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- Convolution_theorem abstract "In mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Versions of the convolution theorem are true for various Fourier-related transforms.Let and be two functions with convolution . (Note that the asterisk denotes convolution in this context, and not multiplication. The tensor product symbol is sometimes used instead.)Let denote the Fourier transform operator, so and are the Fourier transforms of and , respectively.Then where denotes point-wise multiplication. It also works the other way around: By applying the inverse Fourier transform , we can write: Note that the relationships above are only valid for the form of the Fourier transform shown in the Proof section below. The transform may be normalised in other ways, in which case constant scaling factors (typically or ) will appear in the relationships above.This theorem also holds for the Laplace transform, the two-sided Laplace transform and, when suitably modified, for the Mellin transform and Hartley transform (see Mellin inversion theorem). It can be extended to the Fourier transform of abstract harmonic analysis defined over locally compact abelian groups.This formulation is especially useful for implementing a numerical convolution on a computer: The standard convolution algorithm has quadratic computational complexity. With the help of the convolution theorem and the fast Fourier transform, the complexity of the convolution can be reduced to O(n log n). This can be exploited to construct fast multiplication algorithms.".
- Convolution_theorem wikiPageExternalLink index.html.
- Convolution_theorem wikiPageID "53268".
- Convolution_theorem wikiPageRevisionID "603597543".
- Convolution_theorem hasPhotoCollection Convolution_theorem.
- Convolution_theorem subject Category:Articles_containing_proofs.
- Convolution_theorem subject Category:Theorems_in_Fourier_analysis.
- Convolution_theorem type Abstraction100002137.
- Convolution_theorem type Communication100033020.
- Convolution_theorem type Message106598915.
- Convolution_theorem type Proposition106750804.
- Convolution_theorem type Statement106722453.
- Convolution_theorem type Theorem106752293.
- Convolution_theorem type TheoremsInFourierAnalysis.
- Convolution_theorem comment "In mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Versions of the convolution theorem are true for various Fourier-related transforms.Let and be two functions with convolution .".
- Convolution_theorem label "Convolution theorem".
- Convolution_theorem label "Teorema da convolução".
- Convolution_theorem label "Teorema de convolución".
- Convolution_theorem label "Teorema di convoluzione".
- Convolution_theorem label "卷积定理".
- Convolution_theorem sameAs Teorema_de_convolución.
- Convolution_theorem sameAs Teorema_di_convoluzione.
- Convolution_theorem sameAs Teorema_da_convolução.
- Convolution_theorem sameAs m.0dzr9.
- Convolution_theorem sameAs Q2638931.
- Convolution_theorem sameAs Q2638931.
- Convolution_theorem sameAs Convolution_theorem.
- Convolution_theorem wasDerivedFrom Convolution_theorem?oldid=603597543.
- Convolution_theorem isPrimaryTopicOf Convolution_theorem.