Approximation of Functions of Bounded p-variation by Euler Means

In this paper we study the Euler means e(n)(q)(f)(x) = Sigma(n)(k=0) ((n)(k) )q(n-k) (1 + q)(-n) S-k(f)(x), q >= 0, n is an element of Z(+), where S-k(f) is the k-th partial trigonometric Fourier sum. For p-absolutely continuous functions (f is an element of C-p, 1 < p < infinity) we consider their approximation by the Euler means in uniform and C-p-metric in terms of moduli of continuity omega(k)(f)(Cp), k is an element of N, and the best approximations by trigonometric polynomials E-n(f)(Cp). One can note the following inequality for different metrics from Theorem 2 parallel to f - e(n)(q)(f)parallel to(infinity) <= C-1 (1 + q)(-n) Sigma(n)(j=0) ((n)(j))q(n - j) E-j(f)(Cp), n is an element of N, which is sharp. Also the following generalization of a result due to C. K. Chui and A. S. Holland is proved. If omega is a modulus of continuity on [0, pi] such that delta integral(pi)(delta) t(-2)omega(t) dt = O (omega(delta)), 1 < p < infinity and f is an element of C-p satisfies two properties 1) omega(2) (f, t)(Cp) <= C-omega(t); 2) integral(pi)(2 pi/(n+1)) t(-1) parallel to phi(x)(t) - phi(x)(t + 2 pi/( n + 1)parallel to(Cp) dt = O (omega(1/n)), where phi(x)(t) = f (x + t)+ f (x - t) - 2f(x), then parallel to e(n)(1)(f) - f parallel to(Cp) <= C-omega(1/n), n is an element of N. Some applications to the approximation in Holder type metrics are given.