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In [1, 2] the classical equiconvergence theorem of Walsh was extended by the application of summability methods in order to enlarge the disk of equiconvergence to regions of equisummability. A further generalization was achieved in [3], where sequences of Hadamard products of a fixed power series with interpolatory polynomials were considered. The aim of this paper is to continue this work by investigating commutators of interpolatory polynomials and Hermite interpolatory polynomials. 0. Introduction Let g be a function holomorphic in the disk D^ := {z G C : \z\ <i^}for some R > 1, let Ln(]g) be the Lagrange interpolatory polynomial to g in the (n + l)-st roots of unity, and denote by S^ the n-th partial sum of the power series expansion of g about 0. Then the classical equiconvergence theorem of J. L. Walsh [8, p. 153] states that lim[Ln(z;g)-S°n(z)]=0 n—+oo compactly in B#2 (i.e., uniformly on compact subsets of D^). In [1, 2] the first author applied certain summability methods in order to enlarge the disk of equiconvergence to regions of equisummability. Roughly speaking, one of his results may be stated as follows. If A is a summability method which sums the geometric series 7(2:) = S^Lo z to 1/(1 — z) compactly in an open set S containing the unit disk D, then the sequence (Ln(' ;g) — S£) is compactly A-summable to 0 in a certain open set £. This set £ always contains the disk D^2, and it depends on S and the singularities of g. Recently, the first two authors [3] generalized this result by replacing the "test function" 7 by an arbitrary power series /, and by considering the sequence (S^ * (Ln(\g) — SfO), where * denotes the Hadamard product of two power series. The aim of this paper is to extend their result in two directions. In [5], Lou considered commutators of interpolatory polynomials, namely ^mnte > P) = m (z\ (3, Ln(-; a, g)) Ln (z] a, Lm(-; /3, g)), (0.1) Received August 23, 1993, revised February 16, 1994. 1991 Mathematics Subject Classification: 30E10, 40A30.

Equisummability of certain sequences of Hadamard products of Taylor sections and interpolatory polynomials