Approximation of functions by linear matrix operators. (English) Zbl 0932.42004

Chui, Charles K. (ed.) et al., Approximation theory IX. Volume 1. Theoretical aspects. Proceedings of the 9th international conference, Nashville, TN, USA, January 3–6, 1998. Nashville, TN: Vanderbilt University Press. Innovations in Applied Mathematics. 263-270 (1998).
The present reviewer [Mat. Vesn. 38, 263-269 (1986; Zbl 0655.42002); ibid. 42, No. 1, 9-10 (1990; Zbl 0725.42004)] obtained some results in \(L_p\)-norm by using Nörlund operators. In this paper, the authors have extended these results by using linear matrix operators given by lower-triangular matrix \(A= (a_{n,k})\) with \(a_{n,k}\geq 0\) and \(\sum^n_{k= 0}a_{n,k}= 1\), under superfluous condition (3.3): \(t H(t)= o(1)\) \((t\to 0+)\), since the proof can be obtained without it. For a proof, a reference may be made to the present reviewer [Acta Math. Hung. 62, No. 1-2, 21-23 (1993; Zbl 0794.42002)].
Earlier, similar results were obtained for continuous functions by the present reviewer [Acta Math. Hung. 52, No. 3-4, 199-205 (1988; Zbl 0704.42004)] in the sup-norm.
Read \(\int^{\pi/n}_0\) for \(\int^{\pi/2}_0\) on page 267.
For the entire collection see [Zbl 0910.00046].
Reviewer: P.Chandra (Ujjain)


42A10 Trigonometric approximation
41A25 Rate of convergence, degree of approximation