Non‐orthogonal expansions on the sphere

Abstract

Discrete families of functions with the property that every function in a certain space can be represented by its formal Fourier series expansion are developed on the sphere. A Fourier-series-type expansion is obviously true if the family is an orthonormal basis of a Hilbert space, but it also can hold in situations where the family is not orthogonal and is 'overcomplete'. Furthermore, all functions in our approach are axisymmetric (depending only on the spherical distance) so that they can be used adequately in (rotation) invariant pseudo-differential equations on the sphere. The three classes of examples particularly studied here are (i) Abel-Poisson frames (ii) Gauss-Weierstrass frames and (iii) frames consisting of locally supported kernel functions. Abel-Poisson frames form families of harmonic functions and provide us with powerful approximation tools in potential theory. Gauss-Weierstrass frames are intimately related to the diffusion equation on the sphere and play an important role in multiscale descriptions of image processing on the sphere. The third class enables us to discuss spherical 'Fourier expansions' by means of 'axisymmetric finite elements'.

A General Hilbert Space Approach to Framelets

“…Since {Φ S J } J is a scaling function, the property of an approximate identity is obviously fulfilled. These two major conditions, i.e., (11) and (12), combined with (6), are the three primary stability conditions for a sequence {Ψ ∧ J (n)} J∈N−1,n∈N0 to generate S-type scaling functions and wavelets.…”

“…∧ J (n)} n∈N0 be a family of sequences in R. Let each sequence be an H-admissible symbol and let them fulfill together the frame condition (6) with the two frame constants 0 < A ≤ B < +∞ and the two summability conditions (11) and (12). Then the symbols generated by equations (8), (9) …”

Locally supported approximate identities on the unit ball

“…The early paper [15] uses a tensor product ansatz to construct locally supported splines on the sphere. In [2,6] axisymmetric locally supported basis functions on the 2-sphere are introduced by use of Bernstein polynomials. Their treatment as foundation for spherical spline interpolation in certain Sobolev spaces is discussed in [14].…”

Biorthogonal Locally Supported Wavelets on the Sphere Based on Zonal Kernel Functions

“…The constructions of the Geomathematics Group in Kaiserslautern ( [15], [19], [20], [8], [9], [11], [12]) are intrinsically based on the specific properties concerning the geometry of the sphere and the theory of 'spherical polynomials', i.e., in the jargon of the geosciences 'spherical harmonics'. Two approaches to spherical wavelets have been established: On the one hand, a continuous wavelet transform (and its discretizations) was obtained by taking particular advantage of the conception of spherical singular integrals.…”