Geodesic
Approximation properties of partial Fourier sums in the $p$-variation metric
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the 20 International Saratov Winter School "Contemporary Problems of Function Theory and Their Applications", Saratov, January 28 — February 1, 2020. Part 2, Tome 200 (2021), pp. 29-35.

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

In this paper, we examine the degree of approximation by Fourier partial sums in the $p$-variational norm. We propose two criteria for convergence of these sums with a given rate in terms of growth of the norms of the differentiated Fourier partial sums. Also, we establish the relationship between the approximation of a function and its conjugate function.
Keywords: function of bounded $p$-variation, partial Fourier sums, derivative, conjugate function.
Mots-clés : convergence rate 
@article{INTO_2021_200_a2,
     author = {S. S. Volosivets and A. A. Tyuleneva},
     title = {Approximation properties of partial {Fourier} sums in the $p$-variation metric},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {29--35},
     publisher = {mathdoc},
     volume = {200},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_200_a2/}
}
TY  - JOUR
AU  - S. S. Volosivets
AU  - A. A. Tyuleneva
TI  - Approximation properties of partial Fourier sums in the $p$-variation metric
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2021
SP  - 29
EP  - 35
VL  - 200
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2021_200_a2/
LA  - ru
ID  - INTO_2021_200_a2
ER  - 
%0 Journal Article
%A S. S. Volosivets
%A A. A. Tyuleneva
%T Approximation properties of partial Fourier sums in the $p$-variation metric
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2021
%P 29-35
%V 200
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2021_200_a2/
%G ru
%F INTO_2021_200_a2
S. S. Volosivets; A. A. Tyuleneva. Approximation properties of partial Fourier sums in the $p$-variation metric. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the 20 International Saratov Winter School "Contemporary Problems of Function Theory and Their Applications", Saratov, January 28 — February 1, 2020. Part 2, Tome 200 (2021), pp. 29-35. http://geodesic.mathdoc.fr/item/INTO_2021_200_a2/

[1] Bari N. K., Trigonometricheskie ryady, GIFML, M., 1961

[2] Bari N. K., Stechkin S. B., “Nailuchshie priblizheniya i differentsialnye svoistva dvukh sopryazhennykh funktsii”, Tryu Mosk. mat. o-va., 5 (1956), 483–522 | Zbl

[3] Volosivets S. S., “Polinomy nailuchshego priblizheniya i sootnosheniya mezhdu modulyami nepreryvnosti v prostranstvakh funktsii ogranichennoi $p$-variatsii”, Izv. vuzov. Mat., 1996, no. 9, 21–26 | Zbl

[4] Terekhin A. P., “Priblizhenie funktsii ogranichennoi $p$-variatsii”, Izv. vuzov. Mat., 1965, no. 2, 171–187 | Zbl

[5] Terekhin A. P., “Integralnye svoistva gladkosti periodicheskikh funktsii ogranichennoi $p$-variatsii”, Mam. zametki., 2:3 (1967), 289–300 | Zbl

[6] Timan A. F., Teoriya priblizheniya funktsii deistvitelnogo peremennogo, GIFML, M., 1960

[7] Butzer P. L., Scherer K., “On the fundamental approximation theorems of D. Jackson, S. N. Bernstein and theorems of M. Zamansky and S. B. Stechkin”, Aequat. Math., 3:1 (1969), 170–185 | DOI | Zbl

[8] DeVore R. A., Lorentz G. G., Constructive Approximation, Springer-Verlag, Berlin–Heidelberg–New York, 1993 | Zbl

[9] Salem R., Zygmund A., “Approximation by partial sums of Fourier series”, Trans. Am. Math. Soc., 59:1 (1946), 14–22 | DOI | MR | Zbl

[10] Sunouchi G.-I., “Derivatives of a polynomial of best approximation”, Jahresber. Dtsch. Math.-Ver., 70 (1968), 165–166 | Zbl

[11] Volosivets S. S., Tyuleneva A. A., “Generalized monotonicity of sequences and functions of bounded $p$-variation”, Acta Math., 82:1-2 (2016), 111–124 | Zbl

[12] Wiener N., “The quadratic variation of a function and its Fourier coefficients”, J. Math. Phys., 3 (1924), 72–94 | DOI | Zbl

[13] Young L. C., “An inequality of Hölder type connected with Stieltjes integration”, Acta Math., 67 (1936), 251–282 | DOI | MR

[14] Zamansky M., “Classes de saturation de certainsprocédes d'approximation des séries de Fourier des fonctions continues et applications à quelques problèmes d'approximation”, Ann. Sci. Ecole Norm. Super., 66 (1949), 19–93 | DOI | Zbl