Constructive polynomial approximation on spheres and projective spaces.
Author:
David L. Ragozin
Journal:
Trans. Amer. Math. Soc. 162 (1971), 157170
MSC:
Primary 41.15; Secondary 46.00
MathSciNet review:
0288468
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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 Ginvariant linear method of polynomial approximation to continuous functions can be modified to yield a linear method which produces better approximations to ktimes 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, McGrawHill
Book Co., New YorkToronto, Ont.London, 1966. MR 0222517
(36 #5568)
 [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 AllUnion Conf., Baku, 1962) Izdat. Akad. Nauk
Azerbaĭdžan. SSR, Baku, 1965, pp. 46–52 (Russian).
MR
0198068 (33 #6227)
 [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, http://dx.doi.org/10.1090/S00029947191415009626
 [4]
Sigurđur
Helgason, Differential geometry and symmetric spaces, Pure and
Applied Mathematics, Vol. XII, Academic Press, New YorkLondon, 1962. MR 0145455
(26 #2986)
 [5]
Sigurđur
Helgason, The Radon transform on Euclidean spaces, compact
twopoint homogeneous spaces and Grassmann manifolds, Acta Math.
113 (1965), 153–180. MR 0172311
(30 #2530)
 [6]
Sheng
Gong, Fourier analysis on unitary groups. IV. On the PeterWeyl
theorem, Chinese Math.—Acta 4 (1963),
351–359. MR 0180627
(31 #4861)
 [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 (26 #530)
 [8]
G.
G. Lorentz, Approximation of functions, Holt, Rinehart and
Winston, New YorkChicago, Ill.Toronto, Ont., 1966. MR 0213785
(35 #4642)
 [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
(32 #310)
 [10]
Katsumi
Nomizu, Invariant affine connections on homogeneous spaces,
Amer. J. Math. 76 (1954), 33–65. MR 0059050
(15,468f)
 [11]
David
L. Ragozin, Approximation theory on 𝑆𝑈(2), J.
Approximation Theory 1 (1968), 464–475. MR 0241877
(39 #3214)
 [12]
David
L. Ragozin, Polynomial approximation on compact
manifolds and homogeneous spaces, Trans. Amer.
Math. Soc. 150
(1970), 41–53. MR 0410210
(53 #13960), http://dx.doi.org/10.1090/S00029947197004102100
 [13]
G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1959. MR 21 #5029.
 [14]
Shinsheng
Tai, Minimum imbeddings of compact symmetric spaces of rank
one, J. Differential Geometry 2 (1968), 55–66.
MR
0231395 (37 #6950)
 [1]
 E. W. Cheney, Introduction to approximation theory, McGrawHill, New York, 1966. MR 36 #5568. MR 0222517 (36:5568)
 [2]
 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 AllUnion Conference (Baku, 1962), Studies of Contemporary Problems in Constructive Theory of Functions, Izdat. Akad. Nauk Azerbaidžan. SSR, Baku, 1965, pp. 4652. (Russian) MR 33 #6227. MR 0198068 (33:6227)
 [3]
 T. H. Gronwall, On the degree of convergence of Laplace series, Trans. Amer. Math. Soc. 15 (1914), 130. MR 1500962
 [4]
 S. Helgason, Differential geometry and symmetric spaces, Pure and Appl. Math., vol. XII, Academic Press, New York, 1962. MR 26 #2986. MR 0145455 (26:2986)
 [5]
 , The Radon transform on Euclidean spaces, compact twopoint homogeneous spaces, and Grassmann manifolds, Acta Math. 113 (1965), 153180. MR 30 #2530. MR 0172311 (30:2530)
 [6]
 Sun Kung, Fourier analysis on unitary groups. IV: On the PeterWeyl theorem, Acta Math. Sinica 13 (1963), 323331 = Chinese Math.Acta 4 (1964), 351359. MR 31 #4861. MR 0180627 (31:4861)
 [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, No. 4, 4753. (Russian) MR 26 #530. MR 0142963 (26:530)
 [8]
 G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York, 1966. MR 35 #4642. MR 0213785 (35:4642)
 [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. 208219. MR 32 #310. MR 0182828 (32:310)
 [10]
 K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 3365. MR 15, 468. MR 0059050 (15:468f)
 [11]
 D. L. Ragozin, Approximation theory on , J. Approximation Theory 1 (1968), 464475. MR 39 #3214. MR 0241877 (39:3214)
 [12]
 , Polynomial approximation on compact manifolds and homogeneous spaces, Trans. Amer. Math. Soc. 150 (1970), 4153. MR 0410210 (53:13960)
 [13]
 G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1959. MR 21 #5029.
 [14]
 S. S. Tai, Minimum imbeddings of compact symmetric spaces of rank one, J. Differential Geometry 2 (1968), 5566. MR 37 #6950. MR 0231395 (37:6950)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
41.15,
46.00
Retrieve articles in all journals
with MSC:
41.15,
46.00
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197102884681
PII:
S 00029947(1971)02884681
Keywords:
Polynomial approximation on spheres,
polynomial approximation on projective spaces,
Jackson estimates,
invariant approximation operators,
homogeneous spaces
Article copyright:
© Copyright 1971
American Mathematical Society
