Extensions of certain Markov inequalities to Toeplitz matrices

K. Castillo; A. Suzuki

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arXiv · 2406.07104

Extensions of certain Markov inequalities to Toeplitz matrices

K. Castillo·A. Suzuki

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Abstract

We establish finite Toeplitz analogues of Markov's inequalities for Hankel determinants and zeros of orthogonal polynomials. We characterise the real symmetric positive-definite Toeplitz matrices whose determinants dominate those of all such matrices below them in the alternating moment order: non-negativity of the diagonal sums of deleted minors is necessary and sufficient. Equality holds only when the matrices coincide, and a quantitative estimate controls the determinant gap. Paraorthogonal zeros in the right half-plane provide a geometric sufficient condition in both parities. For differentiable finite Hermitian moment data, we derive individual and relative angular-velocity identities. For real symmetric data, these identities yield a finite comparison of all upper zeros under simultaneous changes in the moments: a semicircle condition at one endpoint is inherited by the other, and every angular inequality is strict when the matrices differ. No regularity assumption on a representing measure is required. Coefficient criteria also establish nearest-zero monotonicity. An exact Rogers-Szeg\H{o} counterexample shows that this need not extend to every upper zero.

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BibTeX RIS

K. Castillo, A. Suzuki. 2024-06-11. Extensions of certain Markov inequalities to Toeplitz matrices. https://arxiv.org/abs/2406.07104

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