By Ralph P.Jr. Boas
This monograph is areport at the current nation of a reasonably coherent selection of difficulties approximately which a monstrous literature has grown up lately. during this literature, the various difficulties have, because it occurs, been analyzed in nice aspect, while different very comparable ones were handled even more superficially. i haven't tried to enhance at the literature via making both distinct displays of each subject. i've got additionally now not aimed toward encyclopedic completeness. i've got, although, mentioned a few attainable generalizations by means of pointing out a few questions; a few of these may possibly without doubt be disposed of in a couple of minutes; a few are most likely fairly tricky. This monograph used to be written on the advice of B. SZ.-NAGY. I take this chance of mentioning that his paper  encouraged the larger a part of the fabric that's offered right here; particularly, it comprises the satisfied inspiration of focusing Y awareness at the multipliers nY-i, x- . R. ASKEY, P. HEYWOOD, M. and S. IZUMI, and S. WAINGER have kindly communicated a few of their fresh effects to me prior to ebook. i'm indebted for assistance on a number of issues to L. S. BOSANQUET, S. M. EDMONDS, G. is going, S. IZUMI, A. ZYGMUND, and particularly to R. ASKEY. My paintings was once supported through the nationwide technological know-how origin lower than can provide GP-314, GP-2491, GP-3940 and GP-5558. Evanston, Illinois, February, 1967 R. P. Boas, Jr. Contents Notations ... § 1. Introduetion three §2. Lemmas .. 7 § three. Theorems with confident or lowering features .
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Extra info for Integrability Theorems for Trigonometric Transforms
1) Partial summation applied to ZVGMUND , p. 12). § 5. Tbe exceptional integral values of the index 31 Note that X-I g(x)eL implies the eonvergence ofIbn (not ofIlbnD with no positivity hypothesis on g, by Dini's test for the eonvergence of the eonjugate series (HEYWOOD ). The next theorem is not symmetrie, but has no positivity hypothesis at all (BoAS ). 22. 1/ Ilbnl < then 00 J" X-I g(x) dx 0+ exists. 23. 20 be extended to Fourier series with positive coeffieients? A partial answer is fumished by the next theorem1).
IZUMI . § 9. Trigonometrie integrals We may expeet theorems for trigonometrie integrals to be similar to theorems for trigonometrie series and to be provable by similar methods. They are likely to be different in detail. Also if, for example, J a(t) eos xt dt, 00 f(x) = o we ean diseuss the integrability of x y - 1 a(x) and of x- Y f(x) in neighborhoods either of 0 or of 00. On the other hand, sinee a(t) = 2 n- 1 J f(x) eos xt dx, 00 o there is no real distinction between, for example, theorems with deereasing a and theorems with deereasing f The following theorems are analogues of some of the theorems of § 4.
31) folIows. 1 to be conneeted with eaeh other. 32. IJ a. ~ Oultimately, and a. are the cosine coefficients oJ J, then J(O)I < 00 if and 'only if L ak log k converges. Jx- 1 IJ(x) - L Let us notice that the eonvergence of ak log k is, when a. , where s. = a1 + a2 + ... + a•. This suggests considering the series n- 1 s. in other cases; we quote the following theorem on eonditional eonvergence1). 33. IJ JE L, a. are the cosine coefficients oJ J, and x Jo IJ(t)1 dt = I) BOSANQUET and OFFORD . 4 Ergebnisse der Mathematik.
Integrability Theorems for Trigonometric Transforms by Ralph P.Jr. Boas