Constructive polynomial approximation on spheres and projective spaces.

Author:
David L. Ragozin

Journal:
Trans. Amer. Math. Soc. **162** (1971), 157-170

MSC:
Primary 41.15; Secondary 46.00

DOI:
https://doi.org/10.1090/S0002-9947-1971-0288468-1

MathSciNet review:
0288468

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Abstract: This paper contains constructive generalizations to functions defined on spheres and projective spaces of the Jackson theorems on polynomial approximation. These results, (3.3) and (4.6), give explicit methods of constructing uniform approximations to smooth functions on these spaces by polynomials, together with error estimates based on the smoothness of the function and the degree of the polynomial. The general method used exploits the fact that each space considered is the orbit of some compact subgroup, *G*, of an orthogonal group acting on a Euclidean space. For such homogeneous spaces a general result (2.1) is proved which shows that a *G*-invariant linear method of polynomial approximation to continuous functions can be modified to yield a linear method which produces better approximations to *k*-times differentiable functions. Jackson type theorems (3.4) are also proved for functions on the unit ball (which is not homogeneous) in a Euclidean space.

**[1]**E. W. Cheney,*Introduction to approximation theory*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0222517****[2]**Arif S. Džafarov,*On the order of the best approximations of the functions continuous on the unit sphere by means of finite spherical sums*, Studies Contemporary Problems Constructive Theory of Functions (Proc. Second All-Union Conf., Baku, 1962) Izdat. Akad. Nauk Azerbaĭdžan. SSR, Baku, 1965, pp. 46–52 (Russian). MR**0198068****[3]**T. H. Gronwall,*On the degree of convergence of Laplace’s series*, Trans. Amer. Math. Soc.**15**(1914), no. 1, 1–30. MR**1500962**, https://doi.org/10.1090/S0002-9947-1914-1500962-6**[4]**SigurÄ‘ur Helgason,*Differential geometry and symmetric spaces*, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR**0145455****[5]**SigurÄ‘ur Helgason,*The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds*, Acta Math.**113**(1965), 153–180. MR**0172311**, https://doi.org/10.1007/BF02391776**[6]**Sheng Gong,*Fourier analysis on unitary groups. IV. On the Peter-Weyl theorem*, Chinese Math.—Acta**4**(1963), 351–359. MR**0180627****[7]**G. G. Kušnirenko,*The approximation of functions defined on the unit sphere by finite spherical sums*, Naučn. Dokl. Vysš. Skoly Fiz.-Mat. Nauki**1958**(1958), no. 4, 47–53 (Russian). MR**0142963****[8]**G. G. Lorentz,*Approximation of functions*, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. MR**0213785****[9]**D. J. Newman and H. S. Shapiro,*Jackson’s theorem in higher dimensions*, On Approximation Theory (Proceedings of Conference in Oberwolfach, 1963), Birkhäuser, Basel., 1964, pp. 208–219. MR**0182828****[10]**Katsumi Nomizu,*Invariant affine connections on homogeneous spaces*, Amer. J. Math.**76**(1954), 33–65. MR**0059050**, https://doi.org/10.2307/2372398**[11]**David L. Ragozin,*Approximation theory on 𝑆𝑈(2)*, J. Approximation Theory**1**(1968), 464–475. MR**0241877****[12]**David L. Ragozin,*Polynomial approximation on compact manifolds and homogeneous spaces*, Trans. Amer. Math. Soc.**150**(1970), 41–53. MR**0410210**, https://doi.org/10.1090/S0002-9947-1970-0410210-0**[13]**G. Szegö,*Orthogonal polynomials*, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1959. MR**21**#5029.**[14]**Shin-sheng Tai,*Minimum imbeddings of compact symmetric spaces of rank one*, J. Differential Geometry**2**(1968), 55–66. MR**0231395**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0288468-1

Keywords:
Polynomial approximation on spheres,
polynomial approximation on projective spaces,
Jackson estimates,
invariant approximation operators,
homogeneous spaces

Article copyright:
© Copyright 1971
American Mathematical Society