"\u5377\u79EF\u5B9A\u7406\u6307\u51FA\uFF0C\u51FD\u6570\u5377\u79EF\u7684\u5085\u91CC\u53F6\u53D8\u6362\u662F\u51FD\u6570\u5085\u91CC\u53F6\u53D8\u6362\u7684\u4E58\u79EF\u3002\u5373\u4E00\u4E2A\u57DF\u4E2D\u7684\u5377\u79EF\u5BF9\u5E94\u4E8E\u53E6\u4E00\u4E2A\u57DF\u4E2D\u7684\u4E58\u79EF\uFF0C\u4F8B\u5982\u65F6\u57DF\u4E2D\u7684\u5377\u79EF\u5BF9\u5E94\u4E8E\u9891\u57DF\u4E2D\u7684\u4E58\u79EF\u3002 \u5176\u4E2D\u8868\u793Af \u7684\u5085\u91CC\u53F6\u53D8\u6362\u3002\u4E0B\u9762\u8FD9\u79CD\u5F62\u5F0F\u4E5F\u6210\u7ACB\uFF1A \u501F\u7531\u5085\u91CC\u53F6\u9006\u53D8\u6362\uFF0C\u4E5F\u53EF\u4EE5\u5199\u6210 \u6CE8\u610F\u4EE5\u4E0A\u7684\u5199\u6CD5\u53EA\u5BF9\u7279\u5B9A\u5F62\u5F0F\u5B9A\u4E49\u7684\u53D8\u6362\u6B63\u786E\uFF0C\u53D8\u6362\u53EF\u80FD\u7531\u5176\u5B83\u65B9\u5F0F\u6B63\u89C4\u5316\uFF0C\u4F7F\u5F97\u4E0A\u9762\u7684\u5173\u7CFB\u5F0F\u4E2D\u51FA\u73B0\u5176\u5B83\u7684\u3002 \u8FD9\u4E00\u5B9A\u7406\u5BF9\u62C9\u666E\u62C9\u65AF\u53D8\u6362\u3001\u53CC\u8FB9\u62C9\u666E\u62C9\u65AF\u53D8\u6362\u3001Z\u53D8\u6362\u3001\u6885\u6797\u53D8\u6362\u548C\uFF08\u53C2\u89C1\uFF09\u7B49\u5404\u79CD\u5085\u91CC\u53F6\u53D8\u6362\u7684\u53D8\u4F53\u540C\u6837\u6210\u7ACB\u3002\u5728\u8C03\u548C\u5206\u6790\u4E2D\u8FD8\u53EF\u4EE5\u63A8\u5E7F\u5230\u5728\u5C40\u90E8\u7D27\u81F4\u7684\u963F\u8D1D\u5C14\u7FA4\u4E0A\u5B9A\u4E49\u7684\u5085\u91CC\u53F6\u53D8\u6362\u3002 \u5229\u7528\u5377\u79EF\u5B9A\u7406\u53EF\u4EE5\u7B80\u5316\u5377\u79EF\u7684\u8FD0\u7B97\u91CF\u3002\u5BF9\u4E8E\u957F\u5EA6\u4E3A\u7684\u5E8F\u5217\uFF0C\u6309\u7167\u5377\u79EF\u7684\u5B9A\u4E49\u8FDB\u884C\u8BA1\u7B97\uFF0C\u9700\u8981\u505A\u7EC4\u5BF9\u4F4D\u4E58\u6CD5\uFF0C\u5176\u8BA1\u7B97\u590D\u6742\u5EA6\u4E3A\uFF1B\u800C\u5229\u7528\u5085\u91CC\u53F6\u53D8\u6362\u5C06\u5E8F\u5217\u53D8\u6362\u5230\u9891\u57DF\u4E0A\u540E\uFF0C\u53EA\u9700\u8981\u4E00\u7EC4\u5BF9\u4F4D\u4E58\u6CD5\uFF0C\u5229\u7528\u5085\u91CC\u53F6\u53D8\u6362\u7684\u5FEB\u901F\u7B97\u6CD5\u4E4B\u540E\uFF0C\u603B\u7684\u8BA1\u7B97\u590D\u6742\u5EA6\u4E3A\u3002\u8FD9\u4E00\u7ED3\u679C\u53EF\u4EE5\u5728\u5FEB\u901F\u4E58\u6CD5\u8BA1\u7B97\u4E2D\u5F97\u5230\u5E94\u7528\u3002"@zh . . "53268"^^ . . . . . . "Convolution theorem"@en . "Teorema de convoluci\u00F3n"@es . . "\u5377\u79EF\u5B9A\u7406"@zh . "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 . . "A time-domain derivation proceeds as follows:\n\n\nA frequency-domain derivation follows from , which indicates that the DTFTs can be written as:\n\n\n\nThe product with is thereby reduced to a discrete-frequency function:\n\nwhere the equivalence of and follows from . Therefore, the equivalence of and requires:"@en . . . "Multi-dimensional derivation of Eq.1"@en . "En matem\u00E0tica, el teorema de convoluci\u00F3 estableix que en determinades circumst\u00E0ncies, la Transformada de Fourier d'una convoluci\u00F3 \u00E9s el producte punt a punt de les transformades de Fourier. En altres paraules, la convoluci\u00F3 en un domini (per exemple el domini temporal) \u00E9s equivalent al producte punt a punt en l'altre domini (\u00E9s a dir domini espectral). Siguin f i g dues funcions la convoluci\u00F3 s'expressa amb .(Nota: l'asterisc en aquest context, indica convoluci\u00F3 i no multiplicaci\u00F3, de vegades s'utilitza tamb\u00E9 el s\u00EDmbol ).Sigui l'operador de la transformada de Fourier, de manera que i s\u00F3n les transformades de Fourier de f i g , respectivament. Llavors on \"\u00B7\" indica producte punt. Tamb\u00E9 es pot afirmar que: Aplicant la transformada inversa de Fourier , podem escriure:"@ca . . . . . . ":"@en . . . . . . . . . "En matem\u00E1tica, el teorema de convoluci\u00F3n establece que, bajo determinadas circunstancias, la transformada de Fourier de una convoluci\u00F3n es el producto punto a punto (o producto Hadamard) de las transformadas. En otras palabras, la convoluci\u00F3n en un dominio (por ejemplo el dominio temporal) es equivalente al producto punto a punto en el otro dominio (es decir dominio espectral). Sean y dos funciones cuya convoluci\u00F3n se expresa con .(notar que el asterisco denota convoluci\u00F3n en este contexto, y no multiplicaci\u00F3n; a veces es utilizado tambi\u00E9n el s\u00EDmbolo ).Sea el operador de la transformada de Fourier, con lo que y son las transformadas de Fourier de f y g, respectivamente. Entonces donde \u00B7 indica producto punto a punto. Tambi\u00E9n puede afirmarse que: Aplicando la transformada inversa de Fourier , podemos escribir:"@es . "\u5377\u79EF\u5B9A\u7406\u6307\u51FA\uFF0C\u51FD\u6570\u5377\u79EF\u7684\u5085\u91CC\u53F6\u53D8\u6362\u662F\u51FD\u6570\u5085\u91CC\u53F6\u53D8\u6362\u7684\u4E58\u79EF\u3002\u5373\u4E00\u4E2A\u57DF\u4E2D\u7684\u5377\u79EF\u5BF9\u5E94\u4E8E\u53E6\u4E00\u4E2A\u57DF\u4E2D\u7684\u4E58\u79EF\uFF0C\u4F8B\u5982\u65F6\u57DF\u4E2D\u7684\u5377\u79EF\u5BF9\u5E94\u4E8E\u9891\u57DF\u4E2D\u7684\u4E58\u79EF\u3002 \u5176\u4E2D\u8868\u793Af \u7684\u5085\u91CC\u53F6\u53D8\u6362\u3002\u4E0B\u9762\u8FD9\u79CD\u5F62\u5F0F\u4E5F\u6210\u7ACB\uFF1A \u501F\u7531\u5085\u91CC\u53F6\u9006\u53D8\u6362\uFF0C\u4E5F\u53EF\u4EE5\u5199\u6210 \u6CE8\u610F\u4EE5\u4E0A\u7684\u5199\u6CD5\u53EA\u5BF9\u7279\u5B9A\u5F62\u5F0F\u5B9A\u4E49\u7684\u53D8\u6362\u6B63\u786E\uFF0C\u53D8\u6362\u53EF\u80FD\u7531\u5176\u5B83\u65B9\u5F0F\u6B63\u89C4\u5316\uFF0C\u4F7F\u5F97\u4E0A\u9762\u7684\u5173\u7CFB\u5F0F\u4E2D\u51FA\u73B0\u5176\u5B83\u7684\u3002 \u8FD9\u4E00\u5B9A\u7406\u5BF9\u62C9\u666E\u62C9\u65AF\u53D8\u6362\u3001\u53CC\u8FB9\u62C9\u666E\u62C9\u65AF\u53D8\u6362\u3001Z\u53D8\u6362\u3001\u6885\u6797\u53D8\u6362\u548C\uFF08\u53C2\u89C1\uFF09\u7B49\u5404\u79CD\u5085\u91CC\u53F6\u53D8\u6362\u7684\u53D8\u4F53\u540C\u6837\u6210\u7ACB\u3002\u5728\u8C03\u548C\u5206\u6790\u4E2D\u8FD8\u53EF\u4EE5\u63A8\u5E7F\u5230\u5728\u5C40\u90E8\u7D27\u81F4\u7684\u963F\u8D1D\u5C14\u7FA4\u4E0A\u5B9A\u4E49\u7684\u5085\u91CC\u53F6\u53D8\u6362\u3002 \u5229\u7528\u5377\u79EF\u5B9A\u7406\u53EF\u4EE5\u7B80\u5316\u5377\u79EF\u7684\u8FD0\u7B97\u91CF\u3002\u5BF9\u4E8E\u957F\u5EA6\u4E3A\u7684\u5E8F\u5217\uFF0C\u6309\u7167\u5377\u79EF\u7684\u5B9A\u4E49\u8FDB\u884C\u8BA1\u7B97\uFF0C\u9700\u8981\u505A\u7EC4\u5BF9\u4F4D\u4E58\u6CD5\uFF0C\u5176\u8BA1\u7B97\u590D\u6742\u5EA6\u4E3A\uFF1B\u800C\u5229\u7528\u5085\u91CC\u53F6\u53D8\u6362\u5C06\u5E8F\u5217\u53D8\u6362\u5230\u9891\u57DF\u4E0A\u540E\uFF0C\u53EA\u9700\u8981\u4E00\u7EC4\u5BF9\u4F4D\u4E58\u6CD5\uFF0C\u5229\u7528\u5085\u91CC\u53F6\u53D8\u6362\u7684\u5FEB\u901F\u7B97\u6CD5\u4E4B\u540E\uFF0C\u603B\u7684\u8BA1\u7B97\u590D\u6742\u5EA6\u4E3A\u3002\u8FD9\u4E00\u7ED3\u679C\u53EF\u4EE5\u5728\u5FEB\u901F\u4E58\u6CD5\u8BA1\u7B97\u4E2D\u5F97\u5230\u5E94\u7528\u3002"@zh . . . . "where denotes point-wise multiplication"@en . . . . . . . . . . . . . . . . . "En matem\u00E0tica, el teorema de convoluci\u00F3 estableix que en determinades circumst\u00E0ncies, la Transformada de Fourier d'una convoluci\u00F3 \u00E9s el producte punt a punt de les transformades de Fourier. En altres paraules, la convoluci\u00F3 en un domini (per exemple el domini temporal) \u00E9s equivalent al producte punt a punt en l'altre domini (\u00E9s a dir domini espectral). Llavors on \"\u00B7\" indica producte punt. Tamb\u00E9 es pot afirmar que: Aplicant la transformada inversa de Fourier , podem escriure:"@ca . "Teorema de convoluci\u00F3"@ca . "1114206769"^^ . . "En matem\u00E1tica, el teorema de convoluci\u00F3n establece que, bajo determinadas circunstancias, la transformada de Fourier de una convoluci\u00F3n es el producto punto a punto (o producto Hadamard) de las transformadas. En otras palabras, la convoluci\u00F3n 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 \u00B7 indica producto punto a punto. Tambi\u00E9n puede afirmarse que: Aplicando la transformada inversa de Fourier , podemos escribir:"@es . "Teorema di convoluzione"@it . . . . . . "#0073CF"@en . "Circular convolution"@en . . . . "6"^^ . . . . "21401"^^ . . . "#F5FFFA"@en . "In matematica, il teorema di convoluzione afferma che sotto opportune condizioni la trasformata di Laplace, cos\u00EC come la trasformata di Fourier della convoluzione di due funzioni \u00E8 il prodotto delle trasformate delle funzioni stesse. Questo teorema ha importanti risvolti nell'analisi dei segnali, in particolare nell'ambito delle reti lineari."@it . . . . . . "Em matem\u00E1tica, o teorema da convolu\u00E7\u00E3o estabelece que, sob condi\u00E7\u00F5es apropriadas, a transformada de Fourier de uma convolu\u00E7\u00E3o de duas fun\u00E7\u00F5es absolutamente integr\u00E1veis \u00E9 igual ao das transformadas de Fourier de cada fun\u00E7\u00E3o. Em outras palavras, convolu\u00E7\u00E3o em um dom\u00EDnio (e.g., no dom\u00EDnio do tempo) equivale a multiplica\u00E7\u00E3o ponto a ponto no outro dom\u00EDnio (e.g., no dom\u00EDnio da frequ\u00EAncia). O teorema \u00E9 verdadeiro para v\u00E1rias transformadas relacionadas \u00E0 transformada de Fourier. Sejam e duas fun\u00E7\u00F5es e sua convolu\u00E7\u00E3o (note-se que o asterisco aqui denota a opera\u00E7\u00E3o de convolu\u00E7\u00E3o, e n\u00E3o de multiplica\u00E7\u00E3o; o s\u00EDmbolo de produto tensorial algumas vezes \u00E9 usado no seu lugar.).Seja o operador transformada de Fourier, tal que e s\u00E3o as transformadas de Fourier de e , respectivamente. Ent\u00E3o onde o s\u00EDmbolo denota multiplica\u00E7\u00E3o ponto a ponto. A rec\u00EDproca tamb\u00E9m \u00E9 verdadeira: A respeito da transformada inversa de Fourier , podemos escrever: Note-se que as f\u00F3rmulas acima s\u00E3o v\u00E1lidas apenas quando a transformada de Fourier aparece na forma mostrada na se\u00E7\u00E3o de abaixo. A transformada pode ser normalizada de outras formas, casos em que fatores de escalamento constantes (tipicamente ou ) aparecer\u00E3o nas f\u00F3rmulas. Este teorema tamb\u00E9m vale para a transformada de Laplace, a transformada de Laplace bilateral e, quando convenientemente modificada, para a transformada de Mellin e para a transformada de Hartley. Ele pode ser estendido para a transformada de Fourier usada em an\u00E1lise de harm\u00F4nicos, definida sobre grupos abelianos localmente compactos. Esta formula\u00E7\u00E3o \u00E9 especialmente \u00FAtil na implementa\u00E7\u00E3o num\u00E9rica da opera\u00E7\u00E3o de convolu\u00E7\u00E3o em um computador digital. O algoritmo padr\u00E3o para c\u00E1lculo da convolu\u00E7\u00E3o tem complexidade computacional quadr\u00E1tica; lan\u00E7ando m\u00E3o do teorema da convolu\u00E7\u00E3o e da transformada r\u00E1pida de Fourier, a complexidade pode ser reduzida a O(n log n). Ele tamb\u00E9m pode ser explorado na constru\u00E7\u00E3o de algoritmos mais r\u00E1pidos para multiplica\u00E7\u00E3o de fun\u00E7\u00F5es."@pt . . . "In matematica, il teorema di convoluzione afferma che sotto opportune condizioni la trasformata di Laplace, cos\u00EC come la trasformata di Fourier della convoluzione di due funzioni \u00E8 il prodotto delle trasformate delle funzioni stesse. Questo teorema ha importanti risvolti nell'analisi dei segnali, in particolare nell'ambito delle reti lineari."@it . . . . "Derivations of Eq.4"@en . . . "Consider functions in Lp-space , with Fourier transforms :\n\nwhere indicates the inner product of : and \n\nThe convolution of and is defined by:\n\n\nAlso:\n\n\nHence by Fubini's theorem we have that so its Fourier transform is defined by the integral formula:\n\n\nNote that and hence by the argument above we may apply Fubini's theorem again :"@en . . . . . "Teorema da convolu\u00E7\u00E3o"@pt . "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 . . . "Derivation of Eq.2"@en . . . . . "Convolution theorem"@en . . . "We can also verify the inverse DTFT of :"@en . . . . . . "Em matem\u00E1tica, o teorema da convolu\u00E7\u00E3o estabelece que, sob condi\u00E7\u00F5es apropriadas, a transformada de Fourier de uma convolu\u00E7\u00E3o de duas fun\u00E7\u00F5es absolutamente integr\u00E1veis \u00E9 igual ao das transformadas de Fourier de cada fun\u00E7\u00E3o. Em outras palavras, convolu\u00E7\u00E3o em um dom\u00EDnio (e.g., no dom\u00EDnio do tempo) equivale a multiplica\u00E7\u00E3o ponto a ponto no outro dom\u00EDnio (e.g., no dom\u00EDnio da frequ\u00EAncia). O teorema \u00E9 verdadeiro para v\u00E1rias transformadas relacionadas \u00E0 transformada de Fourier. onde o s\u00EDmbolo denota multiplica\u00E7\u00E3o ponto a ponto. A rec\u00EDproca tamb\u00E9m \u00E9 verdadeira:"@pt .