Zeros of Multiple Orthogonal Polynomials: Location and Interlacing
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909996217794560 |
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| author | Kozhan, Rostyslav Vaktnäs, Marcus |
| author_facet | Kozhan, Rostyslav Vaktnäs, Marcus |
| contents | We prove a criterion on the possible locations of zeros of type I and type II multiple orthogonal polynomials in terms of normality of degree $1$ Christoffel transforms. We provide another criterion in terms of degree $2$ Christoffel transforms for establishing zero interlacing of the neighbouring multiple orthogonal polynomials of type I and type II.
We apply these criteria to establish zero location and interlacing of type I multiple orthogonal polynomials for Nikishin systems. Additionally, we recover the known results on zero location and interlacing for type I multiple orthogonal polynomials for Angelesco systems, as well as for type II multiple orthogonal polynomials for Angelesco and AT systems.
Finally, we demonstrate that normality of the higher order Christoffel transforms is naturally related to the zeros of the Wronskians of consecutive orthogonal polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15122 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zeros of Multiple Orthogonal Polynomials: Location and Interlacing Kozhan, Rostyslav Vaktnäs, Marcus Classical Analysis and ODEs Spectral Theory 42C05, 41A21, 26C10 We prove a criterion on the possible locations of zeros of type I and type II multiple orthogonal polynomials in terms of normality of degree $1$ Christoffel transforms. We provide another criterion in terms of degree $2$ Christoffel transforms for establishing zero interlacing of the neighbouring multiple orthogonal polynomials of type I and type II. We apply these criteria to establish zero location and interlacing of type I multiple orthogonal polynomials for Nikishin systems. Additionally, we recover the known results on zero location and interlacing for type I multiple orthogonal polynomials for Angelesco systems, as well as for type II multiple orthogonal polynomials for Angelesco and AT systems. Finally, we demonstrate that normality of the higher order Christoffel transforms is naturally related to the zeros of the Wronskians of consecutive orthogonal polynomials. |
| title | Zeros of Multiple Orthogonal Polynomials: Location and Interlacing |
| topic | Classical Analysis and ODEs Spectral Theory 42C05, 41A21, 26C10 |
| url | https://arxiv.org/abs/2503.15122 |