Local harmonic analysis on spheres
A continuous expansion of the form \(f(x)=\int^{\infty}_{0}f_{\lambda}(x)h(\lambda)d\lambda\) is established for functions or distributions supported on a compact set K that is small in the sense that no pair of antipodal points can both belong to K in the sphere S n, valid in a neighbourhood \(\Omega\) of K consisting of the complement of the antipodal points of K, where \(f_{\lambda}(x)\) are eigenfunctions of the Laplacian of \(\Omega\) with eigenvalues ((n-1)/2) 2-\(\lambda\) 2 and h(\(\lambda)\) can be given explicitly as follows. When n is odd, \[ h(\lambda)=1/(2\pi)^{\sigma +1}\gamma_ n\prod^{\sigma -1}_{k=0}(\lambda \quad 2-k\quad 2),\quad \gamma_ n=(n-2)(n-4)...1 \] and when n is even \[ h(\lambda)=\lambda /(2\pi)^{\sigma +1/2}\gamma_ n\prod^{\sigma -3/2}_{k=0}(\lambda \quad 2-(k+1/2)\quad 2)h_ 1(\lambda),\quad \gamma_ n=(n-2)(n-4)...2 \] where \(h_ 1(\lambda)=1/\pi \int^{\pi}_{0}(\sin \lambda_ s/\sin s/2)ds\), for \(\sigma =(n-1)/2\) and \(\lambda =\sigma +k\), \(k=0,1,2,...\). The function \(f_{\lambda}(x)\) can be given by \(f_{\lambda}(x)=\int_{K}\phi_{\lambda}(d(x,y))f(y)dy\) where d(x,y) denotes the distance on the sphere S n and \(\phi_{\lambda}\) are certain zonal functions. When n is odd, the function \(f_{\lambda}(x)\) can be uniquely characterized by analyticity and growth conditions in \(\lambda\) of Paley-Wiener type. The expansion is applied to study functions of the Laplacian and operators that commute with the Laplacian.
scientific article; zbMATH DE number 4061876
scientific article