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Fourier Transformation : Even Function

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

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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 ...

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