Approximation in weighted spaces of vector functions

Abstract

In this paper, we present the duality theory for general weighted space of vector functions. We mention that a characterization of the dual of a weighted space of vector functions in the particular case $V \subset C^{+} (X)$ is mentioned by J. B. Prolla in [6]. Also, we extend de Branges lemma in this new setting for convex cones of a weighted spaces of vector functions (Theorem 4.2). Using this theorem, we find various approximations results for weighted spaces of vector functions: Theorems 4.2-4.6 as well as Corollary 4.3. We mention also that a brief version of this paper, in the particular case $V \subset C^{+} (X)$, is presented in [3], Chapter 2, subparagraph 2.5.

Keywords

• Nachbin family
• weighted space
• antialgebraic set

References

1. L. De Branges: The Stone-Weierstrass theorem, Proc. Amer. Math. Soc., 10 (5) (1959), 822–824.
2. I. Bucur, G. Pâltineanu: De Branges type lemma and approximation in weighted spaces, Mediterranean J. Math., (to appear).
3. I. Bucur, G. Pâltineanu: Topics in the uniform approximation of continuous functions, Birkhauser (2020).
4. L. Nachbin: Weigthed approximation for algebras and modules of continuous functions: real and self-adjoint complex cases, Ann. of Math., 81 (1965), 289–302.
5. L. Nachbin: Elements of approximation theory, D. Van Nostrand, Princeton (1967).
6. J. B. Prolla: Bishop’s generalized Stone-Weierstrass theorem for weighted spaces, Math. Anal., 191 (4) (1971), 283–289.
7. W. H. Summers: Dual spaces of weighted spaces, Trans. Amer. Math. Soc., 151 (1) (1970), 323–333.