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

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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 = = + , 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 = + , where ( ) g and ( ) k 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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This note was uploaded on 09/23/2009 for the course MATH 200 taught by Professor Gur during the Spring '09 term at Accreditation Commission for Acupuncture and Oriental Medicine.

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

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