The Wiener Transform on the Besicovitch Spaces

نویسندگان

  • CHRISTOPHER HEIL
  • C. HEIL
چکیده

In his fundamental research on generalized harmonic analysis, Wiener proved that the integrated Fourier transform defined by Wf(γ) = ∫ f(t) (e−2πiγt − χ[−1,1](t))/(−2πit) dt is an isometry from a nonlinear space of functions of bounded average quadratic power into a nonlinear space of functions of bounded quadratic variation. We consider this Wiener transform on the larger, linear, Besicovitch spaces Bp,q(R) defined by the norm ‖f‖Bp,q = (∫∞ 0 ( 1 2T ∫ T −T |f(t)|p dt )q/p dT T )1/q . We prove that W maps Bp,q(R) continuously into the homogeneous Besov space Ḃ 1/p′ p′,q (R) for 1 < p ≤ 2 and 1 < q ≤ ∞, and is a topological isomorphism when p = 2.

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تاریخ انتشار 1999