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On a conjecture of F. Móricz and X. L. Shi》.pdf
JOURNAL OF APPKOXIMATION THEORY 62. 257-261 ( 1990) On a Conjecture of F. Mbricz and X. L. Shi s. P. ZHOU* Let .f‘(.~, ~1) be a continuous function, 2rr-periodic in each variable, in symbols f’~ C,, x 2n. The paritial moduli of continuity of f’ are dlelined for ii0 by cl~,(.J S) = sup max VI-u + U, .I!) -.f(,~, .r)I, 1111-SC) \. 1’ and Moreover, the Lipschitz class Lip(z, [I), where x, fl E (0, 11. is defined to be Lip(sc, fi)= j.f‘~C~~~~~:(~),(.f;~)=O(fi~) and ~1,(.Lb)=:0(6”)i. The corresponding conjugate function of ,f‘(x) is I f’(x+u, j~-v)+,f’(~u-u, ?~-1.))pOt;du. Concerning the approximation to continuous functions by Cesaro means of double conjugate series, F. Moricz and X. L. Shi [I] raised two conjectures, one of which is as follows. Conjecture MS [I, p. 360, Conjecture 21. There exists a function ,f‘~ Lip( 1, 1) such that the estimate cannot hold. We now show that the answer to Conjecture MS is affirmative * Present address: Department of Mathematics, Statistics and Computing Science. Dalhouw University. Halifax, NS. Canada B3H 355. 257 258 S. P. ZHOL The following lemmas are needed. Puyf: Evidently on the other hand. by Abel transform where
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