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[parent] approximating Fourier integrals with discrete Fourier transforms (Derivation)
ApproximatingFourierIntegralsWithDiscreteFourierTransforms

"approximating Fourier integrals with discrete Fourier transforms" is owned by stevecheng.
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See Also: discrete time Fourier transform in relation with continuous time Fourier transform


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Cross-references: partial sums, multiples, plus, coefficient, entails, expand, inner product, discrete, positive semi-definite inner product, Fourier coefficient, analysis, summation, conditionally convergent, sufficient, Fourier series, points, Riemann sum, vector, cube, unit, substitution, algorithm, discrete Fourier transform, approximation, domain, Fourier integral, compact rectangle, function, Fourier transform, continuous function

This is version 6 of approximating Fourier integrals with discrete Fourier transforms, born on 2006-12-18, modified 2007-06-17.
Object id is 8637, canonical name is ApproximatingFourierIntegralsWithDiscreteFourierTransforms.
Accessed 1654 times total.

Classification:
AMS MSC: 42A38 (Fourier analysis :: Fourier analysis in one variable :: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type)
 42B10 (Fourier analysis :: Fourier analysis in several variables :: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type)
 65T50 (Numerical analysis :: Numerical methods in Fourier analysis :: Discrete and fast Fourier transforms)
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Excellent by fernsanz on 2007-06-26 15:36:21
Really brilliant entry. This builds the bridge between rigorous mathematics and aliasing as used by electronic engineers.

I have been involved for a time now in obtaining a similar result relating the Fourier transform in discrete and continuous times, but this issue seems more complicated that it might seem.

What seems clear is that, as discrete time transforms involve a countable number of points -hence of zero measure-, one has to stick to continuous functions, because otherwise one always can alter those countable many points without changing the continuous transform at all but dramatically changing the discrete transform (in particular one can always carry that countable many points to zero so obtaining a zero discrete transform).
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