A concept of generalized discrepancy, which involves pseudodifferential operators to give a criterion of equidistributed pointsets, is developed on the sphere. A simply structured formula in terms of elementary functions is established for the computation of the generalized discrepancy. With the help of this formula five kinds of point systems on the sphere, namely lattices in polar coordinates, transformed two-dimensional sequences, rotations on the sphere, triangulations, and "sum of three squares sequence," are investigated. Quantitative tests are done, and the results are compared with one another. Our calculations exhibit different orders of convergence of the generalized discrepancy for different types of point systems.AMS subject classifications. 33C55, 40A10, 47G30, 62E25, 86-08 PII. S1064827595281344 Introduction.Of practical importance is the problem of generating equidistributed pointsets on the sphere. For that reason the concept of generalized discrepancy, which involves pseudodifferential operators to give a quantifying criterion of equidistributed pointsets, is of great interest. In this paper an explicit formula in terms of elementary functions is developed for the generalized discrepancy. Essential tools are Sobolev space structures and pseudodifferential operator techniques. It is mentioned that an optimal pointset may be obtained by minimizing the generalized discrepancy. But, in spite of the elementary representation of the generalized discrepancy, this is a nonlinear optimization problem which will not be discussed here. Our investigations, however, show that there are many promising ways to generate point systems on the sphere such that the discrepancy becomes small. To be specific, we distinguish five kinds of point systems on the sphere: lattices in polar coordinates, transformed two-dimensional sequences, rotations on the sphere, triangulations, and "sum of three squares sequence." By using our developed formulas, the five classes of point systems are described, and their discrepancies are explicitly calculated for increasing numbers of points. The results show different orders of convergence indicated by the generalized discrepancy. Furthermore, our computations enable us to give a quantitative comparison between the different point systems. It is somewhat surprising that certain types of transformed sequences yield the best results. Nevertheless, there are special other pointsets which provide us with better results for comparable numbers of points. For instance, the soccer ball (C 60 ) leads us to the best result in all our considered pointsets of about 60 points.The problem of generating a large number of "equidistributed points" on the sphere has many applications in various fields of computation, particularly in geoscience and medicine. The advantage of equidistributed point systems lies in the fact that relatively few samplings of the data are needed, and approximate integration can be simply performed by computation of a mean value, i.e., the arithmetical mean.

The basic theory of spherical singular integrals is recapitulated. Criteria are given for measuring the space-frequency localization of functions on the sphere. The trade-off between "space localization" on the sphere and "frequency localization" in terms of spherical harmonics is described in form of an "uncertainty principle." A continuous version of spherical multiresolution is introduced, starting from continuous wavelet transform corresponding to spherical wavelets with vanishing moments up to a certain order. The wavelet transform is characterized by least-squares properties. Scale discretization enables us to construct spherical counterparts of P(acket)-scale discretized and D(aubechies)-scale discretized wavelets. It is shown that singular integral operators forming a semigroup of contraction operators of class (C 0 ) (like Abel-Poisson or Gauß-Weierstraß operators) lead in canonical way to pyramid algorithms. Fully discretized wavelet transforms are obtained via approximate integration rules on the sphere. Finally applications to (geo-)physical reality are discussed in more detail. A combined method is proposed for approximating the "low frequency parts" of a physical quantity by spherical harmonics and the "high frequency parts" by spherical wavelets. The particular significance of this combined concept is motivated for the situation of today's physical geodesy, viz. the determination of the high frequency parts of the earth's gravitational potential under explicit knowledge of the lower order part in terms of a spherical harmonic expansion.

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Spherical spline functions are introduced by use of Green's surface functions with respect to the (Laplace‐)Beltrami operator of the (unit) sphere. Natural (spherical) spline functions are used to interpolate data discretely given on the sphere. A method is presented that allows the smoothing of irregularities in measured values or experimental data. Extensions of Peano's theorem and Sard's theory of best approximation to the spherical case are given by integral formulas. Schoenberg's theorem is transcribed into spherical nomenclature.