Given a sequence of orthogonal polynomials <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(p_n)_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> with respect to a positive measure in the real line, we study the real zeros of finite combinations of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$K+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> consecutive orthogonal polynomials of the form <Equation ID="Equ38"> <EquationSource Format="TEX">$ q_n(x)=\sum _{j=0}^K\gamma _jp_{n-j}(x),\quad n\ge K, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>q</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>K</mi> </munderover> <msub> <mi>γ</mi> <mi>j</mi> </msub> <msub> <mi>p</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>n</mi> <mo>≥</mo> <mi>K</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma _j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$j=0,\cdots ,K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation>, are real numbers with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\gamma _0=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\gamma _K\not =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mi>K</mi> </msub> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> (which do not depend on n). We prove that for every positive measure <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> there always exists a sequence of orthogonal polynomials with respect to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> such that all the zeros of the polynomial <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$q_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>q</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> above are real and simple for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$n\ge n_0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <msub> <mi>n</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$n_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>n</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> is a positive integer depending on K and the <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\gamma _j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation>’s.

Zeros of Linear Combinations of Orthogonal Polynomials

期刊论文
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2026年6月18日

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Antonio J. Durán

摘要

Given a sequence of orthogonal polynomials $(p_n)_n$ ${(p_{n})}_{n}$ with respect to a positive measure in the real line, we study the real zeros of finite combinations of $K+1$ $K + 1$ consecutive orthogonal polynomials of the form $ q_n(x)=\sum _{j=0}^K\gamma _jp_{n-j}(x),\quad n\ge K, $ $q_{n} (x) = \sum_{j = 0}^{K} γ_{j} p_{n - j} (x), n \geq K,$ where $\gamma _j$ $γ_{j}$ , $j=0,\cdots ,K$ $j = 0, \dots, K$ , are real numbers with $\gamma _0=1$ $γ_{0} = 1$ , $\gamma _K\not =0$ $γ_{K} \neq 0$ (which do not depend on n). We prove that for every positive measure $\mu $ $μ$ there always exists a sequence of orthogonal polynomials with respect to $\mu $ $μ$ such that all the zeros of the polynomial $q_n$ $q_{n}$ above are real and simple for $n\ge n_0$ $n \geq n_{0}$ , where $n_0$ $n_{0}$ is a positive integer depending on K and the $\gamma _j$ $γ_{j}$ ’s.