Besicovitch Meets Wiener—fourier Expansions and Fractal Measures
نویسندگان
چکیده
(3) v = fdju + vc for the decomposition into discrete (ƒ dju as above) and continuous vc parts. Let F = û = J2k' + J'e' dvc{y) be the Fourier transform of the measure v. Then Wiener's theorem says (2) continues to hold. This means that the Fourier transform vc of the continuous portion of the measure does not contribute to the Bohr mean of |F|. We will interpret Wiener's theorem to mean that every finite measure is in some sense zerodimensional, but the discrete part is the significant zero-dimensional part. At the other extreme, the «-dimensional theory is just the Plancherel formula, which we write
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