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De Marchi, Stefano; Usevich, Konstantin

On certain multivariate Vandermonde determinants whose variables separate. (English) Zbl 1303.15008

Linear Algebra Appl. 449, 17-27 (2014).
Summary: We prove that for almost square tensor product grids and certain sets of bivariate polynomials the Vandermonde determinant can be factored into a product of univariate Vandermonde determinants. This result generalizes the conjecture [L. Bos, S. De Marchi and S. Waldron, “On the Vandermonde determinant of padua-like points”, Dolomites Res. Notes Approx. 2, 1–15 (2009), http://journals.padovauniversitypress.it/dolomites/volumes/Volume-2]. As a special case, we apply the result to Padua and Padua-like points.

Cited in 1 Review
Cited in 3 Documents

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
15B99 Special matrices
41A05 Interpolation in approximation theory

Keywords:

multivariate Vandermonde determinant; Padua points; tensor product grid; polynomial matrices; semiseparable matrices
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References:

[1] Bos, L.; Caliari, M.; De Marchi, S.; Vianello, M.; Xu, Y., Bivariate Lagrange interpolation at the Padua points: the generating curve approach, J. Approx. Theory, 143, 15-25 (2006) · Zbl 1113.41001
[2] Bos, L.; De Marchi, S.; Vianello, M.; Xu, Y., Bivariate Lagrange interpolation at the Padua points: the ideal theory approach, Numer. Math., 108, 1, 43-57 (2007) · Zbl 1126.41002
[3] Bos, L.; De Marchi, S.; Waldron, S., On the Vandermonde determinant of Padua-like points, Dolomites Res. Notes Approx., 2, 1-15 (2009)
[4] Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge University Press · Zbl 0729.15001
[5] Vandebril, R.; Van Barel, M.; Mastronardi, N., Matrix Computations & Semiseparable Matrices I: Linear Systems (2007), Johns Hopkins University Press
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