{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T23:28:43Z","timestamp":1648682923181},"reference-count":27,"publisher":"World Scientific Pub Co Pte Lt","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Wavelets Multiresolut Inf. Process."],"published-print":{"date-parts":[[2014,9]]},"abstract":"<jats:p> In this paper, we discuss the best approximation of functions on the sphere by spherical polynomials and the approximation by the Fourier partial summation operators and the Vall\u00e9e-Poussin operators, on a Sobolev space with a Gaussian measure in the probabilistic case setting, and get the probabilistic error estimation. We show that in the probabilistic case setting, the Fourier partial summation operators and the Vall\u00e9e-Poussin operators are the order optimal linear operators in the L<jats:sub>q<\/jats:sub> space for 1 \u2264 q \u2264 \u221e, but the spherical polynomial spaces are not order optimal in the L<jats:sub>q<\/jats:sub> space for 2 &lt; q \u2264 \u221e. This is completely different from the situation in the average case setting, which the spherical polynomial spaces are order optimal in the L<jats:sub>q<\/jats:sub> space for 1 \u2264 q &lt; \u221e. Also, in the L<jats:sub>q<\/jats:sub> space for 1 \u2264 q \u2264 \u221e, worst-case order optimal subspaces are also order optimal in the probabilistic case setting. <\/jats:p>","DOI":"10.1142\/s0219691314610128","type":"journal-article","created":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T02:43:00Z","timestamp":1394592180000},"page":"1461012","source":"Crossref","is-referenced-by-count":2,"title":["Approximation of functions on the sphere on a Sobolev space with a Gaussian measure in the probabilistic case setting"],"prefix":"10.1142","volume":"12","author":[{"given":"Heping","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, BCMIIS, Capital Normal University, Beijing 100048, P. R. China"}]},{"given":"Xuebo","family":"Zhai","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Zaozhuang University, Zaozhuang 277160, Shandong, P. R. 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