Summation of the Universal Series on the Chebyshev Polynomials

Abstract

The Universal functional series have been studied by many authors since 1906, the year when the Hungarian mathematician, Michael Fekete first considered the universal power series in the real domain. Trigonometric universal series were built by D. E. Menshov (1945), they were also researched, for example, by J. Edge (1970), and N. Pogosyan (1983). In the complex domain the existence of the universal power series was proved by A. I. Seleznyov, C.K. Chui, M.N. Parnes (1971), V. Nestoridis (1996), and by other authors. Various authors have also studied other universal functional series. The property of universality of a functional series is the approximation of the function of a certain class by partial sums of this series. This property is a generalization of the well-known S.N. Mergelyan theorem (1952) on the approximation of analytic functions by polynomials on compact sets. The present work demonstrates the existence of the universal series on the Chebyshev polynomials. W. Luh (1976) summarized the universal property of a power series in case of its matrix transformations. In some sense the analogues of this generalization were obtained by the first author of this work (1990, coauthored: 2012, 2013) for certain functional series. In this paper the aforementioned generalization is extended to series on the Chebyshev polynomials by summation of the universal series. Namely, we have constructed special sums related to the series on the Chebyshev polynomials possessing the property of universality, that is, any function of a certain class on compact sets, taken in a specific way, uniformly approach by these sums. The construction of the sums is carried out by the method of matrix transformation that was previously used by the first author in the construction of specific sums for other series; but unlike them, in researched sams in the matrix transformation the columns are taken without gaps.

Keywords

• universal series
• Chebyshev polynomials
• uniform approximation
• summation