On the degree of approximation of continuous functions by using linear operators on their Fourier series. (English) Zbl 0958.42001

The authors apply lower triangular nonnegative summability matrices to the partial sums of the Fourier expansion of a continuous function on \([0,\pi]\). They assume that the row-sums are 1 and estimate the distance between the function and this summability transform, by means of the modulus of continuity of the function and various combinations of the elements of the matrix. As special cases they obtain earlier estimates which were obtained under the additional assumption that the elements in each row of the matrix are nondecreasing. Among the earlier estimates are some by Bernstein, Jackson and Alexits. The proofs are quite straightforward.


42A10 Trigonometric approximation
41A25 Rate of convergence, degree of approximation
41A35 Approximation by operators (in particular, by integral operators)