On Gaussian quadrature formulas for the Chebyshev weight (Q1288885)

The names of Gauss and Chebyshev, great and illustrious as they are, appear in the mathematics literature quite a good number of times. Particularly, Chebyshev polynomials and Gaussian quadrature are well-known independently and to connect them up in a meaningful manner is a task worthy of note. In the paper under review the author has studied in depth and shown that the Chebyshev weight \(W(x)= (1-x^2)^{-1/2}\) is the only one possessing the property (up to a linear transformation) that the solutions of a specific extremal problem are identical for any values of two parameters \((m,p\geq 1)\). The special conditions required for this are characterized through two theorems with detailed proof, a remark and five auxiliary lemmas, three of which are followed by brief justification also. As a generalization of the earlier research results by the same author, the present contribution is an important addition to certain approximation theories and their applications.