Interpolating wavelets on the sphere

Nikolai I. Chernykh

Ural mathematical journal, Tome 5 (2019) no. 2, pp. 3-12

Résumé

There are several works where bases of wavelets on the sphere (mainly orthogonal and wavelet-like bases) were constructed. In all such constructions, the authors seek to preserve the most important properties of classical wavelets including constructions on the basis of the lifting-scheme. In the present paper, we propose one more construction of wavelets on the sphere. Although two of three systems of wavelets constructed in this paper are orthogonal, we are more interested in their interpolation properties. Our main idea consists in a special double expansion of the unit sphere in $\mathbb{R}^3$ such that any continuous function on this sphere defined in spherical coordinates is easily mapped into a $2\pi$-periodic function on the plane. After that everything becomes simple, since the classical scheme of the tensor product of one-dimensional bases of functional spaces works to construct bases of spaces of functions of several variables.

Détail

Keywords: Wavelets, Multiresolution analysis, Scaling functions, Interpolating wavelets, Best approximation, Trigonometric polynomials.