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Maxim R. Burke

Publications and source records attributed to Maxim R. Burke.

5 recordsLinked to original sources
arXiv2025

Comonotone approximation and interpolation by entire functions II

A theorem of Hoischen states that given a positive continuous function $\varepsilon:\mathbb{R}\to\mathbb{R}$, an integer $n\geq 0$, and a closed discrete set $E\subseteq\mathbb{R}$, any $C^n$ function $f:\mathbb{R}\to\mathbb{R}$ can be approximated by an entire function $g$ so that for $k=0,\dots,n$, and $x\in\mathbb{R}$, $|D^{k}g(x)-D^{k}f(x)|<\varepsilon(x)$, and if $x\in E$ then $D^{k}g(x)=D^{k}f(x)$. The approximating function $g$ is entire and hence piecewise monotone. Building on earlier work, for $n\leq 3$, we determine conditions under which when $f$ is piecewise monotone we can choose $g$ to be comonotone with $f$ (increasing and decreasing on the same intervals), and under which the derivatives of $g$ can be taken to be comonotone with the corresponding derivatives of $f$ if the latter are piecewise monotone. The proof for $n\leq 3$ establishes the theorem for all $n$, assuming a conjecture (shown in previous work with Haris and Madhavendra to hold for $n\leq 3$) regarding the set of $2(n+1)$-tuples $(f(0),Df(0),\dots,D^nf(0),f(1),Df(1),\dots,D^nf(1))$ of the values at the endpoints of the derivatives of a $C^n$ function $f$ on $[0,1]$ for which $D^nf$ is increasing and not constant.

math.CA↗
arXiv2025

Repeated integrals of increasing functions

Motivated by a problem on comonotone approximation of $C^n$ functions by entire functions, for increasing functions $f\colon[0,1]\to[0,1]$, we characterize the possible values of $(a,b,c)$, where $a=I(f)(1)$, $b=I^2(f)(1)$, $c=I^3(f)(1)$ ($I$ is the integral operator $I(f)(x)=\int_0^xf(t)\,dt$), as those which satisfy the conditions $0\leq a\leq 1$, $a^2/2\leq b\leq a/2$, $2b^2\leq 3ac$, $a^2 + 4b^2 + 6c\leq 6ac +2ab+2b$, and $0\leq c\leq a/6$. Our main theorem states that if $a,b,c$ are real numbers for which the inequalities are strict, then there is a function $f$ satisfying $a=I(f)(1)$, $b=I^2(f)(1)$, $c=I^3(f)(1)$ which is $C^\infty$ with $f(0)=0$, $f(1)=1$, $Df(x)>0$ for $0<x<1$, and whose derivatives $D^jf(0)$ and $D^jf(1)$, $j\geq 1$, are arbitrary as long as they are consistent with the increasing nature of $f$. The construction of $f$ proceeds by starting with a continuous parametrization $s\mapsto ρ_s\in C^\infty([0,1])$ defined on an open subset of $\mathbb{R}^4$, and composing with successive continuous transversals through the open set to fix the values of $I^j(ρ_s)(1)$ for $j=0,1,2,3$. Addressing the aforementioned problem on comonotone approximation, we examine the set $V_n\subseteq\mathbb{R}^{2(n+1)}$ of possible values $D^jf(0)$, $D^jf(1)$, $j=0,\dots,n$, of the derivatives of a $C^n$ function at the endpoints when $D^nf$ is increasing but not constant. We make a conjecture about the nature of this set and prove our conjecture for $n\leq 3$ as a consequence of the theorem mentioned above.

math.CA↗
arXiv2025

Vector liftings for products of probability spaces and measurable modifications of stochastic processes

We investigate the properties of linear primitive liftings $ρ\colon \mathcal{L}^p(μ)\to \mathcal{L}^p(μ)$ for probability spaces $(X,Σ,μ)$, which are linear maps selecting a representative from each class for almost everywhere equality. We call them vector liftings. They have the advantage over liftings or linear liftings that they exist for all $p\in[0,\infty]$, not only for $p=\infty$. Their relationship to products is still not clear, but we establish existence of (strong) product vector liftings for products of two factors. The vector liftings which are $2$-marginals with respect to a suitable product yielded a characterization of stochastic processes having a measurable modification modelled on one discovered by Musiał, and led to a proof of the characterization that does not use liftings. The improvement relies on results on extending a measure by null sets that apply when the family of new null sets is an ideal of a $σ$-algebra larger than the domain of the measure, but not necessarily an ideal of the power set. These allow us to reduce or eliminate completeness assumptions on the measures from several of our results.

math.PR↗
arXiv2004

Hechler's theorem for the null ideal

We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler's classical result in the theory of forcing, and the statement of the theorem for the meager ideal has been already proved by Bartoszynski and the author.

math.LO↗