Compiling essential results for non-quasianalytic ultradistribution spaces and Colombeau versions of generalized ultradistribution algebras, we analyze strong B- and strong R-association of a generalized ultradistribution <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2097_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$[(f_\varepsilon )]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mi>ε</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. The strong association of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2097_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$[(f_\varepsilon )]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mi>ε</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> with a Komatsu-type ultradistribution T, with an additional assumption on regularity of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2097_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$[(f_\varepsilon )]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mi>ε</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> of Beurling, respectively, Roumieu type, implies that T is an ultradifferentiable function of Beurling, Roumieu type, respectively. We show that under suitable conditions, a weakly negligible net <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2097_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$(f_\varepsilon )_{\varepsilon \in (0,1)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mi>ε</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>ε</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> is a negligible net in the sense of generalized ultradistributions. Furthermore, we prove that a translation-invariant generalized ultradistribution g is equal to a generalized constant in both types of generalized ultradistribution algebras.

Topology and regularity for generalized ultradistribution algebras

期刊论文
发表日期：

2025年7月2日

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Stevan Pilipović,
Dragana Risteski,
Dimitris Scarpalézos,
Milica Žigić

摘要

Compiling essential results for non-quasianalytic ultradistribution spaces and Colombeau versions of generalized ultradistribution algebras, we analyze strong B- and strong R-association of a generalized ultradistribution $[(f_\varepsilon )]$ $[(f_{ε})]$ . The strong association of $[(f_\varepsilon )]$ $[(f_{ε})]$ with a Komatsu-type ultradistribution T, with an additional assumption on regularity of $[(f_\varepsilon )]$ $[(f_{ε})]$ of Beurling, respectively, Roumieu type, implies that T is an ultradifferentiable function of Beurling, Roumieu type, respectively. We show that under suitable conditions, a weakly negligible net $(f_\varepsilon )_{\varepsilon \in (0,1)}$ ${(f_{ε})}_{ε \in (0, 1)}$ is a negligible net in the sense of generalized ultradistributions. Furthermore, we prove that a translation-invariant generalized ultradistribution g is equal to a generalized constant in both types of generalized ultradistribution algebras.