A function h(x) is positive or zero for all values of x. Assume h(x) is even. If the Fourier transformation of h(x) is H(u) show that...
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Q:
A function h (x) is positive or zero for all values of x. Assume h(x) is even .If the Fourier transformation of h(x) is H(u) show that,

For all u,
|H(u)| <= |H(0)|

(Note <= mean less than equal to)
Show that as

Solution:

Let's consider the Fourier transform of a function h(x):
( 1)
where function h(x) must comply with the condition
( 2)
(otherwise the integral (1) doesn't exist)
In general, (u) is a ...

Solution Summary

The Fourier Transformation of an Even Function is investigated. The solution is detailed and well presented.

... By the convention we use, the inverse Fourier transform to the time domain of the ... Since is an even function, and is an even function for even m and odd for odd ...

... The Fourier transform is given by,. ... is even and the product of even and odd functions is zero ... Proof: Since cosine is an even function and sine is an odd function...

... To find the Fourier transform of the function we ... represent the original function as an inverse transform: ... Note that is an even function with respect to ...

... Q F (k ) = (1.7) ( ) D 2p k 2 + K 2 - ax The Fourier transform of a decaying exponential e where a > 0 is ... ax cos (x )dx is an even function, therefore: On the ...

... So h(t) is odd while g(t) is even. The Fourier transform of x(t) is therefore: ¥ ¥ ¥. ... F é (t )ù= 2 ò x (t )cos (wt )dt x êú ëû 0 5. The function. ...

Fourier transform in distribution sense. ... a) Let's consider a test function D; we have to ... 3) Let's put ( 4) where ( 5) ( 6) Since even expression, the ...

... For some reason you chose to prove these properties of the Fourier transform using the Gaussian ... A. The Gaussian function is hard to integrate B. Even if you ...

... the function below: Expand the function in an ... integral representation is actually the Fourier transform format which I ...Even if we would go writing it initially ...

... Eignevalues and eigenvectors of the Fourier transform are investigated ... F 2 is λ = 1 and the eigenfunctions are all square-integrable,even functions, ie f ...

... If you want the output Fourier Transform data to be ... the information in a new worksheet or even new workbook. ... Column H using the IMABS function to display the ...