Unbounded Widom factors for orthogonal and residual polynomials
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909746646220800 |
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| author | Alpan, Gökalp |
| author_facet | Alpan, Gökalp |
| contents | We study Widom factors for (a) monic orthogonal polynomials in $L^2$ with respect to the equilibrium measure of a compact set $K\subset\mathbb{R}$ and (b) residual polynomials normalized at an exterior point. Using weakly equilibrium Cantor sets $K(γ)$, we prove:
(1) Given any sequence $(c_n)$ with subexponential growth, there exists a non-polar Cantor set $K(γ)$, depending on $(c_n)$, such that the $L^2$ Widom factors of the associated orthogonal polynomials (with respect to the equilibrium measure of $K(γ)$) exceed $c_n$ for every $n$.
(2) For the same $K(γ)$ built from $(c_n)$ and each exterior point $x_0\in\mathbb{R}\setminus K(γ)$, the residual Widom factors satisfy power-type lower bounds with a Harnack-distance exponent $τ_{x_0}\in(0,1)$: they are bounded below by $c_n^{τ_{x_0}}$ for all degrees when $x_0$ lies in an unbounded gap, and along a subsequence of degrees when $x_0$ lies in a bounded gap. Consequently, if, in addition, $(c_n)$ is monotone increasing and unbounded, then the sequence of residual Widom factors is unbounded for every $x_0\in\mathbb{R}\setminus K(γ)$.
The proofs combine inverse-image constructions, capacity comparisons for period-$n$ sets, harmonic-measure representations for differences of Green functions, and alternation principles on nested approximants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15131 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unbounded Widom factors for orthogonal and residual polynomials Alpan, Gökalp Complex Variables Classical Analysis and ODEs Primary 41A17, Secondary 41A44, 42C05, 33C45 We study Widom factors for (a) monic orthogonal polynomials in $L^2$ with respect to the equilibrium measure of a compact set $K\subset\mathbb{R}$ and (b) residual polynomials normalized at an exterior point. Using weakly equilibrium Cantor sets $K(γ)$, we prove: (1) Given any sequence $(c_n)$ with subexponential growth, there exists a non-polar Cantor set $K(γ)$, depending on $(c_n)$, such that the $L^2$ Widom factors of the associated orthogonal polynomials (with respect to the equilibrium measure of $K(γ)$) exceed $c_n$ for every $n$. (2) For the same $K(γ)$ built from $(c_n)$ and each exterior point $x_0\in\mathbb{R}\setminus K(γ)$, the residual Widom factors satisfy power-type lower bounds with a Harnack-distance exponent $τ_{x_0}\in(0,1)$: they are bounded below by $c_n^{τ_{x_0}}$ for all degrees when $x_0$ lies in an unbounded gap, and along a subsequence of degrees when $x_0$ lies in a bounded gap. Consequently, if, in addition, $(c_n)$ is monotone increasing and unbounded, then the sequence of residual Widom factors is unbounded for every $x_0\in\mathbb{R}\setminus K(γ)$. The proofs combine inverse-image constructions, capacity comparisons for period-$n$ sets, harmonic-measure representations for differences of Green functions, and alternation principles on nested approximants. |
| title | Unbounded Widom factors for orthogonal and residual polynomials |
| topic | Complex Variables Classical Analysis and ODEs Primary 41A17, Secondary 41A44, 42C05, 33C45 |
| url | https://arxiv.org/abs/2508.15131 |