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(1). This problem involves Fourier Transforms and the function: [1, [x] <1 0, 1x1 ≥1 1 f(x) = (a). Compute the Fourier Transform F(k)=√√√f(x)exp(-ikx)dx. (This is the symmetric convention for the 2л that was introduced in class. You can use whatever convention you prefer, as long as you are consistent.) 00 S g(x')h(x − x')dx'. -00 By direct integration, find the convolution of f(x) with itself, i.e. compute (ƒ* f)(x). (b). Define the convolution of g(x) with h(x) to be (g* h)(x) 1 √2л =

(1). This problem involves Fourier Transforms and the function: [1, [x] <1 0, 1x1 ≥1 1 f(x) = (a). Compute the Fourier Transform F(k)=√√√f(x)exp(-ikx)dx. (This is the symmetric convention for the 2л that was introduced in class. You can use whatever convention you prefer, as long as you are consistent.) 00 S g(x')h(x − x')dx'. -00 By direct integration, find the convolution of f(x) with itself, i.e. compute (ƒ* f)(x). (b). Define the convolution of g(x) with h(x) to be (g* h)(x) 1 √2л =

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 48E
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(1). This problem involves Fourier Transforms and the function: ¸ [x] <1 [0, [x] ≥1 f(x) = (a). Compute the Fourier Transform F(k)=f(x)exp(-ikx)dx. (This is the symmetric convention for the 2л that was introduced in class. You can use whatever convention you prefer, as long as you are consistent.) -00 (b). Define the convolution of g(x) with h(x) to be (g* h)(x) = √27 S 8(x')h(x − x')dx'. -00 By direct integration, find the convolution of ƒ(x) with itself, i.e. compute (ƒ * ƒ)(x). (c). Find the Fourier Transform of the convolution of f(x) with itself, i.e. compute the Fourier Transform of (ƒ * f)(x). (Hint: this step does not require any further integration.) (d). Use these results and Parseval's Theorem to evaluate the integral 1-S sin kdk. = 4 -00 kª
Transcribed Image Text:(1). This problem involves Fourier Transforms and the function: ¸ [x] <1 [0, [x] ≥1 f(x) = (a). Compute the Fourier Transform F(k)=f(x)exp(-ikx)dx. (This is the symmetric convention for the 2л that was introduced in class. You can use whatever convention you prefer, as long as you are consistent.) -00 (b). Define the convolution of g(x) with h(x) to be (g* h)(x) = √27 S 8(x')h(x − x')dx'. -00 By direct integration, find the convolution of ƒ(x) with itself, i.e. compute (ƒ * ƒ)(x). (c). Find the Fourier Transform of the convolution of f(x) with itself, i.e. compute the Fourier Transform of (ƒ * f)(x). (Hint: this step does not require any further integration.) (d). Use these results and Parseval's Theorem to evaluate the integral 1-S sin kdk. = 4 -00 kª
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