Approximation theory in the space of sections of a vector bundle

Author:
David Handel

Journal:
Trans. Amer. Math. Soc. **256** (1979), 383-394

MSC:
Primary 55R25; Secondary 41A65, 55R40

DOI:
https://doi.org/10.1090/S0002-9947-1979-0546924-X

MathSciNet review:
546924

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a real *m*-plane bundle and *S* an *n*-dimensional subspace of the space of sections of *E*. *S* is said to be *k*-regular if whenever are distinct points of *B* and , , there exists a such that for . It is proved that if *E* has a Riemannian metric and *B* is compact Hausdorff with at least points, then *S* is *k*-regular if and only if for each , the set of best approximations to by elements of *S* has dimension at most *n* - *km*. This extends a classical theorem of Haar, Kolmogorov, and Rubinstein (the case of the product line bundle). Complex and quaternionic analogues of the above are obtained simultaneously. Existence and nonexistence of *k*-regular subspaces of a given dimension are obtained in special cases via cohomological methods involving configuration spaces. For example, if *E* is the product real -plane bundle over a 2-dimensional disk, then contains a *k*-regular subspace of dimension , but not one of dimension , where denotes the number of ones in the dyadic expansion of *k*.

**[1]**N. I. Achieser,*Theory of approximation*, Translated by Charles J. Hyman, Frederick Ungar Publishing Co., New York, 1956. MR**0095369****[2]**K. Borsuk,*On the 𝑘-independent subsets of the Euclidean space and of the Hilbert space*, Bull. Acad. Polon. Sci. Cl. III.**5**(1957), 351–356, XXIX (English, with Russian summary). MR**0088710****[3]**F. R. Cohen and D. Handel,*𝑘-regular embeddings of the plane*, Proc. Amer. Math. Soc.**72**(1978), no. 1, 201–204. MR**524347**, https://doi.org/10.1090/S0002-9939-1978-0524347-1**[4]**F. R. Cohen, M. E. Mahowald, and R. J. Milgram,*The stable decomposition for the double loop space of a sphere*, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 225–228. MR**520543****[5]**David Handel,*An embedding theorem for real projective spaces*, Topology**7**(1968), 125–130. MR**0227998**, https://doi.org/10.1016/0040-9383(68)90020-7**[6]**David Handel,*Obstructions to 3-regular embeddings*, Houston J. Math.**5**(1979), no. 3, 339–343. MR**559974****[7]**-,*Some existence and non-existence theorems for k-regular maps*, Fund. Math. (to appear).**[8]**David Handel and Jack Segal,*On 𝑘-regular embeddings of spaces in Euclidean space*, Fund. Math.**106**(1980), no. 3, 231–237. MR**584495**, https://doi.org/10.4064/fm-106-3-231-237**[9]**Dusa McDuff,*Configuration spaces of positive and negative particles*, Topology**14**(1975), 91–107. MR**0358766**, https://doi.org/10.1016/0040-9383(75)90038-5**[10]**Ivan Singer,*Best approximation in normed linear spaces by elements of linear subspaces*, Translated from the Romanian by Radu Georgescu. Die Grundlehren der mathematischen Wissenschaften, Band 171, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New York-Berlin, 1970. MR**0270044****[11]**Wen Tsün Wu,*On the realization of complexes in euclidean spaces. II*, Acta Math. Sinica**7**(1957), 79–101 (Chinese, with English summary). MR**0097056**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
55R25,
41A65,
55R40

Retrieve articles in all journals with MSC: 55R25, 41A65, 55R40

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1979-0546924-X

Keywords:
Vector bundles,
best approximation of sections,
*k*-regular subspaces of sections,
configuration spaces,
characteristic classes

Article copyright:
© Copyright 1979
American Mathematical Society