Holland, Anthony Stanley Bassett

Sahney, Badri Narayan

Sidon, Jean

English

Holland Anthony Stanley Bassett

Sahney Badri Narayan

Sidon Jean

University of Szeged

Acta Sei. Math., 38 (1976), 69—105 On degree of approximation of a class of functions by means of Fourier series A. S. B. HOLLAND, B. N. SAHNEY, and J. TZIMBALARIO 1. Let / be periodic with period 2n, and integrable in the Lebesgue sense. The Fourier series associated with / at the point x, is given by CL 00 (1.1) f ( x ) % -rf + 2 (an C0S nX + K sin nx). n = l If {/?„} is a sequence of positive constants, such that Pn = P0+P1+ -+Pn-~ °° as n - 0 0 0 0 then a given series 2 cn with the sequence of partial sums {$„} is said to be Norlund n = 0 summable ( N , p„) to 5, provided that 1 " T„ = 2Pn-k*k •* s as n - co. rn k=0 We call T„ the (N, ¿>„)-mean or Norlund mean of 2 cn • I n the following we assume that the Norlund means are regular, more precisely, we assume that (1.2) 0 < np„ ̂  cP„ for n= 1, 2 , . . . , and p0 > 0. 2. The following theorem on the degree of approximation of a func t ion / i Lip a, by the (C, <5)-means of its Fourier series, is due to G . ALEXITS [1]. T h e o r e m A. If a periodic function fdLip a for 0 < a ^ l , then the degree of approximation of the (C, d)-means of its Fourier series for 0<a<<5^ 1 is given by max |/(*)-<x<«(*)| = o ( - U 0 S I S 2 I \ n ) Received April 12, 1975. 70 A. S. B. Holland, B. N. Sahney, J. Tzimbalario and for 0<as5sl, is given by max | / ( * ) - a « > ( * ) | = 0 0SX32JI log« where o„ are the (C, 8)-means of the partial sums of (1.1). Let C*[0,2n] denote the class of all continuous functions on [0, 2iz], periodic and of period 2n. The object of this paper is to prove the following theorem. T h e o r e m . If co(t) is the modulus of continuity off£C*[0, In], then the degree of approximation of f by the Norlund means of the Fourier series for f is given by En= m a x | / ( i ) ( 0 1 0 S ( S 2 l t where T„ are the (N, p„)-means of the Fourier series for f If we deal with Cesaro means of order 8 and consider a function / £ Lip a, 0 < a ^ l , then our Theorem reduces to Theorem A. P r o o f . „ „ Sinlfc + y | i r , ( * ) - / ( * ) = / { / ( * + 0 + / ( * - 0 - 2 / ( * ) } ¿ P n - i dt. sin • If we write <p(t) = \if(x + t) + $ f ( x - t ) - f ( x ) \ then it is clear that q>{t) ^ ©(0, and therefore, 1 IT In \ f ( x ) - T n ( x ) \ ^ - - f (0(t) nP„ J t/2 Z P„-kSin(k + l/2)t k=0 dt + 1 f a>it) ZPn-kSinkt k=0 r <0 it) nP„ J t/2 n njn ' ZPn-k cos kt k = 0 dt = Now = I i + I i + I s , say. " l n co (0 » „ i /2 izjn Zp„-ksm(_k+l/2)t k=o dt = ^ p - Zp»-kik+mtdt = I k=0 = f "(Ocit ZoPn-kik+ 1/2) = o [ ^ L ] a > | i J kZkpn-k = 0(co(\/n)). On degree of approximation of a class of functions by means of Fourier series 71 By (1.2), consequi Now Similarly which is dominated by the bound for /2 . Adding the bounds for Ix, /2 , I3 we have the desired result. 3. R e m a r k s . It may be interesting to know the answers to the following questions: (i) Can our Theorem be extended to matrix summability? (ii) Can the result be extended to differentiated Fourier series? (iii) Can the result be extended to some other series, viz. Legendre series, ultraspherical, Bessel series, etc? [1] G. ALEXITS, Über Die Annäherung einer stetigen Funktion durch die Cesároschen Mittel ihrer Fourrierreihe, Math. Annalen, 100 (1928), 264—277. [2] G . ALEXITS, Convergence Problems of Orthogonal Series, Pergamon Press (1961). [3] V. A. ANDRIENKO, The approximation of functions by Fejér means, Sibirsk. Mat. Z., 9 (1968), 3—12. [4] H. BERENS, On the saturation problem for the Cesáro means of Fourier series, Acta Math. Acad. Sei. Hungar., 21 (1970) , 9 5 — 9 9 . [5] J . S . BYRNES, L 2 approximation with trigonometric /i-nomials, J. Approx. Theory, 9 (1973) , 373—379: References 72 A. S. B. Holland, B. N. Sahney, J. Tzimbalario: On degree of approximation [6] L . MCFADDEN, Absolute Nörlund summability, Duke Math. J., 9 (1942) , 2 0 7 — 1 6 8 . [7] A . ZYGMUND, Trigonometric Series, Vols. I & I I combined, Cambridge University Press (Cam-bridge, 1968). A. S. B. HOLLAND A N D B. N. SAHNEY J. TZIMBALARIO DEPARTMENT O F MATHEMATICS A N D STATISTICS DEPARTMENT OF MATHEMATICS THE UNIVERSITY O F CALGARY THE UNIVERSITY O F ALBERTA CALGARY, ALBERTA, CANADA T2N 1N4 EDMONTON, ALBERTA, CANADA T6G 2G1 

On degree of approximation of a class of functions by means of Fourier series

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On degree of approximation of a class of functions by means of Fourier series

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