Geodesic
Divergence of the Fourier series by generalized Haar systems at points of continuity of a~function
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2016), pp. 49-68.

Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain a connection between the Dirichlet kernels and partial Fourier sums by generalized Haar and Walsh (Price) systems. Based on this, we establish an interrelation between convergence of the Fourier series by generalized Haar and Walsh (Price) systems. For any unbounded sequence we construct a model of continuous function on a group (and even on a segment $[0,1]$), whose Fourier series by generalized Haar system generated by this sequence, diverges at some point.
Keywords: Abelian group, modified segment $[0,1]$, continuity on modified segment $[0,1]$, systems of characters, Price's systems, generalized Haar's systems. 
@article{IVM_2016_1_a4,
     author = {V. I. Shcherbakov},
     title = {Divergence of the {Fourier} series by generalized {Haar} systems at points of continuity of a~function},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {49--68},
     publisher = {mathdoc},
     number = {1},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2016_1_a4/}
}
TY  - JOUR
AU  - V. I. Shcherbakov
TI  - Divergence of the Fourier series by generalized Haar systems at points of continuity of a~function
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2016
SP  - 49
EP  - 68
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2016_1_a4/
LA  - ru
ID  - IVM_2016_1_a4
ER  - 
%0 Journal Article
%A V. I. Shcherbakov
%T Divergence of the Fourier series by generalized Haar systems at points of continuity of a~function
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2016
%P 49-68
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2016_1_a4/
%G ru
%F IVM_2016_1_a4
V. I. Shcherbakov. Divergence of the Fourier series by generalized Haar systems at points of continuity of a~function. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2016), pp. 49-68. http://geodesic.mathdoc.fr/item/IVM_2016_1_a4/

[1] Walsh J. L., “A closed set of normal orthogonal functions”, Amer. J. Math., 45:1 (1923), 5–24 | DOI | MR | Zbl

[2] Paley R. E. A. C., “A remarkable series of orthogonal functions. I”, Proc. London Math. Soc., 34 (1932), 241–264 | DOI | MR | Zbl

[3] Paley R. E. A. C., “A remarkable series of orthogonal functions. II”, Proc. London Math. Soc., 34 (1932), 265–279 | DOI | MR | Zbl

[4] Chrestenson H. E., “A class of generalized Walsh functions”, Pacific J. Math., 5:1 (1955), 17–31 | DOI | MR | Zbl

[5] Price J. J., “Certain group of orthonormal step functions”, Canadian J. Math., 9:3 (1957), 417–425 | MR

[6] Vilenkin N. Ya., “Ob odnom klasse polnykh ortogonalnykh sistem”, Izv. AN SSSR, ser. matem., 11:4 (1947), 363–400 | MR | Zbl

[7] Haar A., “Zur Theorie der orthogonalen Functionensysteme”, Math. Ann., 69:3 (1910), 331–371 | DOI | MR | Zbl

[8] Scherbakov V. I., “O potochechnoi skhodimosti ryadov Fure po multiplikativnym sistemam”, Vestn. MGU, ser. matem., mekh., 1983, no. 2, 37–42

[9] Onneweer C. W., Waterman D., “Uniform convergence of Fourier series on groups”, Michigan Math. J., 18:3 (1971), 265–273 | DOI | MR | Zbl

[10] Scherbakov V. I., “Priznak Dini–Lipshitsa i skhodimost ryadov Fure po multiplikativnym sistemam”, Analysis Math., 10:1 (1984), 134–150

[11] Golubov B. I., Rubinshtein A. I., “Ob odnom klasse sistem skhodimosti”, Matem. sb., 71(113):1 (1966), 96–115 | MR | Zbl

[12] Golubov B. I., “Ob odnom klasse polnykh ortogonalnykh sistem”, Sib. matem. zhurn., 9:2 (1968), 297–314 | MR | Zbl

[13] Golubov B. I., Efimov A. B., Skvortsov B. A., Ryady i preobrazovaniya Uolsha. Teoriya i primenenie, Nauka, M., 1987 | MR

[14] Lukomskii S. F., “O ryadakh Khaara na kompaktnoi nul-mernoi gruppe”, Izv. Saratovsk. un-ta, ser. matem., mekh., inform., 9:1 (2009), 14–19