Fourier Theory by B. Clarke

By B. Clarke

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Band Limited Functions and Shannon's Sampling Theorem. The Fourier transform variable has the role of frequency and f ^(ξ) is referred to as the frequency representation of f(x). If f ^(ξ) = 0 for |ξ | > ξc > 0, then f(x) is called a band-limited function and ξ c is called the cut-off frequency. Many functions from science and technology, are band-limited. For example, human hearing is assumed to be limited to frequencies below about 20 kHz. Therefore the acoustic signals recorded on compact discs are limited to a bandwidth of 22 kHz.

When f = g we obtain Parsevals' identity, ∞ – ∫∞ |f( ∞ y) |2 1 dy = |f ^(ξ)|2dξ . 2π – ∫∞ Examples. |x| < a  1; . 1. Let f(x) = χ a(x) =   0; otherwise 2 sin a ξ Then f^(ξ) = , and by Parsevals' identity, ξ ∞ ∞ 1 sin a ξ2 2 ∫ |χa(x) | dx = 2π – ∫∞  ξ  dξ –∞ – or – ∞  2 sin a ξ  dξ = 4 πa   ξ   ∞  sin a ξ dξ = π a.  ξ    ∫∞ ∫∞ 2 2 54 16. Plancherels' and Parsevals' Identities. 2. Let f(x) = e – a|x|, f^(ξ) = 2a + ξ2 a2 ∞ – ∫( ∞ e – a|x| 1 2a2 = a π or ∞ – ∫∞ ) ∞ – ∫∞ 2dx and by Parsevals' identity 1 = 2π ∞ 2a 2  ∫  2 2 dξ –∞ a + ξ  1 dξ (a2 + ξ 2)2 π 1 d ξ = .

Applications to Differential Equations. f^(ξ)e – ξy ; ξ > 0 u^ (ξ, y) =   f^(ξ)e ξy ; ξ < 0 = f^(ξ)e – ξ|y|. ^ y  , by the convolution theorem, Since e – ξ|y| =  2 2   π (x + y ) y  u(x, y) =  *f  = 2 2  π (x + y )  ∞ – ∫∞ y f(s) ds. π ((x – s) 2 + y 2) 3. The Heat Equation. The heat equation, ∂u ∂2u – κ = 0, ∂t ∂x 2 – ∞ < x < ∞, t > 0 for u(x, t) a function of two variables, with initial condition u(x, t) = f(x), – ∞ < x < ∞ for f ∈ L1 (R), f continuous and bounded, can be solved using Fourier transforms.

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