A spherical Wiener-Plancherel formula
Wiener developed a modification of the Fourier transform in one dimension, \(s(\gamma)\), that converges for functions whose square integral can diverge linearly, \[ \sup_{T>0} \frac 1 T \int_{- T}^{T} | f(t)| ^ 2 dt < +\infty. \] The formula for \(s\) is \[ s(\gamma) = \int_{| t| <1} f(t) \frac{e^{-i \gamma t} - 1}{i t} dt + \int_{| t| >1} f(t) \frac{e^{-i \gamma t}}{-i t} dt, \] and the generalization of the Plancherel formula is: \[ \lim_{T \rightarrow \infty} \frac 1 T \int_{-T}^ T | f(t)| ^ 2 dt = \lim_{\epsilon \rightarrow 0} \frac 1{4\pi \epsilon}\int_{- \infty}^{\infty} | s(\gamma + \epsilon)- s(\gamma - \epsilon)| ^ 2 d\gamma, \] where this means that if either limit exists, the other exists and they agree. In \(d\)-dimensional space, a Wiener-Plancherel formula takes the form \[ \lim_{T \rightarrow \infty} \frac 1{| A_ T| } \int_{A_ T} | f(t)| ^ 2 dt = \lim_{\epsilon \rightarrow 0} \frac{\nu}{\phi(\epsilon)} \int_{R^ d} | \nabla_{\epsilon} s(\gamma)| ^ 2 d\gamma, \] where \(\{A_ T\}\) is some increasing family of sets whose union is \(R^ d\), \(| A_ T| \) is the Lebesgue measure of \(A_ T\), \(\phi(\epsilon)\) is a function which converges to \(0\) as \(\epsilon\) converges to \(0\), \(\nu\) is a normalizing constant, \(s\) is a formal solution of \(Ls = \widehat f\), \(L\) is a differential operator and \(\nabla_{\epsilon}\) is a difference operator associated with \(L\). In \textit{J. Benedetto, G. Benke} and \textit{W. Evans} [Adv. Appl. Math. 10, No. 4, 457-487 (1989; Zbl 0703.42008)] this was proved when the sets \(\{A_ T\}\) are rectangles, \(L = \partial^ n/\partial \gamma_ 1 \dots \partial \gamma_ n\), and \(\nabla_{\epsilon}\) is a discrete symmetric difference operator. The convergence criterion is roughly that the limit exists for every increasing sequence of rectangles of fixed shape and all the limits are the same. \textit{J. Benedetto} [SIAM J. Math. Anal. 22, No. 4, 1110-1130 (1991; Zbl 0731.42015)] has considered a spherical case where \(L\) is an iterated Laplacian and \(\nabla_{\epsilon}\) is a spherically symmetrized second order difference operator. \textit{Lau} and \textit{Wang} [preprint] gave another spherical formula by replacing the difference operator by an integral operator \(W_{\epsilon}\) with a right hand side of \[ \lim_{\epsilon \rightarrow 0} \frac{\nu}{\phi(\epsilon)} \int_{R^ d} | W_{\epsilon}f(\gamma)| ^ 2 d\gamma. \] The author proves a \(d\)-dimensional Wiener-Plancherel formula of the following form: for \(n \geq 2k > d/2 \geq 1\), \[ \lim_{R \rightarrow \infty} \frac 1{R^ d} \int_{B_ R} | f(t)| ^ 2 dt = \lim_{\epsilon \rightarrow 0} \frac {\nu}{\epsilon^{4k-d}} \int_{R^ d} | \nabla_{\epsilon}^ ns_ k(\gamma)| ^ 2 d\gamma, \] where \(s_ k\) is an \(s\)-function associated with the differential operator \[ (-2\pi i)^{2k} \left( \partial^{2k}/\partial\gamma_ 1^{2k} + \dots + \partial^{2k}/\partial \gamma_ d^{2k}\right), \] \(\nabla_{\epsilon}^ n\) is an \(n\)th order discrete difference operator. The \(s\) function is obtained by integrating \(f\) against the function \[ \frac{e^{-2\pi i \gamma \cdot t} - \sum_{j=0}^{2k-1} \frac{(-2\pi i\gamma \cdot t)^ j}{j!}}{| t| ^{2k}} \] for \(| t| \leq 1\) and against \(e^{-2\pi i \gamma \cdot t}/| t| ^{2k}\) for \(| t| > 1\).
scientific article; zbMATH DE number 150704
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