<p>Holomorphic functions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f \in \mathscr {H}({\textbf {C}}\setminus I)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <mi mathvariant="bold">C</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are represented by an “integral" over the jump of the associated distribution along the branch cut <i>I</i>. The “integral" is the evaluation of the Cauchy kernel in the sense of the duality between the locally convex spaces <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {B}_1({\textbf {R}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">B</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {D}'_{L^1,-1}({\textbf {R}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="script">D</mi> </mrow> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mo>,</mo> <mo>-</mo> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. New Hilbert transforms of distributions are given.</p>

A distributional version of Cauchy’s integral formula with applications to the Hilbert transformation

期刊论文
出版模式：

开放获取
发表日期：

2026年3月16日

查看全文

Norbert Ortner,
Christian Pfeifer,
Peter Wagner

摘要

Holomorphic functions $f \in \mathscr {H}({\textbf {C}}\setminus I)$ $f \in H (C \ I)$ are represented by an “integral" over the jump of the associated distribution along the branch cut I. The “integral" is the evaluation of the Cauchy kernel in the sense of the duality between the locally convex spaces $\mathcal {B}_1({\textbf {R}})$ $B_{1} (R)$ and $\mathcal {D}'_{L^1,-1}({\textbf {R}})$ $D_{L^{1}, - 1}^{'} (R)$ . New Hilbert transforms of distributions are given.