Geodesic
Sequences of Composition Operators in Spaces of Functions of Bounded $\Phi$-Variation
Matematičeskie zametki, Tome 85 (2009) no. 3, pp. 330-341.

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

The main results of the paper are contained in Theorems 1 and 2. Theorem 1 presents necessary and sufficient conditions for a sequence of functions $h_n\colon\langle c,d\rangle\to\langle a,b\rangle$, $n=1,2,\dots$, to have bounded sequences of $\Psi$-variations $\{V_\Psi(\langle c,d\rangle;f\circ h_n)\}_{n=1}^\infty$ evaluated for the compositions of an arbitrary function $f\colon\langle a,b\rangle\to\mathbb R$ with finite $\Phi$-variation and the functions $h_n$. In Theorem \ref{t2:u330}, the same is done for a sequence of functions $h_n\colon\mathbb R\to\mathbb R$, $n=1,2,\dots$, and the sequence of $\Psi$-variations $\{V_\Psi(\langle a,b\rangle;h_n\circ f)\}_{n=1}^\infty$.
Keywords: composition operator, $\varphi$-function, modulus of continuity, Lipschitz function, Hölder property.
Mots-clés : $\Phi$-variation 
@article{MZM_2009_85_3_a1,
     author = {O. E. Galkin},
     title = {Sequences of {Composition} {Operators} in {Spaces} of {Functions} of {Bounded} $\Phi${-Variation}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {330--341},
     publisher = {mathdoc},
     volume = {85},
     number = {3},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2009_85_3_a1/}
}
TY  - JOUR
AU  - O. E. Galkin
TI  - Sequences of Composition Operators in Spaces of Functions of Bounded $\Phi$-Variation
JO  - Matematičeskie zametki
PY  - 2009
SP  - 330
EP  - 341
VL  - 85
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2009_85_3_a1/
LA  - ru
ID  - MZM_2009_85_3_a1
ER  - 
%0 Journal Article
%A O. E. Galkin
%T Sequences of Composition Operators in Spaces of Functions of Bounded $\Phi$-Variation
%J Matematičeskie zametki
%D 2009
%P 330-341
%V 85
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2009_85_3_a1/
%G ru
%F MZM_2009_85_3_a1
O. E. Galkin. Sequences of Composition Operators in Spaces of Functions of Bounded $\Phi$-Variation. Matematičeskie zametki, Tome 85 (2009) no. 3, pp. 330-341. http://geodesic.mathdoc.fr/item/MZM_2009_85_3_a1/

[1] J. Musielak, W. Orlicz, “On generalized variations. I”, Studia Math., 18 (1959), 11–41 | MR | Zbl

[2] R. Lésniewicz, W. Orlicz, “On generalized variations. II”, Studia Math., 45 (1973), 71–109 | MR | Zbl

[3] M. Josephy, “Composing functions of bounded variation”, Proc. Amer. Math. Soc., 83:2 (1981), 354–356 | DOI | MR | Zbl

[4] C. Jordan, “Sur la série de Fourier”, C. R. Acad. Sci. Paris, 92:5 (1881), 228–230 | Zbl

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

[6] J. Marcinkiewicz, “On a class of functions and their Fourier series”, Collected Papers, Państwowe Wydawnictwo Naukowe, Warsawa, 1964, 36–41 | MR | Zbl

[7] L. C. Young, “Inequalities connected with bounded $p$-th power variation in the Wiener sense and with integrated Lipschitz conditions”, Proc. London Math. Soc. (2), 43 (1937), 449–467 | DOI | Zbl

[8] L. C. Young, “Sur une généralisation de la notion de variation de puissance $p$-ième bornée au sens de N. Wiener, et sur la convergence des séries de Fourier”, C. R. Acad. Sci. Paris, 204:7 (1937), 470–472 | Zbl

[9] V. V. Chistyakov, O. E. Galkin, “Mappings of bounded $\Phi$-variation with arbitrary function $\Phi$”, J. Dynam. Control Systems, 4:2 (1998), 217–247 | DOI | MR | Zbl

[10] J. Ciemnoczołowski, W. Orlicz, “Composing functions of bounded $\varphi$-variation”, Proc. Amer. Math. Soc., 96:3 (1986), 431–436 | DOI | MR | Zbl