We consider the problem of finding a best uniform approximation to the standard monomial on the unit ball in C 2 by polynomials of lower degree with complex coefficients. We reduce the problem to a onedimensional weighted minimization problem on an interval. In a sense, the corresponding extremal polynomials are uniform counterparts of the classical orthogonal Jacobi polynomials. They can be represented by means of special conformal mappings on the so-called comb-like domains. In these terms, the value of the minimal deviation and the representation for a polynomial of best approximation for the original problem are given. Furthermore, we derive asymptotics for the minimal deviation.
P = sup(z 1 ,z 2 )∈B |P (z 1 , z 2 )|, B = {(z 1 , z 2 ) ∈ C 2 : |z 1 | 2 + |z 2 | 2 ≤ 1}.
The main result of this work is a parametric description of the spectral surfaces of a class of periodic 5-diagonal matrices, related to the strong moment problem. This class is a self-adjoint twin of the class of CMV matrices. Jointly they form the simplest possible classes of 5-diagonal matrices.
The main result of this work is a parametric description of the spectral sets of a class of periodic 5-diagonal matrices, related to the strong moment problem. This class is a self-adjoint twin of the class of CMV matrices. Both are hidden in the simplest possible class of regular 5-diagonal matrices, this fact we also show here. Classification (2010). 30E05, 30F15, 47B39, 46E22.
Mathematics Subject
We consider the classical problem of finding the best uniform approximation by polynomials of 1/(x − a) 2 , where a > 1 is given, on the interval [−1, 1]. First, using symbolic computation tools we derive the explicit expressions of the polynomials of best approximation of low degrees and then give a parametric solution of the problem in terms of elliptic functions. Symbolic computation is invoked then once more to derive a recurrence relation for the coefficients of the polynomials of best uniform approximation based on a Pell-type equation satisfied by the solutions.
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