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=== MD FFT ===
A [[fast Fourier transform]] (FFT) is an algorithm to compute the discrete Fourier transform (DFT) and its inverse. An FFT computes the DFT and produces exactly the same result as evaluating the DFT definition directly; the only difference is that an FFT is much faster. (In the presence of [[round-off error]], many FFT algorithms are also much more accurate than evaluating the DFT definition directly).There are many different FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory. See more in [[Fast Fourier transform|FFT]].
=== MD DFT ===
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== Multidimensional Laplace transform ==
The multidimensional [[Laplace transform]] is useful for the solution of boundary value problems. Boundary value problems in two or more variables characterized by partial differential equations can be solved by a direct use of the Laplace transform.<ref name=":0">{{Cite journal|title = Theorems on multidimensional laplace transform for solution of boundary value problems|journal = Computers & Mathematics with Applications|date = 1989-01-01|pages = 1033–1056|volume = 18|issue = 12|doi = 10.1016/0898-1221(89)90031-X|first1 = Joyati|last1 = Debnath|first2 = R. S.|last2 = Dahiya|doi-access = free}}</ref> The Laplace transform for an M-dimensional case is defined<ref name=":0"/> as
<math> F(s_1,s_2,\ldots,s_n) = \int_{0}^{\infty} \cdots \int_{0}^{\infty}
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<math> F(z_1,z_2)= \sum_{n_1=-\infty}^{\infty} \sum_{n_2=-\infty}^{\infty} f(n_1,n_2) z_1^{-n_1} z_2^{-n_2} </math>
The Fourier transform is a special case of the Z transform evaluated along the [[unit circle]] (in 1D) and unit bi-circle (in 2D). i.e. at
<math display="inline"> z=e^{iw} </math> where z and w are vectors.
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