By Z. Ditzian, V. Totik (auth.)
The topic of this ebook is the creation and alertness of a brand new degree for smoothness offunctions. notwithstanding we have now either formerly released a few articles during this path, the consequences given listed below are new. a lot of the paintings was once performed in the summertime of 1984 in Edmonton after we consolidated previous principles and labored out many of the info of the textual content. It took one other yr and a part to enhance and varnish a number of the theorems. We show our gratitude to Paul Nevai and Richard Varga for his or her encouragement. We thank NSERC of Canada for its invaluable aid. We additionally thank Christine Fischer and Laura Heiland for his or her cautious typing of our manuscript. z. Ditzian V. Totik CONTENTS creation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 half I. THE MODULUS OF SMOOTHNESS bankruptcy 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1. Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2. dialogue of a few stipulations on cp(x). . . . • . . . . . . . • . . • . . • • . eight . . . • . 1.3. Examples of assorted Step-Weight services cp(x) . . • . . • . . • . . • . . . nine . . • bankruptcy 2. The K-Functional and the Modulus of Continuity ... . ... 10 2.1. The Equivalence Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 10 . . . . . . . . . 2.2. the higher Estimate, Kr.tp(f, tr)p ~ Mw;(f, t)p, Case I . . . . . . . . . . . . 12 . . . 2.3. the higher Estimate of the K-Functional, the opposite instances. . . . . . . . . . sixteen . 2.4. The decrease Estimate for the K-Functional. . . . . . . . . . . . . . . . . . . 20 . . . . . bankruptcy three. K-Functionals and Moduli of Smoothness, different kinds. 24 3.1. A changed K-Functional . . . . . . . . . . . . . . . . . . . . . . . . . . 24 . . . . . . . . . . 3.2. ahead and Backward modifications. . . . . . . . . . . . . . . . . . . . . . 26 . . . . . . . 3.3. Main-Part Modulus of Smoothness. . . . . . . . . . . . . . . . . . . . . . 28 . . . . . . .
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To complete the proof it is enough to show 2'W(t} ~ RQ(C,J,tB} for some B < 2- 1 +/1 independent of I and t for which we need the following lemma. 6. [a,cl (III - where M depends on p but not on I, a, b, c, or 15. C])' 32 3. K-Functionals and Moduli of Smoothness, Other Forms This was proved many times before; we state it only to emphasize that in the construction the constant M is independent of many of the parameters. We now write W(t) ~ sup Lt*/4rs;u If cp(x) = Jx(1 - x) (A = 1) for instance, any C < 1/4 would satisfy w;(f, t)p = n~(c,f, t)p for t small enough. 4. For CI. > 0, w;(f, t)p = O(ta) and n~(c,f, t)p = OW) are equivalent, and also w;(f, t)p '" t a and n~(c,f, t)p '" t a are equivalent. 5. For CI. > ° and cp(x) = (1 - X 2 )1/2 IIAh",fIILp[-1+r2h2,1-r2h2] = w;(f, t)p = OW) and O(ha) are equivalent and also w;(f, t)p '" t a and IIA~",fIILp[-1+r2h2,1-r2h2] '" h a are equivalent. The advantages of the main-part modulus are best illustrated by these corollaries. 00, r = 2 and q>(x) = Jx. Then for some f E LP(R+) PROOF. For p = 00 we choose f(x) = (logxfl for 0 < x ::; 1/2, f(x) = 0 for x> 1 and f(X)E C2. Choosing Xo = t 2, we can write w;(f, t)oo ;::: If(x o - tq>(xo)) - 4f(xo) + f(xo + tq>(x o)) I = 1-2(logt2 fl + (log2t 2 f l l '" Ilogtl- l . On the other hand we also have Q;(f, t)oo::; sup sup ILl~Jxf(x)1 '" sup (hJx)2f"(~J, O