<p>We show that every Blaschke sequence containing at most one zero can be transformed into an interpolating Blaschke sequence by using an arbitrarily small perturbation. More precisely, for a Blaschke sequence <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\{z_n\}_{n=1}^{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation> containing at most one zero and any <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varepsilon > 0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> there is an interpolating Blaschke sequence <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{\omega _n\}_{n=1}^{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>ω</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation> such that for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1 \le n < \infty ,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>n</mi> <mo><</mo> <mi>∞</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> (1) <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$|z_n| = |\omega _n|,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">|</mo> <mo>=</mo> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ω</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">|</mo> <mo>,</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (2) and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$| \arg z_n - \arg \omega _n| < \varepsilon .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mo>arg</mo> </mrow> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo>-</mo> <mo>arg</mo> <msub> <mi>ω</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">|</mo> <mo><</mo> <mi>ε</mi> <mo>.</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> This result is a refinement of Naftalevič’s Theorem.</p>

Interpolating Blaschke Sequence and Perturbation

期刊论文
发表日期：

2026年5月20日

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Liulan Li,
Chitin Hon,
Tao Qian,
Shilin Wang

摘要

We show that every Blaschke sequence containing at most one zero can be transformed into an interpolating Blaschke sequence by using an arbitrarily small perturbation. More precisely, for a Blaschke sequence $\{z_n\}_{n=1}^{\infty }$ ${z_{n}}_{n = 1}^{\infty}$ containing at most one zero and any $\varepsilon > 0,$ $ε > 0,$ there is an interpolating Blaschke sequence $\{\omega _n\}_{n=1}^{\infty }$ ${ω_{n}}_{n = 1}^{\infty}$ such that for $1 \le n < \infty ,$ $1 \leq n < \infty,$ (1) $|z_n| = |\omega _n|,$ $| z_{n} | = | ω_{n} |,$ (2) and $| \arg z_n - \arg \omega _n| < \varepsilon .$ $| arg z_{n} - arg ω_{n} | < ε .$ This result is a refinement of Naftalevič’s Theorem.