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In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Other versions of the convolution theorem are applicable to various Fourier-related transforms.

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  • Convolution theorem (en)
  • Teorema de convolució (ca)
  • Teorema de convolución (es)
  • Teorema di convoluzione (it)
  • Teorema da convolução (pt)
  • 卷积定理 (zh)
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  • In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Other versions of the convolution theorem are applicable to various Fourier-related transforms. (en)
  • In matematica, il teorema di convoluzione afferma che sotto opportune condizioni la trasformata di Laplace, così come la trasformata di Fourier della convoluzione di due funzioni è il prodotto delle trasformate delle funzioni stesse. Questo teorema ha importanti risvolti nell'analisi dei segnali, in particolare nell'ambito delle reti lineari. (it)
  • 卷积定理指出，函数卷积的傅里叶变换是函数傅里叶变换的乘积。即一个域中的卷积对应于另一个域中的乘积，例如时域中的卷积对应于频域中的乘积。 其中表示f 的傅里叶变换。下面这种形式也成立： 借由傅里叶逆变换，也可以写成 注意以上的写法只对特定形式定义的变换正确，变换可能由其它方式正规化，使得上面的关系式中出现其它的。 这一定理对拉普拉斯变换、双边拉普拉斯变换、Z变换、梅林变换和（参见）等各种傅里叶变换的变体同样成立。在调和分析中还可以推广到在局部紧致的阿贝尔群上定义的傅里叶变换。 利用卷积定理可以简化卷积的运算量。对于长度为的序列，按照卷积的定义进行计算，需要做组对位乘法，其计算复杂度为；而利用傅里叶变换将序列变换到频域上后，只需要一组对位乘法，利用傅里叶变换的快速算法之后，总的计算复杂度为。这一结果可以在快速乘法计算中得到应用。 (zh)
  • En matemàtica, el teorema de convolució estableix que en determinades circumstàncies, la Transformada de Fourier d'una convolució és el producte punt a punt de les transformades de Fourier. En altres paraules, la convolució en un domini (per exemple el domini temporal) és equivalent al producte punt a punt en l'altre domini (és a dir domini espectral). Llavors on "·" indica producte punt. També es pot afirmar que: Aplicant la transformada inversa de Fourier , podem escriure: (ca)
  • En matemática, el teorema de convolución establece que, bajo determinadas circunstancias, la transformada de Fourier de una convolución es el producto punto a punto (o producto Hadamard) de las transformadas. En otras palabras, la convolución en un dominio (por ejemplo el dominio temporal) es equivalente al producto punto a punto en el otro dominio (es decir dominio espectral). Entonces donde · indica producto punto a punto. También puede afirmarse que: Aplicando la transformada inversa de Fourier , podemos escribir: (es)
  • Em matemática, o teorema da convolução estabelece que, sob condições apropriadas, a transformada de Fourier de uma convolução de duas funções absolutamente integráveis é igual ao das transformadas de Fourier de cada função. Em outras palavras, convolução em um domínio (e.g., no domínio do tempo) equivale a multiplicação ponto a ponto no outro domínio (e.g., no domínio da frequência). O teorema é verdadeiro para várias transformadas relacionadas à transformada de Fourier. onde o símbolo denota multiplicação ponto a ponto. A recíproca também é verdadeira: (pt)
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  • We can also verify the inverse DTFT of : (en)
  • A time-domain derivation proceeds as follows: A frequency-domain derivation follows from , which indicates that the DTFTs can be written as: The product with is thereby reduced to a discrete-frequency function: where the equivalence of and follows from . Therefore, the equivalence of and requires: (en)
  • Consider functions in Lp-space , with Fourier transforms : where indicates the inner product of : and The convolution of and is defined by: Also: Hence by Fubini's theorem we have that so its Fourier transform is defined by the integral formula: Note that and hence by the argument above we may apply Fubini's theorem again : (en)
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