On the oscillations of the modulus of Rudin-Shapiro polynomials around the middle of their ranges
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866918153296019456 |
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| author | Erdélyi, Tamás |
| author_facet | Erdélyi, Tamás |
| contents | Let either $R_k(t) := |P_k(e^{it})|^2$ or $R_k(t) := |Q_k(e^{it})|^2$, where $P_k$ and $Q_k$ are the usual Rudin-Shapiro polynomials of degree $n-1$ with $n=2^k$. The graphs of the trigonometric polynomials $R_k$ on the period suggest many zeros of $R_k(t)-n$ in a dense fashion on the period. Let ${\Cal N}(I,R_k-n)$ denote the number of zeros, counted with multiplicities, of the trigonometric polynomial $R_k-n$ in an interval $I := [α,β] \subset [0,2π)$. Improving earlier results proved only for the interval $I := [0,2π)$, in this paper we show that $$\frac{n|I|}{8π} - \frac{2}π (2n\log n)^{1/2} - 1 \leq N(I,R_k-n) \leq \frac{n|I|}π + \frac{8}π(2n\log n)^{1/2}\,,\qquad k \geq 2\,,$$ for every interval $I := [α,β] \subset [0,2π)$, where $|I| = β-α$ denotes the length of the interval $I$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_04385 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the oscillations of the modulus of Rudin-Shapiro polynomials around the middle of their ranges Erdélyi, Tamás Classical Analysis and ODEs 11C08, 41A17, 26C10, 30C15 Let either $R_k(t) := |P_k(e^{it})|^2$ or $R_k(t) := |Q_k(e^{it})|^2$, where $P_k$ and $Q_k$ are the usual Rudin-Shapiro polynomials of degree $n-1$ with $n=2^k$. The graphs of the trigonometric polynomials $R_k$ on the period suggest many zeros of $R_k(t)-n$ in a dense fashion on the period. Let ${\Cal N}(I,R_k-n)$ denote the number of zeros, counted with multiplicities, of the trigonometric polynomial $R_k-n$ in an interval $I := [α,β] \subset [0,2π)$. Improving earlier results proved only for the interval $I := [0,2π)$, in this paper we show that $$\frac{n|I|}{8π} - \frac{2}π (2n\log n)^{1/2} - 1 \leq N(I,R_k-n) \leq \frac{n|I|}π + \frac{8}π(2n\log n)^{1/2}\,,\qquad k \geq 2\,,$$ for every interval $I := [α,β] \subset [0,2π)$, where $|I| = β-α$ denotes the length of the interval $I$. |
| title | On the oscillations of the modulus of Rudin-Shapiro polynomials around the middle of their ranges |
| topic | Classical Analysis and ODEs 11C08, 41A17, 26C10, 30C15 |
| url | https://arxiv.org/abs/2210.04385 |