we get

= ∑ k = 0 ∞ ( 2 π a k ¯ r 2 k ) a k = 2 π ∑ k = 0 ∞ | a k | 2 r 2 k

Using this formula, it is possible to show that

∑ k = 0 ∞ | a k | 2 r 2 k ≤ M r 2 where M r = sup { | f ( z ) | : | z | = r }

This is done by using the integral

∫ 0 2 π | f ( r e i ϑ ) | 2 d ϑ ≤ 2 π | m a x ϑ ∈ [ 0 , 2 π ) ( f ( r e i ϑ ) ) | 2 = 2 π | m a x | z | = r ( f ( z ) ) | 2 = 2 π ( M r ) 2

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