Unbounded banded matrices, shifted positive bidiagonal factorizations, and mixed-type multiple orthogonality
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866910009493815296 |
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| author | Branquinho, Amílcar Foulquié-Moreno, Ana Mañas, Manuel |
| author_facet | Branquinho, Amílcar Foulquié-Moreno, Ana Mañas, Manuel |
| contents | This work extends Favard-type spectral representations for banded matrices $T$ beyond the bounded setting. It assumes that, for every $N\in\mathbb N_0$, there exists a shift $s_N\ge 0$ such that the shifted truncation $A_N:= T^{[N]}+s_N I_{N+1}$ admits a positive bidiagonal factorization (PBF). Allowing $s_N$ to depend on $N$ leads to a natural recentering step: the discrete Gauss-type quadrature measures associated with $A_N$ are translated by $x\mapsto x-s_N$, producing a uniformly bounded family of distribution functions. Combining moment stabilization for banded truncations with Helly-type compactness theorems yields a limiting matrix-valued measure, together with a Favard-type spectral representation and the corresponding mixed-type multiple biorthogonality relations. As a consequence, the classical Favard theorem for (possibly unbounded) Jacobi matrices is recovered as a special case. Indeed, for a tridiagonal $J$ with positive sub- and superdiagonals, each truncation $J^{[N]}$ admits a shift $s_N\ge 0$ such that $J^{[N]}+s_N I_{N+1}$ is oscillatory and therefore admits a PBF. The preceding construction then produces the usual spectral measure for $J$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12453 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unbounded banded matrices, shifted positive bidiagonal factorizations, and mixed-type multiple orthogonality Branquinho, Amílcar Foulquié-Moreno, Ana Mañas, Manuel Classical Analysis and ODEs Mathematical Physics 33C47, 42C05, 47B36, 47A10, 15B48 This work extends Favard-type spectral representations for banded matrices $T$ beyond the bounded setting. It assumes that, for every $N\in\mathbb N_0$, there exists a shift $s_N\ge 0$ such that the shifted truncation $A_N:= T^{[N]}+s_N I_{N+1}$ admits a positive bidiagonal factorization (PBF). Allowing $s_N$ to depend on $N$ leads to a natural recentering step: the discrete Gauss-type quadrature measures associated with $A_N$ are translated by $x\mapsto x-s_N$, producing a uniformly bounded family of distribution functions. Combining moment stabilization for banded truncations with Helly-type compactness theorems yields a limiting matrix-valued measure, together with a Favard-type spectral representation and the corresponding mixed-type multiple biorthogonality relations. As a consequence, the classical Favard theorem for (possibly unbounded) Jacobi matrices is recovered as a special case. Indeed, for a tridiagonal $J$ with positive sub- and superdiagonals, each truncation $J^{[N]}$ admits a shift $s_N\ge 0$ such that $J^{[N]}+s_N I_{N+1}$ is oscillatory and therefore admits a PBF. The preceding construction then produces the usual spectral measure for $J$. |
| title | Unbounded banded matrices, shifted positive bidiagonal factorizations, and mixed-type multiple orthogonality |
| topic | Classical Analysis and ODEs Mathematical Physics 33C47, 42C05, 47B36, 47A10, 15B48 |
| url | https://arxiv.org/abs/2601.12453 |