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H12 - H12 Spectral Analysis 1 A function that satisfies the...

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Unformatted text preview: H12 Spectral Analysis 1. A function that satisfies the equation, ( ) ( ) f x f x k p = + , all x , 0, 1, 2, ... k = ± ± , is called periodic with period p , if p is the smallest number such that the equation holds for all x . Virtually any periodic function ( ) f x may be written as a series in the form [ ] ( ) cos(2 / ) sin(2 / ) r r r f x a r x p b r x p π π ∞ = = + ∑ , where , are constants which may be determined from the form of . 1 1 , , ..., , ,... , a a b b ( ) g x 2. A non-periodic function can be written as { } ( ) ( ) cos ( )sin f x g x k x d ω ω ω ω ∞ = + ∫ ω , where ( ) g ω and ( ) k ω may be determined from the form of ( ) f x . The required conditions is, roughly speaking, that ( ) f x must “die away” as x → ±∞ . More precisely, if ( ) f x is “absolutely integrable”, i.e. if | ( ) | f x dx ∞ −∞ < ∞ ∫ . 3. Let be a sequence of n numbers....
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H12 - H12 Spectral Analysis 1 A function that satisfies the...

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