Chebyshev polynomials corresponding to a vanishing weight
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909208454103040 |
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| author | Bergman, Alex Rubin, Olof |
| author_facet | Bergman, Alex Rubin, Olof |
| contents | We consider weighted Chebyshev polynomials on the unit circle corresponding to a weight of the form $(z-1)^s$ where $s>0$. For integer values of $s$ this corresponds to prescribing a zero of the polynomial on the boundary. As such, we extend findings of Lachance, Saff and Varga, to non-integer $s$. Using this generalisation, we are able to relate Chebyshev polynomials on lemniscates and other, more established, categories of Chebyshev polynomials. An essential part of our proof involves the broadening of the Erdős--Lax inequality to encompass powers of polynomials. We believe that this particular result holds significance in its own right. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02047 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Chebyshev polynomials corresponding to a vanishing weight Bergman, Alex Rubin, Olof Complex Variables Classical Analysis and ODEs 41A50, 30C10, 30A10, 26D05, 41A17 We consider weighted Chebyshev polynomials on the unit circle corresponding to a weight of the form $(z-1)^s$ where $s>0$. For integer values of $s$ this corresponds to prescribing a zero of the polynomial on the boundary. As such, we extend findings of Lachance, Saff and Varga, to non-integer $s$. Using this generalisation, we are able to relate Chebyshev polynomials on lemniscates and other, more established, categories of Chebyshev polynomials. An essential part of our proof involves the broadening of the Erdős--Lax inequality to encompass powers of polynomials. We believe that this particular result holds significance in its own right. |
| title | Chebyshev polynomials corresponding to a vanishing weight |
| topic | Complex Variables Classical Analysis and ODEs 41A50, 30C10, 30A10, 26D05, 41A17 |
| url | https://arxiv.org/abs/2309.02047 |