On Polynomial Approximation of Functions on Hilbert Space

Some algebras of symmetric analytic functions and their spectra

“…Symmetric polynomials on p (with respect to G) and L p [0, 1] (with respect to the group of measurepreserving permutations on [0, 1]) for 1 p < ∞ were first studied by Nemirovski and Semenov in [12]. Symmetric polynomials on p (with respect to G) and L p [0, 1] (with respect to the group of measurepreserving permutations on [0, 1]) for 1 p < ∞ were first studied by Nemirovski and Semenov in [12].…”

Representation of spectra of algebras of block-symmetric analytic functions of bounded type

“…The action of this group has a natural extension to the action on the algebra H b (X) of analytic functions of bounded type on X. Invariants of this representation of S ∞ are so-called symmetric analytic functions of bounded type on X. The algebras of symmetric analytic functions H bs (X) were investigated by many authors ( [1,2,9]). In particular, it is known that H bs ( p ) admits an algebraic basis for 1 ≤ p < ∞.…”

Local smoothing of uniformly smooth maps

“…It readily follows that every uniformly continuous Hilbert-valued map defined on an arbitrary subset of a Hilbert space can be uniformly approximated by a uniformly Lipschitz map. The approximation of uniformly continuous real functions defined on balls in Hilbert spaces and L p -spaces by polynomials and by elements of some classes of uniformly smooth functions was also studied by Nemirovskii and Semenov [4]- [7]. This topic was further developed in the author's papers [8]- [17].…”