Geodesic
A short and simple proof of the Jurkat--Waterman theorem on conjugate functions
Funkcionalʹnyj analiz i ego priloženiâ, Tome 51 (2017) no. 2, pp. 87-91.

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

It is well known that certain properties of continuous functions on the circle T related to the Fourier expansion can be improved by a change of variable, i.e., by a homeomorphism of the circle onto itself. One of the results in this area is the Jurkat–Waterman theorem on conjugate functions, which improves the classical Bohr–Pál theorem. In the present work we propose a short and technically very simple proof of the Jurkat–Waterman theorem. Our approach yields a stronger result.
Keywords: Fourier series, superposition operators, conjugate functions. 
@article{FAA_2017_51_2_a8,
     author = {V. V. Lebedev},
     title = {A short and simple proof of the {Jurkat--Waterman} theorem on conjugate functions},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {87--91},
     publisher = {mathdoc},
     volume = {51},
     number = {2},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2017_51_2_a8/}
}
TY  - JOUR
AU  - V. V. Lebedev
TI  - A short and simple proof of the Jurkat--Waterman theorem on conjugate functions
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2017
SP  - 87
EP  - 91
VL  - 51
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2017_51_2_a8/
LA  - ru
ID  - FAA_2017_51_2_a8
ER  - 
%0 Journal Article
%A V. V. Lebedev
%T A short and simple proof of the Jurkat--Waterman theorem on conjugate functions
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2017
%P 87-91
%V 51
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2017_51_2_a8/
%G ru
%F FAA_2017_51_2_a8
V. V. Lebedev. A short and simple proof of the Jurkat--Waterman theorem on conjugate functions. Funkcionalʹnyj analiz i ego priloženiâ, Tome 51 (2017) no. 2, pp. 87-91. http://geodesic.mathdoc.fr/item/FAA_2017_51_2_a8/

[1] H. Bohr, Acta Sci. Math. (Szeged), 7 (1935), 129–135

[2] J. Pál, C. R. Acad. Sci Paris, 158 (1914), 101–103 | Zbl

[3] R. Salem, Bull. Amer. Math. Soc., 50:8 (1944), 579–580 | DOI | MR | Zbl

[4] N. K. Bari, Trigonometricheskie ryady, Fizmatgiz, M., 1961 | MR

[5] W. Jurkat, D. Waterman, Complex Variables Theory Appl., 12:1–4 (1989), 67–70 | DOI | MR | Zbl

[6] C. Goffman, T. Nishiura, D. Waterman, Homeomorphisms in Analysis, Math. Surveys and Monographs, 54, Amer. Math. Soc., Providence, RI, 1997 | MR | Zbl

[7] N. S. Stylianopoulos, E. Wegert, Ann. Univ. Mariae Curie-Sklodowska, Sect. A., 56 (2002), 97–103 | MR | Zbl

[8] A. Di Bucchianico, J. Math. Anal. Appl., 156:1 (1991), 253–273 | DOI | MR | Zbl

[9] J.-P. Kahane, Y. Katznelson, Studies in Pure Math., to the Memory of Paul Turán, Académiai Kiadó, Budapest, 1983, 395–410 | DOI

[10] A. M. Olevskii, UMN, 40:3(243) (1985), 157–193 | MR | Zbl

[11] V. V. Lebedev, Matem. sb., 181:8 (1990), 1099–1113 | Zbl

[12] V. Lebedev, Studia Math., 231:1 (2015), 73–81 | MR | Zbl

[13] A. A. Saakyan, Matem. sb., 110(152):4(12) (1979), 597–608 | MR

[14] A. M. Olevskii, Dokl. AN SSSR, 256:2 (1981), 284–288 | MR