Geodesic
Properties of Riemann sums for functions representable by a trigonometric series with monotone coefficients
Sbornik. Mathematics, Tome 26 (1975) no. 3, pp. 331-347 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study properties of Riemann sums $$ R_n(\varphi,a)=\frac{2\pi}n\sum_{k=0}^{n-1}\varphi\biggl(2\pi\frac{k+a}n\biggr),\qquad0\leqslant a\leqslant1, $$ for functions representable as the sum of a trigonometric series with monotone (or convex) coefficients. We consider two basic problems: 1) the connection between the behavior of these sums and the rate of decrease of the coefficients of the series; 2) the limit properties of the ratio of a coefficient of the series, considered as an integral, to a corresponding Riemann sum of higher order. Bibliography: 4 titles. 
@article{SM_1975_26_3_a2,
     author = {A. Yu. Petrovich},
     title = {Properties of {Riemann} sums for functions representable by a~trigonometric series with monotone coefficients},
     journal = {Sbornik. Mathematics},
     pages = {331--347},
     year = {1975},
     volume = {26},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1975_26_3_a2/}
}
TY  - JOUR
AU  - A. Yu. Petrovich
TI  - Properties of Riemann sums for functions representable by a trigonometric series with monotone coefficients
JO  - Sbornik. Mathematics
PY  - 1975
SP  - 331
EP  - 347
VL  - 26
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_1975_26_3_a2/
LA  - en
ID  - SM_1975_26_3_a2
ER  - 
%0 Journal Article
%A A. Yu. Petrovich
%T Properties of Riemann sums for functions representable by a trigonometric series with monotone coefficients
%J Sbornik. Mathematics
%D 1975
%P 331-347
%V 26
%N 3
%U http://geodesic.mathdoc.fr/item/SM_1975_26_3_a2/
%G en
%F SM_1975_26_3_a2
A. Yu. Petrovich. Properties of Riemann sums for functions representable by a trigonometric series with monotone coefficients. Sbornik. Mathematics, Tome 26 (1975) no. 3, pp. 331-347. http://geodesic.mathdoc.fr/item/SM_1975_26_3_a2/

[1] N. K. Bari, Trigonometricheskie ryady, Fizmatgiz, Moskva, 1961 | MR

[2] P. L. Ulyanov, “Primenenie $A$-integrirovaniya k odnomu klassu trigonometricheskikh ryadov”, Matem. sb., 35(77) (1954), 469–490 | MR

[3] A. Yu. Petrovich, “O skhodimosti podposledovatelnostei rimanovskikh summ”, Matem. zametki, 16:4 (1974), 645–656 | Zbl

[4] L. Fejer, “Trigonometrische Reihen und Potenzreihen mit mehrfach monotoner Koeffizientenfolge”, Trans. Amer. Math. Soc., 39 (1936), 18–59 | DOI | MR | Zbl