Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Polynomial approximation on compact manifolds and homogeneous spaces


Author: David L. Ragozin
Journal: Trans. Amer. Math. Soc. 150 (1970), 41-53
MSC: Primary 41A65
DOI: https://doi.org/10.1090/S0002-9947-1970-0410210-0
MathSciNet review: 0410210
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove several theorems which relate the smoothness of a function, $ f$, defined on a compact $ {C^\infty }$-submanifold of a Euclidean space to the rate at which the error in the best uniform approximation to $ f$ by polynomials of degree at most $ n$ tends to zero.


References [Enhancements On Off] (What's this?)

  • [1] A. S. Džafarov, On the order of the best approximations of the functions continuous on the unit sphere by means of finite spherical sums, Proc. Second All-Union Conference (Baku, 1962), Studies Contemporary Problems Constructive Theory of Functions, Izdat. Akad. Nauk Azerbaĭdžan. SSR, Baku, 1965, pp. 46-52. (Russian) MR 33 #6227. MR 0198068 (33:6227)
  • [2] Sun Kung, Fourier analysis on unitary groups. IV: On the Peter-Weyl theorem, Acta Math. Sinica 13 (1963), 323-331=Chinese Math.-Acta 4 (1964), 351-359. MR 31 #4861. MR 0180627 (31:4861)
  • [3] T. H. Gronwall, On the degree of convergence of Laplace series, Trans. Amer. Math. Soc. 15 (1914), 1-30. MR 1500962
  • [4] G. G. Kušnirenko, The approximation of functions defined on the unit sphere by finite spherical sums, Naučn. Dokl. Vysš. Skoly Fiz.-Mat. Nauki 1958, no. 4, 47-53. (Russian) MR 26 #530. MR 0142963 (26:530)
  • [5] G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York, 1966. MR 35 #4642. MR 0213785 (35:4642)
  • [6] G. D. Mostow, Equivariant embeddings in Euclidean space, Ann. of Math. (2) 65 (1957), 432-446. MR 19, 291. MR 0087037 (19:291c)
  • [7] J. Nash, Real algebraic manifolds, Ann. of Math. (2) 56 (1952), 405-421. MR 14, 403. MR 0050928 (14:403b)
  • [8] D. J. Newman and L. Raymon, A class of curves on which polynomials approximate efficiently, Proc. Amer. Math. Soc. 19 (1968), 595-599. MR 37 #653. MR 0225056 (37:653)
  • [9] D. J. Newman and H. S. Shapiro, Jackson's theorem in higher dimensions (With discussion), Proc. Conference on Approximation Theory (Oberwolfach, 1963), Birkhäuser, Basel, 1964, pp. 208-219. MR 32 #310. MR 0182828 (32:310)
  • [10] K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33-65. MR 15, 468. MR 0059050 (15:468f)
  • [11] D. L. Ragozin, Approximation theory on compact manifolds and Lie groups, with applications to harmonic analysis, PhD. Thesis, Harvard University, Cambridge, Mass., 1967.
  • [12] -, Constructive polynomial approximation on spheres and projective spaces, Trans. Amer. Math. Soc. (to appear). MR 0288468 (44:5666)
  • [13] A. F. Timan, Theory of approximation of functions of a real variable, Fizmatgiz, Moscow, 1960; English transl., Internat. Series of Monographs in Pure and Appl. Math., vol. 34, Macmillan, New York, 1963. MR 22 #8257; MR 33 #465. MR 0117478 (22:8257)
  • [14] H. Whitney, Elementary structure of real algebraic varieties, Ann. of Math. (2) 66 (1957), 545-556. MR 20 #2342. MR 0095844 (20:2342)
  • [15] O. Zariski and P. Samuel, Commutative algebra. Vol. II, The University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #11006. MR 0120249 (22:11006)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 41A65

Retrieve articles in all journals with MSC: 41A65


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0410210-0
Keywords: Polynomial approximation, Jackson theorems, $ n$-width, function spaces on manifolds, differential operators, homogeneous spaces, Bernstein's inequality
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society