On the representation of smooth functions on the sphere using finitely many bits

Abstract

Listen

Translate article

We discuss the construction of a parsimonious representation of smooth functions on the Euclidean sphere using finitely many bits, in the sense of metric entropy. The smoothness of the functions is measured by Besov spaces. The bit representation is obtained by uniform quantization on the values of a polynomial operator at scattered sites on the sphere. For each cap, one can identify a certain number of bits, commensurable with the local smoothness of the target function on that cap and the volume of that cap, and obtained using the values of the polynomial operator near that cap. The polynomial operator is calculated using either spherical harmonic coefficients or, in the case of uniform approximation, values of the function at scattered sites on the sphere. The localization properties of the polynomial operator are demonstrated by a characterization of local smoothness of the target function near a point in terms of the values of these operators near the point in question.