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Hermite's formula for vector polynomial interpolation with applications to structured matrices

Georg Heinig, Fadhel Al‐Musallam

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Abstract

The classical Hermite formula for polynomial interpolation is generalized to interpolation of vector polynomials (tangential interpolation). The formula exhibits a relation between the matricial homogeneous problem and the nonhomogeneous vector problem. As applications, a formula for the inverse of a generalized Vandermonde matrix is presented and it is shown that the well-known formulas for the inverse of a Toeplitz matrix can be obtained as a special case of the generalized Hermite formula.

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What this paper is about

The classical Hermite formula for polynomial interpolation is generalized to interpolation of vector polynomials (tangential interpolation). The formula exhibits a relation between the matricial homogeneous problem and the nonhomogeneous vector problem. As applications, a formula for the inverse of a generalized Vandermonde matrix is presented and it is shown that the well-known formulas for the inverse of a Toeplitz matrix can be obtained as a special case of the generalized Hermite formula.

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OpenAlex reports 3 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

The classical Hermite formula for polynomial interpolation is generalized to interpolation of vector polynomials (tangential interpolation). The formula exhibits a relation between the matricial homogeneous problem and the nonhomogeneous vector problem. As applications, a formula for the inverse of a generalized Vandermonde matrix is presented and it is shown that the well-known formulas for the inverse of a Toeplitz matrix can be obtained as a special case of the generalized Hermite formula.

Key concepts: Vandermonde matrix, Mathematics, Hermite interpolation, Birkhoff interpolation, Toeplitz matrix, Polynomial interpolation, Interpolation (computer graphics), Trigonometric interpolation

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