Let F be the Cauchy transform of the self-similar measure <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1754_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> defined by <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1754_Article_IEq2.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="150" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu =\frac{1}{m}\sum _{j=0}^{m-1} \mu \circ S_j^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> <msubsup> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mi>μ</mi> <mo>∘</mo> <msubsup> <mi>S</mi> <mi>j</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1754_Article_IEq3.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="212" /> </InlineMediaObject> <EquationSource Format="TEX">$S_jz=e^{2\pi ij/m}+{r (z-e^{2\pi ij/m})}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>j</mi> </msub> <mi>z</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>j</mi> <mo stretchy="false">/</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>j</mi> <mo stretchy="false">/</mo> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1754_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$0<r<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>r</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. The Taylor coefficients <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1754_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$\{b_{nm-1}\}_{n=1}^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>b</mi> <mrow> <mi>n</mi> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation> of F near origin were studied in [<CitationRef CitationID="CR11">11</CitationRef>, <CitationRef CitationID="CR12">12</CitationRef>]. In this paper, we study the asymptotic formulation of the successive coefficients <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1754_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="167" /> </InlineMediaObject> <EquationSource Format="TEX">$\{b_{(n+1)m-1}-b_{nm-1}\}_{n=1}^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>b</mi> <mrow> <mi>n</mi> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1754_Article_IEq7.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> be the Hausdorff dimension of the support of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1754_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1754_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="204" /> </InlineMediaObject> <EquationSource Format="TEX">$R_m=\min \{|z|:z\in \textrm{supp}\;\mu \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mo movablelimits="true">min</mo> <mrow> <mo stretchy="false">{</mo> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> <mo>:</mo> <mi>z</mi> <mo>∈</mo> <mtext>supp</mtext> <mspace width="0.277778em" /> <mi>μ</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We give the set of accumulation points for <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1754_Article_IEq10.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="236" /> </InlineMediaObject> <EquationSource Format="TEX">$\{R_m^{mn}(mn)^\alpha (b_{(n+1)m-1}-b_{nm-1})\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msubsup> <mi>R</mi> <mi>m</mi> <mrow> <mi mathvariant="italic">mn</mi> </mrow> </msubsup> <msup> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mi>α</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>b</mi> <mrow> <mi>n</mi> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and answer whether <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1754_Article_IEq11.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$b_{mn-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> is monotonic at infinity. For the case of <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1754_Article_IEq12.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$m=4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1754_Article_IEq13.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{9}{50}\le r\le \frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>9</mn> <mn>50</mn> </mfrac> <mo>≤</mo> <mi>r</mi> <mo>≤</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, we prove that the Laurent coefficient <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1754_Article_IEq14.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$a_{mn+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mrow> <mi>m</mi> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> of F in <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1754_Article_IEq15.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$|z|>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is eventually decreasing.

Successive Coefficients for Cauchy Transforms of Some Self-Similar Measures

期刊论文
发表日期：

2025年7月11日

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Sheng-Jian Li,
Wu-Yi Pan

摘要

Let F be the Cauchy transform of the self-similar measure $\mu $ $μ$ defined by $\mu =\frac{1}{m}\sum _{j=0}^{m-1} \mu \circ S_j^{-1}$ $μ = \frac{1}{m} \sum_{j = 0}^{m - 1} μ \circ S_{j}^{- 1}$ , where $S_jz=e^{2\pi ij/m}+{r (z-e^{2\pi ij/m})}$ $S_{j} z = e^{2 π i j / m} + r (z - e^{2 π i j / m})$ with $0<r<1$ $0 < r < 1$ . The Taylor coefficients $\{b_{nm-1}\}_{n=1}^\infty $ ${b_{n m - 1}}_{n = 1}^{\infty}$ of F near origin were studied in [11, 12]. In this paper, we study the asymptotic formulation of the successive coefficients $\{b_{(n+1)m-1}-b_{nm-1}\}_{n=1}^\infty $ ${b_{(n + 1) m - 1} - b_{n m - 1}}_{n = 1}^{\infty}$ . Let $\alpha $ $α$ be the Hausdorff dimension of the support of $\mu $ $μ$ and $R_m=\min \{|z|:z\in \textrm{supp}\;\mu \}$ $R_{m} = min {| z | : z \in supp μ}$ . We give the set of accumulation points for $\{R_m^{mn}(mn)^\alpha (b_{(n+1)m-1}-b_{nm-1})\}$ ${R_{m}^{mn} {(m n)}^{α} (b_{(n + 1) m - 1} - b_{n m - 1})}$ and answer whether $b_{mn-1}$ $b_{m n - 1}$ is monotonic at infinity. For the case of $m=4$ $m = 4$ and $\frac{9}{50}\le r\le \frac{1}{2}$ $\frac{9}{50} \leq r \leq \frac{1}{2}$ , we prove that the Laurent coefficient $a_{mn+1}$ $a_{m n + 1}$ of F in $|z|>1$ $| z | > 1$ is eventually decreasing.