FourierTest1 - Topics for Fourier Analysis Test Rules: All...

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Topics for Fourier Analysis Test Rules: All of the following topics can be answered. Proofs of the theorems, or demonstrations of the problems. A perfect grade will be 50. All problems and theorems are worth 10 points, except the uncertainty principle, which is worth 20. Theorem 0.1 (Differentiation) Let f ( t ) L 2 ( R ) and let ˆ f ( s ) L 2 ( R ) be its Fourier transform. If ( - is ) ˆ f ( s ) L 2 ( R ) , then it follows that f ± ( t ) exists as an element of L 2 ( R ) , and that its Fourier transform is F ( f ± ( t )) = ( - is ) ˆ f ( s ) . Theorem 0.2 (Translation) If f ( t ) L 2 ( R ) and ˆ f ( s ) is its Fourier trans- form then F ( T a ( f ( t )) = F ( f ( t - a )) = ˆ f ( s ) e isa . (1) Theorem 0.3 (Convolution Theorem) If f ( t ) and g ( t ) are elements of L 2 ( R ) , then the convolution f * g ( t ) is also an element of L 2 ( R ) . More importantly, the Fourier transform of f * g is given simply by ˆ f ( s g ( s ) , or pointwise multiplication of the corresponding Fourier transforms. Mathematically we have
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This note was uploaded on 01/06/2011 for the course MAP 4413 taught by Professor Olsen during the Spring '10 term at University of Florida.

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FourierTest1 - Topics for Fourier Analysis Test Rules: All...

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