The Wiener Transform on the Besicovitch Spaces
نویسندگان
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.
منابع مشابه
Good modulating sequences for the ergodic Hilbert transform
This article investigates classes of bounded sequences of complex numbers that are universally good for the ergodic Hilbert transform in Lp-spaces, 2 ≤ p ≤ ∞. The class of bounded Besicovitch sequences satisfying a rate condition is among such sequence classes.
متن کاملSharp Continuity Results for the Short-time Fourier Transform and for Localization Operators
We completely characterize the boundedness on Wiener amalgam spaces of the short-time Fourier transform (STFT), and on both L and Wiener amalgam spaces of a special class of pseudodifferential operators, called localization operators. Precisely, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces W (L, L) are given and their sharpness is shown. Localization operators a...
متن کاملCellular Automata in the Cantor, Besicovitch, and Weyl Topological Spaces
The Besicovitch and Weyl pseudometrics on the space AŸ of biinfinite sequences measure the density of differences in either the central or arbitrary segments of given sequences. The Besicovitch and Weyl spaces are obtained from A by factoring through the equivalence of zero distance. Cellular automata are considered as dynamical systems on the Besicovitch and Weyl spaces and their topological a...
متن کاملPaley-wiener Theorem for Line Bundles over Compact Symmetric Spaces
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2: Riemannian Symmetric Spaces and Related Structure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
متن کاملA Paley-wiener Theorem for the Spherical Laplace Transform on Causal Symmetric Spaces of Rank 1 Nils Byrial Andersen and Gestur Olafsson
We formulate and prove a topological Paley-Wiener theorem for the normalized spherical Laplace transform deened on the rank 1 causal sym
متن کامل