Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)
GÁT, G., On the Fejér kernel functions with respect to the Walsh-Paley system
On the Fejér kernel functions with respect to the Walsh-Paley system 107 Lemma 2. Suppose that s,t,n E N , x G It \ It+i • If s < t < |n| ; then |ür n(,+i, i 2.(:c)| < c2 s+ t . On the other hand, if t < s < \n\, we have if x - x te t £ I s, if x - x te t G I s. Proof. If s <t, then for all k G N by (1) and (2) we have \D k(x)\ < — thus in this case |A' n( s + i)(x)| < c2 s+ i. On the other hand, let |n| > s > t. Then t D n(s + iy + j(x) = u n(s+i)+ J(x) ]T(n (s+1 ) + j) kr k(x) k0 = w ni. +D + J-(s) (^ /jk2 k-j t2 t \k= 0 This implies that 2 s -1 j= 0 2 3 —1 /t-1 j= 0 \k=0 t1 k = E ^wE-^ 2 J o j " • ijs — 1 1 í-1 1 ji=0,ij:t,i=0,...,s-l k—0 jt= 0 since ji=0 jt =0 That is, 2 s —1 K n( a +D i 2s(x) = -u n í,+i)(x) Y u j(x)j t2 t j= 0 0 • if x - x te t £ / s, w n(s + i)(x)2 s+i_ 1 if x - x te t G / s. •
