Lipschitz symmetric functions on Banach spaces with symmetric bases

We investigate Lipschitz symmetric functions on a Banach space $X$ with a symmetric basis. We consider power symmetric polynomials on $\ell_1$ and show that they are Lipschitz on the unbounded subset consisting of vectors $x\in \ell_1$ such that $|x_n|\le 1.$ Using functions $\max$ and $\min$ and tropical polynomials of several variables, we constructed a large family of Lipschitz symmetric functions on the Banach space $c_0$ which can be described as a semiring of compositions of tropical polynomials over $c_0$.

Approximations of Symmetric Functions on Banach Spaces with Symmetric Bases

This paper is devoted to studying approximations of symmetric continuous functions by symmetric analytic functions on a Banach space X with a symmetric basis. We obtain some positive results for the case when X admits a separating polynomial using a symmetrization operator. However, even in this case, there is a counter-example because the symmetrization operator is well defined only on a narrow, proper subspace of the space of analytic functions on X. For X=c0, we introduce ε-slice G-analytic functions that have a behavior similar to G-analytic functions at points x∈c0 such that all coordinates of x are greater than ε, and we prove a theorem on approximations of uniformly continuous functions on c0 by ε-slice G-analytic functions.

Duality preserving Gray maps for codes over rings

Given a finite ring [Formula: see text] which is a free left module over a subring [Formula: see text] of [Formula: see text], two types of [Formula: see text]-bases, pseudo-self-dual bases (similar to trace orthogonal bases) and symmetric bases, are defined which in turn are used to define duality preserving maps from codes over [Formula: see text] to codes over [Formula: see text]. Both types of bases are generalizations of similar concepts for fields. Many illustrative examples are given to shed light on the advantages to such mappings as well as their abundance.

On Perfectly Homogeneous Bases in Quasi-Banach Spaces

For0<p<∞the unit vector basis ofℓphas the property of perfect homogeneity: it is equivalent to all its normalized block basic sequences, that is, perfectly homogeneous bases are a special case of symmetric bases. For Banach spaces, a classical result of Zippin (1966) proved that perfectly homogeneous bases are equivalent to either the canonicalc0-basis or the canonicalℓp-basis for some1≤p<∞. In this note, we show that (a relaxed form of) perfect homogeneity characterizes the unit vector bases ofℓpfor0<p<1as well amongst bases in nonlocally convex quasi-Banach spaces.