2Ш

Seuola norm. sup. Ksa, II, s. 3, 43—46 (1932),- dies. Ebl. 8, 310] ul»r die liche Konvergenz auf die (С, a)-Konvergenz (ос > 0). L. СеааН (Вокщть).

Misra , M. L.: The snmmabitity (A) of the sueeessively derived series of a Fourier series and its conjugate series. Duke math. J. 14, 167—177 (1947).

let /(ж) be a function integrable in (—я, + ж) and denote by a^, Ъ^ its Fourier

oo

constants . Write F{r,x) = | «q + JE («» cos nx + 5„ sin nx) r« and F(r, x) =

n~l oo

^ ф^ COS nx — «и sin nx) 1^. If there is a polynomial P(^) of degree ^ ^— 1 and

a number I such that çp^.(f) =o(«*) for t -^ 0, (p^{t) = ]{x + t)—P(t) + (—1)^ Uix—t) —Pi—t)] — 21 fi/k ! then I is called the ib-th generalized symmetric derivative of ihe function f(x). The author proves:

( 1 ) К J^{t) dt=:^o{t^+% then lim ^^F{x, r) =1;

t

( 2 ) If J^if) dt = o{t^ + Î), %(0 = iix -i-t)- fix) - {-If [fix-1) -fix)l

0

я

- ^ Y{x, r)----2^ / ^д. it) -^ ctg -^dt =0, where e = arc sin (1 — x).

e

These theorems generalize, for arbitrary positive integers k, those of Fatou [Acta math., Uppsala 30, 335-—400 (1906)] and Plessner pSkiitteilungen Math. Sem. Gießen Nr. 10, 1—36 (1923)]. G. G. Lorentz (Tübingen).

Boas , R. P. jr.: Inequalities for the coefficients of trigonometric jpolynomials. Proc. Akad. Wet. Amsterdam 50, 492—495 (1947).

A series of inequalities for the coefficients of a trigonometric polyno-

+ n

mial fit) = ^ aj^ exp ikt, a_j. = a^., are proved. If Ж = max /(^) and

— n

1 /' TT

I — I \^fit)\ dt, then for every fe > 0, (1) j%j ^M cos , ^ , where p is the larg-

Ô

est odd integer <nfk, (2) ja^l + Kj see {жЦ1п/к']-{-2)) ^M, andforJfe>%/2

( 3 ) j%| 4- f I«fcl ^(l + ij/2)/, &> n/2. These inequalities are generalizations or improvements of those of J. С van der Corput and С Visser [Proc. Akad. Wet. Amsterdam 49, 383—392 (1946)], and (2) is closely related to the ineqnality \aj^\ <щ cos {лЩп/к} 4- 2)), ^ > 0, fit) ^ 0, due to E. v. Egervary and O. Szasz [Math. Z. 27, 641—652 (1928)]. The method of proof consists in defining a non-

decreasing ait) such that the convolution f fix—t) doc ikt) is a simple trigono-

0

metric poljmomial depending upon «^ and a^, G. 6L Lorentz ^Tübingen).

Snejder (Schneider), A.A.: Über Eeihen nach Walshsehen Funktionen mit monotonen Koeffizienten. Izvestija Akad. Kauk SSSR, Ber. mat. 12, 179—192 (1948) [Russisch].

Für das nonnierte Othogonalsystem (Pn,{x), 0 < a; < 1, von Walsh (dieses System entsteht aus dem Orthogonalsystem von Rademacher durch ständigung; vgl. Kaezmarz und Steinhaus, Theorie der Orthogonalreihen, Warszawa-Lwöw 1935; dies. Zbl. 13, 9) wird die Konvergenz von Beihen

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( * ) ^Cn9n(^)f ^n monotœi geg&i Null abnebm,end

untersucht . Biese hängt naturgemäß von der Anordnung der Funktionen fix) im В^^ш ab: ^i einer solchen Anordnung (von Baîey) ist |ede Reihe {♦) überall