Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

NAGY, K., Norm convergenc e of Fejér means of certain functions with respect to respect to UDMD product systems

Norm convergence of Fejér means of certain functions with respect to UDMD product systems KÁROLY NAGY Abstract. In this paper we investigate the norm convergence of Feje'r means of functions belonging to Lipschitz classes in that case when the orthonormal system is the unitary dyadic martingale différence system (UDMD system). We give an estimation of the order of norm convergence. The resuit of the paper shows a sharp constrast between the corresponding known Statement which relatives to the ordinary Walsh-Paley system. Introduction Let N denote the set of natural numbers, P denote the set of positiv intergers, and A = {0,1}. For each m G N let [mS

J\j G N) represent the binar y coefficient of m, that is, oo m = £ ™>

{j

)2

j (™

(i

) G A, j G N). 3=0 Let (0,A) be a measure space with À(f2) = 1 and $ : = , j G N) be a sequence of A-measurable functions on Cl which are a.e. [À] bounded by 1. The product system generated by $ is the sequence of functions : = , m G N) defined by oo =n *f' 3=0 for m G N. Each is a finite product of </>

n's, ipo(

x) = 1 for \ip

m\ < 1 for m G N, and 4>

n = ip2

n f°

r G N. Let $ (ip

m,m G N) be any orthonormal system on 0. The Fourier coefficients of a function / G A) are defined by {ÎAm}-= S f'KdX (m G N). u Research supported by the Hungárián National Foundation for Scientific Research (OTKA), grant no. F007347