12Regularity of Linear Methods of Summation of Fourier SeriesChapter 1and, hence, t t Nk (t) = �sin(n + k + 2) sin(n �k) 2 2 t /2 sin2 . 2 (2.21)Applying twice the Abel transformation, according to which, for arbitrary real u0 , u1 , . . . and v0 , v1 , . . . and any n �N, one hasn n� kuk vk =k=0 k=0(uk �uk+1 )Vk + un Vn ,Vk =i=0vi ,(2.22)and setting = [(n + 1)/2], we obtainn n�Un (t; 螞) =k=0 �Dk (t)螖k =k=0 n�Mk (t)螖2 + Mn (t)螖n k=k=0Mk (t)螖2 k+k=Nk (t)螖2 + Mn (t)螖 . k(2.23)It follows from (2.20) and (2.21) that, for any 未 > 0, the functions |Mk (t)| and 未 |Nk (t)| are bounded on the segment [未, ] by sin� . Thus, 2 mn (未) = max |Un (t; 螞)| �sin未は��未 2n�|螖2 | + 螖 kk=0.(2.24)Let us verify that, under condition (2.9), the right-hand side of (2.24) tends to zero as n 鈫�� Sincen i=nn k+1 �(n �k) > i n(k + 1)/2if 0 �k < ,(n �k)/2 if �k �n,relation (2.9) yields� n�(k �k=01)|螖2 | k< K1 ,k=(n �k)|螖2 | �K2 . k(2.25)
