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Some remarks on multivariate Chebyshev polynomials. (English) Zbl 0871.41022
Let \(S\) be the triangular planar set \(S=\{ (x,y):x\geqslant 0, y \geqslant 0, 0\leqslant x+y \leqslant 1\} .\) D. J. Newman and Y. Xu [Constructive Approximation 9, 543-546 (1993; Zbl 0780.41004)] found an analogue of Chebyshev polynomials \(T_{n}\) deviated least from zero on \([-1,1]\) for \(S.\) The author investigates a possibility to generalize an extremal property of Chebyshev polynomials (maximal growth outside the interval). This is the case for polynomials of degree 1, with replacing zero dimensional point evaluation by one-dimensional norms on diagonal parallels, \(x+y=s\geqslant 1.\) Uniqueness of Chebyshev polynomials in such a case is proved, too.
Secondly, the author shows that polynomials \(T_{n}(x+y)\) are unique deviating least from zero among all polynomials with ”leading term” \[ \sum_{i+j}a_{i,j}x^{i}y^{j}=2^{n-1}(x+y)^{n} \] on \(\{ (x,y): xy\geqslant 0, x+y = s, -1\leqslant s \leqslant 1\}.\)

MSC:
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
41A10 Approximation by polynomials
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[1] Eier, R.; Lidl, R., A class of orthogonal polynomials in k variables, Math ann., 260, 93-99, (1982) · Zbl 0474.33009
[2] Koornwinder, T.H., Two-variable analogues of the classical orthogonal polynomials, (), 435-495
[3] Rivlin, T.J.; Shapiro, H.S., A unified approach to certain problems of approximation and minimization, J. soc. indust. applied math., 9, 670-699, (1961) · Zbl 0111.06103
[4] Newman, D.J.; Xu, Y., Tchebycheff polynomials on a triangular region, Constr. approx., 543-546, (1993) · Zbl 0780.41004
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