arXiv subjects

S. V. Butler

Publications and source records attributed to S. V. Butler.

2 recordsLinked to original sources

arXiv2026

Image transformations, Markov operators, and sample median

(I.) We consider generalizations of an iterated function system and the associated Markov operators. A Markov operator, defined on the space of (deficient) topological measures on a locally compact space, is an infinite convex linear combination of adjoints of (d-) image transformations. Restricted to measures, this Markov-Feller operator has a nonlinear dual operator given by an infinite convex linear combination of (conic) quasi-homomorphisms. If (d-) image transformations are contractions with respect to the Kantorovich-Rubinstein metric, a Markov operator has the unique invariant (deficient) topological measure. Taking a compact space, finitely many inverses of contractions as image transformations, and restricting the Markov operator to measures gives the classical result from the theory of fractals. There are various relations between Markov operator and the iterated function system where adjoints of (d-) image transformations are contractions on the compact metric space of $\{0,1\}$-valued (deficient) topological measures. For instance, the invariant (deficient) topological measure is the composition of the fixed point of the IFS and the basic (d-) image transformation. (II.) We define a generalized distribution of the sample median (g.d.s.m.) for continuous proper maps using an image transformation. We show that the g.d.s.m. and the inverse on the sample median are equivariant under solid variables, a large collection of transformations. On $\mathbb{R}^n$ such transformations include rotations, translations, symmetries, stretching, projections, monotone maps, etc. (III.) We show that a (signed) topological measure on a locally compact space with the covering dimension $\dim X \le 1$ is a (signed) Radon measure.

math.FA↗

arXiv2025

Quasi-linear maps and image transformations

Conic quasi-linear maps are nonlinear operators from $C_0(X)$ to a normed linear space $E$ which preserve nonnegative linear combinations on positive cones generated by single functions; quasi-linear maps are linear on singly generated subalgebras. While nonlinear, a quasi-linear map is bounded iff it is continuous. $E = \mathbb{R}$ gives quasi-integrals, which correspond to (deficient) topological measures - nonsubadditive set functions generalizing measures. Like image measures $\mu \circ u^{-1}$, (d-) image transformations move (deficient) topological measures from one space to another, generalizing $u^{-1}$. We give criteria for a (d-) image transformation to be $u^{-1}$ for some proper continuous function. We study the interrelationships between (conic) quasi-linear maps, quasi-integrals, (deficient) topological measures and (d-) image transformations when $E = C_0(Y), X, Y$ are locally compact. (Conic) quasi-homomorphisms behave like homomorphisms on singly generated subalgebras or cones. We show that (conic) quasi-homomorphisms are in 1-1 correspondence with (d-) image transformations and with certain continuous proper functions. We give criteria for a (conic) quasi-linear map to be a (conic) quasi-homomorphism, and for the latter to be an algebra homomorphism. Any conic quasi-linear map or bounded quasi-linear map is a composition of an algebra homomorphism with the basic quasi-linear map, and we give criteria for the latter to be linear. We study the adjoints of (d-) image transformations and (conic) quasi-linear maps; for (conic) quasi-homomorphisms they give Markov-Feller operators with nonlinear duals.

math.FA↗