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Fourier Analysis

Complex Fourier Series

Please see the attached file for the fully formatted problem. Let Y: R --> R be the periodic function whose restriction to [0,1] is X (0,1/2) - X(1/2,1) Y is an odd function. S 1--> 0 Y(x) cos 2pi kx dx = S 1/2-->-1/2 Y(x) cos 2pi kx dx Vk Conclude the the complex Fourier Series...can be expressed in the form...

Eignevalues and Eigenvectors of the Fourier Transform

Problem attached. "Eigenvalues and Eigenvectors of the Fourier Transform" Recall that the Fourier transform F is a linear one-to-one transformation from L2 (?cc, cc) onto itself. Let .. be an element of L2(?cc,cc). Let..= , the Fourier transform of.., be defined by ..... It is clear that ..... are square-integrable fu

Fourier Transform : Equivalent Width

"Equivalent Widths" Suppose we define for a square-integrable function f(t) and its Fourier transform ..... the equivalent width as .... and the equivalent Fourier width as .... a) Show that ..... is independent of the function f, and determine the value of this const. b) Determine the equivalent width and the equiva

Convolution of Fourier Transforms: Associativity Law

Please see the attached file for the fully formatted problems. The convolution of two functions f and g is defined to be the new function f * g. whenever the integral converges. (a)Is it true that f*g = g*f? Why? (b) If F(k) = (i//) f°° exp (?ikx) f(x) dx and G(k) = (i//) f°° exp (?ikx) g(x) dx are the Fourier trans

Stretching and Compressing Functions in Fourier Transform

If f(x) is a Gaussian with unit area - show that the scaled and stretched function 1/a * f(x/a) also has unit area - that's the hardest part. The other parts (along with a detailed explanation of this one) are in an attachment as both mathcad v.11 and in an html file - they're the same thing - but if you don't have mathcad y

Inner Product : Fourier Transform

Please see the attached file for the fully formatted problem. 2. Let p be a fixed and given square integrable function, i.e. 0 < S g(x)g(x) dx = ||g^2|| < inf The function g must vanish as |x| ---> inf. Consequently, one can think of g as a function whose non-zero values are concentrated in a small set around the origin x

Fourier Analysis Relationship Between Spectral Components

(a) Explain the relationship between the spectral components indicated above and the corresponding graph showing the motion of the surface of the motor plotted as a function of time. What do they represent? (b) Given the amplitudes indicated in the above diagram, copy and complete the table below for the amplitude of the vib