Amenability by Paterson A.L.T.

By Paterson A.L.T.

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E + (K1) ^)> and x € [0,°o) k\ is fixed, and proved some approximation properties. In this paper, motivated by the recent work of Derriennic [1] on modi­ fied Bernstein polynomials introduced by Durrmeyer [2] for functions integrable on [ 0, 1 ] , we propose a sequence of modified Szafsz operators defined on the space of integrable functions on [0,«>) as oo (Mn,xf){t) = Mn,x(f{y);t) = n Σ oo P n,lSt) k=o where t,x€ [0,°°) and x is fixed. \ P n,k{y) f (x+^)^> ( K 2 ) ° Clearly, (M f){t) is a linear posi­ tive operator.

Amer. Math. Soc. 65 (1949) 372-414. 6. G. Goes, BK-Raume und Matrix transformation für Fourierkoeffizienten, Math. Zeit. 70(1959), 345-371. 7. F. Harmuth, Transmission of Information by Orthogonal Functions, Springer Verlag, Berlin, 1972. 8. F. Harmuth, Sequence theory — Foundations and Applications, Academic Press, New York, 1977. 9. A. Jastrebova, On the approximation of functions satisfying a Lipschitz condition by the arithmetic means of their Walsh Fourier series, Amer. Math. Soc. Trans. Ser.

7) So we consider the case 0 < a < 1 . 7) 4 R e a 3 = a Re {p2 4- 3c 2 -2(l -a) 2 βγ } . (t) a r e i n c r e a s i n g on --t ' _ we [0,2π] and μ . ( 2 π ) - μ . ( 0 ) = 1, i- We a l s o have 2π σ and = 2 [ e"™* d p , ( t ) , n j 0 n = 1, 2 , . . , 2π Pn = 2J e~int dM2(t) , n=l,2 0 Now ( 2 . 8 ) becomes 2TT 2π 4 R e a 3 = 2a 0 cos It -8a(l -a) j ! d\x(t) 2π + 6a ° cos £ d y x ( t ) cos It 2 - d\il(t) | 2π s i n t dv1(t)\ 2 \ , 1,2. 56 D. A. Brannan and T. S. Tana < 2a 2π 2ir 2π cos It d]i2(t) + 6a cos It d]\ ( t ) + 8a(l - a ) u o 2 2TT = 2a j l - 2 sin2tdy ( £ ) + 3 - 6 sin t d]i1 it) π sin^dy^t) + 4(1 - a ) j I s i n t d\\ ^t) 1 j .

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