The Fourier Transform of a function f(x) can be defined as:
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^f(x)=1√2π∫∞−∞e−ikxf(x)dx |
So,
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^f′(x)=1√2π∫∞−∞e−ikxf′(x)dxsubstitue f(x) by f′(x) |
Using integration by parts, let
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dv=f′(x)dxv=f(x)u=e−ikxdu=−ike−ikxdx |
Now,
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1√2π∫∞−∞e−ikxf′(x)dx=1√2π∫∞−∞udv=1√2π(uv|∞−∞−∫∞−∞vdu)=1√2π(e−ikxf(x)|∞−∞+ik∫∞−∞e−ikxf(x)dx) |
Suppose f(x)→0 as x→±∞,
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1√2π∫∞−∞e−ikxf′(x)dx=ik1√2π∫∞−∞e−ikxf(x)dx∴ |
Telescoping,
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\begin{aligned}
\widehat{f^{(n)}} &= (ik)^{n} \cdot \widehat{f} \\
\end{aligned}
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\Box
Source: Computational Methods for Data Analysis.
# posted by rot13(Unafba Pune) @ 8:12 PM
