<p>We develop a theory of abstract intermediate function spaces on a compact convex set <i>X</i> and study the behaviour of multipliers and centers of these spaces. In particular, we provide some criteria for coincidence of the center with the space of multipliers and a general theorem on boundary integral representation of multipliers. We apply the general theory in several concrete cases, among others to strongly affine Baire functions, to the space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1249_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$A_f(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of fragmented affine functions, to the space <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1249_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$(A_f(X))^\mu $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>μ</mi> </msup> </math></EquationSource> </InlineEquation>, the monotone sequential closure of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1249_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$A_f(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, to their natural subspaces formed by Borel functions, or, in some special cases, to the space of all strongly affine functions. In addition, we prove that the space <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1249_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$(A_f(X))^\mu $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>μ</mi> </msup> </math></EquationSource> </InlineEquation> is determined by extreme points and provide a large number of illustrating examples and counterexamples.</p>

Boundary integral representation of multipliers of fragmented affine functions and other intermediate function spaces

期刊论文
发表日期：

2025年5月31日

查看全文

Ondřej F. K. Kalenda,
Jakub Rondoš,
Jiří Spurný

摘要

We develop a theory of abstract intermediate function spaces on a compact convex set X and study the behaviour of multipliers and centers of these spaces. In particular, we provide some criteria for coincidence of the center with the space of multipliers and a general theorem on boundary integral representation of multipliers. We apply the general theory in several concrete cases, among others to strongly affine Baire functions, to the space $A_f(X)$ $A_{f} (X)$ of fragmented affine functions, to the space $(A_f(X))^\mu $ ${(A_{f} (X))}^{μ}$ , the monotone sequential closure of $A_f(X)$ $A_{f} (X)$ , to their natural subspaces formed by Borel functions, or, in some special cases, to the space of all strongly affine functions. In addition, we prove that the space $(A_f(X))^\mu $ ${(A_{f} (X))}^{μ}$ is determined by extreme points and provide a large number of illustrating examples and counterexamples.