Teorema de Plancherel

Em matemática, o Teorema de Plancherel é um resultado em análise harmónica, primeiramente demonstrado por Michel Plancherel.[1] Na sua forma mais simples estabelece que se uma função f é tanto elemento de L¹(R) quanto elemento de L²(R), então sua Transformada de Fourier também está em L²(R) e possui a mesma norma L². Em particular, a Transformada de Fourier é uma aplicação isométrica. Isto implica que a Transformada de Fourier restrita a L¹(R) ∩ L²(R) tem uma única extensão para um operador isométrico linear L²(R) →L²(R).

Aqui a versão de Plancherel relaciona espaço de funções na linha dos reais. O teorema é válido em versões mais abstratas, por exemplo, em grupos abelianos localmente compactos. Ainda mais genericamente, esta é uma versão do Teorema de Plancherel que faz sentido para grupos localmente compactos não-cumutativos satisfazendo certas presunções técnicas.Este é tema de analisa harmonica não-cumutativa.

A unicidade da Transformada de Fourier é frequentemente chamada de Teorema de Parseval nos campos da ciência e engenharia, baseada na resultado anterior (mas menos genêrico) que era usado para provar a unicidade da série de Fourier.

Referências

  1. ↑ Plancherel, Michel (1910) "Contribution a l'etude de la representation d'une fonction arbitraire par les integrales définies," Rendiconti del Circolo Matematico di Palermo, vol. 30, pages 298-335
  • J. Dixmier, Les C*-algèbres et leurs Represéntations, Gauthier Villars, 1969
  • K. Yosida, Functional Analysis, Springer Verlag, 1968
  • Este artigo foi traduzido de sua versão da Wikipédia em inglês.

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