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Approximation in \(L_2\) Sobolev spaces on the 2-sphere by quasi-interpolation. (English) Zbl 0986.41009

The authors consider radial quasi-interpolation by polynomials on the unit sphere in \(\mathbb{R}^3\), and present error estimates for quasi-interpolation in Sobolev spaces. Using equiangular grid points, a discrete Fourier transform on the sphere is studied. The authors show how the discrete Fourier series is a special case of the general framework, and give error bounds for the discrete Fourier projection in Sobolev spaces. Some applications in spherical wavelet analysis are discussed too.

MSC:

41A30 Approximation by other special function classes
41A63 Multidimensional problems
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
65T50 Numerical methods for discrete and fast Fourier transforms
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[1] Abramowitz, M. and Stegun, I.A. (1964).Handbook of Mathematical Functions, National Bureau of Standards, Dover Publications. · Zbl 0171.38503
[2] Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, G. (1953).Higher Transcendental Functions, The Bateman Manuscript Project, Volume 2, McGraw-Hill.
[3] Driscoll, J.R. and Healy, D.M. (1994). Computing Fourier transforms and convolutions on the 2-sphere,Advances in Applied Mathematics,15, 202–250. · Zbl 0801.65141 · doi:10.1006/aama.1994.1008
[4] Fasshauer, G.E. and Schumaker, L.L. (1998). Scattered data fitting on the sphere.Mathematical methods for curves and surfaces, II (Lillehammer, 1997), 117–166.Innov. Appl. Math., Vanderbilt University Press, Nashville, TN. · Zbl 0904.65015
[5] Freeden, W., Gervens, T., and Schreiner, M. (1998).Constructive Approximation on the Sphere, Claredon Press. · Zbl 0896.65092
[6] Freeden, W. and Windheuser, U. (1997). Combined spherical harmonic and wavelet expansion-a future concept in earth’s gravitational determination,Appl. Comput. Harmon. Anal.,4, 1–37. · Zbl 0865.42029 · doi:10.1006/acha.1996.0192
[7] Kushpel, A.K. and Levesley, J. (2000). Quasi-interpolation on the 2-sphere using radial polynomials,J. Approx. Theory,102, 141–154. · Zbl 0949.41003 · doi:10.1006/jath.1999.3373
[8] Muller, C. (1966). Spherical harmonics,Lecture Notes in Mathematics, 17, Springer Verlag. · Zbl 0138.05101
[9] Orzag, S.A. (1970). Transform method for the calculation of vector-coupled sums: applications to spectral form of the vorticity equation,J. Atmosph. Sci.,27, 890–895. · doi:10.1175/1520-0469(1970)027<0890:TMFTCO>2.0.CO;2
[10] Potts, D., Steidl, G., and Tasche, M. (1996). Kernels of spherical harmonics and spherical frames, inAdvanced Topics in Multivariate Approximation, Fontanella, F., et al., Eds., World Science Publishing, River Edge, NJ. · Zbl 1273.42031
[11] Powell, M.J.D. (1992). The theory of radial basis functions in 1990, inAdvances in Numerical Analysis II: Wavelets, Subdivision, and Radial Basis Functions, Light, W.A., Ed., Oxford University Press Oxford.
[12] Reimer, M. (2000). Hyperinterpolation on the sphere at the minimal projection order,J. Approx. Theory,104, 272–286. · Zbl 0959.41001 · doi:10.1006/jath.2000.3454
[13] Schreiner, M. (1996). A pyramid scheme for spherical wavelets,AGTM Report 170, University of Kaiserslauten. · Zbl 0863.68077
[14] Sloan, I.H. (1997). Interpolation and hyperinterpolation on the sphere,Multivariate Approximation., (Witten-Bommerholz, 1996), 255–268.Math. Res.,101, Akademie Verlag, Berlin. · Zbl 0894.41005
[15] Szegö, G. (1959).Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, RI. · Zbl 0089.27501
[16] Jakob-Chien, R., Hack, J.J., and Williamson, D.L. (1995). Spectral transform solutions to shallow water test sets,J. Comp. Physics,119, 164–187. · Zbl 0878.76059 · doi:10.1006/jcph.1995.1125
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