On the invariance of certain classes of functions under composition

Authors:
Milton Chaika and Daniel Waterman

Journal:
Proc. Amer. Math. Soc. **43** (1974), 345-348

MSC:
Primary 26A16; Secondary 42A20

DOI:
https://doi.org/10.1090/S0002-9939-1974-0330367-9

MathSciNet review:
0330367

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Abstract | References | Similar Articles | Additional Information

Abstract: Certain classes of functions are mapped into themselves by any change of variable. For some classes of this type which are of interest in the study of Fourier series, it is shown that the necessary and sufficient condition that be in the class for each of that class is that .

**[1]**A. Baernstein and D. Waterman,*Functions whose Fourier series converge uniformly for every change of variable*, Indiana Univ. Math. J. (to appear). MR**0310523 (46:9621)****[2]**A. M. Garsia and S. Sawyer,*On some classes of continuous functions with convergent Fourier series*, J. Math. Mech.**13**(1964), 589-601. MR**33**#7777. MR**0199634 (33:7777)****[3]**C. Goffman,*Everywhere convergence of Fourier series*, Indiana Univ. Math. J.**20**(1970/71), 107-112. MR**42**#4941. MR**0270048 (42:4941)****[4]**C. Goffman and D. Waterman,*Functions whose Fourier series converge for every change of variable*, Proc. Amer. Math. Soc.**19**(1968), 80-86. MR**36**#4245. MR**0221193 (36:4245)****[5]**C. O. Kiselman,*On the Garsia-Sawyer condition for uniform convergence of Fourier series*, Uppsala University, Dept. of Mathematics, Report No. 25, 1971.**[6]**D. Waterman,*On convergence of Fourier series of functions of generalized bounded variation*, Studia Math.**44**(1972), 107-117. MR**0310525 (46:9623)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0330367-9

Keywords:
Lipschitz classes,
compositions,
convergence of Fourier series,
generalized bounded variation

Article copyright:
© Copyright 1974
American Mathematical Society