×

Fundamental sets of continuous functions on spheres. (English) Zbl 0886.41016

Let \(S^m\) and \(S^\infty\) denote the unit spheres in \(\mathbb{R}^{m+1}\) and \(\ell^2\), respectively. The authors look for functions \(f\) in \(C[- 1,1]\) such that the family of functions \(x\mapsto f(\langle x,v\rangle)\), where \(v\) runs over \(S^m\), is fundamental in the space \(C(S^m)\). They also consider this problem for \(C(S^\infty)\) when this space is given the topology of uniform convergence on compact sets.

MSC:

41A30 Approximation by other special function classes
41A45 Approximation by arbitrary linear expressions
41A63 Multidimensional problems
42A65 Completeness of sets of functions in one variable harmonic analysis
42A82 Positive definite functions in one variable harmonic analysis
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R. Askey (1975): Orthogonal Polynomials and Special Functions. Regional Conference Series in Applied Mathematics, Vol. 21. Philadelphia: SIAM. · Zbl 0298.33008
[2] S. Chanillo, B. Muckenhoupt (1993): Weak Type Estimates for Césaro Sums of Jacobi Polynomial Series. Memoirs, Amer. Math. Soc., No. 487. Providence, RI. · Zbl 0773.40001
[3] N. Dunford, J. T. Schwartz (1953): ”Linear Operators, Part I. General Theory. New York: Interscience. · Zbl 0084.10402
[4] E. Kogbetliantz (1924):Recherches sur la summabilité des séries ultrasphériques par la méthod des moyennes arithmetique. J. Math. Pures Appl.,3:107–187. · JFM 50.0207.05
[5] C. Müller (1966): Spherical Harmonics. Lecture Notes in Mathematics, Vol. 17. Berlin: Springer-Verlag. · Zbl 0138.05101
[6] I. J. Schoenberg (1942):Positive definite functions on spheres. Duke Math. J.,9:96–108. · Zbl 0063.06808 · doi:10.1215/S0012-7094-42-00908-6
[7] E. M. Stein, andG. Weiss (1971): Introduction to Fourier Analysis on Euclidean Spaces. Princeton, NJ: Princeton University Press. · Zbl 0232.42007
[8] Xingping Sun (1994):The fundamentality of translates of a continuous function on spheres. Namer. Algorithms,8: 131–134. · Zbl 0819.43003 · doi:10.1007/BF02145700
[9] G. Szegö (1959): Orthogonal Polynomials. Amer. Math. Soc. Colloquium Publ., Vol. XXIII New York.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.