On the representation of smooth functions on the sphere using finitely many bits
The paper deals with approximation of smooth functions defined on the \(q\)-dimensional Euclidean sphere \(S^q\) by polynomial operators. Let \({\mathcal P}_l(t)\) be the Legendre polynomial of degree \(l\) and let \(h(t)\) be some \([0, \infty) \to {\mathbb R}\) function vanishing for sufficiently large \(t\). For \(t \in {\mathbb R}\) and \(n=0,1,\ldots\), let \[ \Phi_n(t)=\sum_{l=0}^\infty h(l/2^n) {d_l^q \over \omega_q}{\mathcal P}_l(t), \] where \(d_l^q\) is the linear dimension of the set of all spherical harmonics of degree \(l\) and \(\omega_q\) is the volume of \(S^q\). Given a sequence \(\{\mu_n\}\) of signed Borel measures on \(S^q\), the summability operator is defined by \[ \sigma_n(f, {\mathbf x})=\int_{S_q} \Phi_n ({\mathbf x}\cdot \xi)\,d\mu_n(\xi). \] The difference \(\tau_n(f)=\sigma_n(f)-\sigma_{n-1}(f)\) is the polynomial frame operator. The most interesting cases are when \(\mu_n\) are the volume measures on \(S^q\) for all \(n\) and when \(\mu_n\) are the so-called M-Z quadrature measures. In the latter case, the author proves that under certain uniformity conditions \(f=\sum_{n=0}^\infty \tau_n(f)\), with the series convergent in the sense of \(L_p(S^q)\). Various properties of this expansion are proved. In particular, by uniform quantization of the values of polynomial operators one can obtain a bit coding of functions \(f\) from the Besov classes, with the number of bits that is within only a logarithmic factor of their metric entropy. Moreover, the coding has a localization property: from the total set of bits one can identify for each spherical cap a certain subset of bits that represents the function on this cap. The cardinality of this subset depends on the local smoothness of the function and the volume of the cap.
scientific article; zbMATH DE number 2175611
scientific article