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Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 2, Pages 131–164 (Mi izv2745)  

On the topological stability of continuous functions in certain spaces related to Fourier series

V. V. Lebedev

Moscow State Institute of Electronics and Mathematics (Technical University)

Abstract: We show that the following properties of a continuous function $f$ on the circle $\mathbb T$ are equivalent: the sequence $\widehat{f\circ h}$ of the Fourier coefficients of the superposition $f\circ h$ belongs to the weak $l^1$ for every homeomorphism $h$ of the circle onto itself; $f$ is a function of bounded quadratic variation. We obtain similar results for spaces of functions whose sequence of Fourier coefficients belongs to the weak $l^p$, $1<p<2$, for spaces $A_p$ of functions $f$ with $\widehat{f}\in l^p$, for the Sobolev spaces $W_2^\lambda$, and for other spaces of functions on $\mathbb T$. Under rather general assumptions on a space $\mathbb X$ of functions on the circle, we give a necessary condition for a given continuous function $f$ to stay in $\mathbb X$ for every change of variable. We also consider the multidimensional case, which is essentially different from the one-dimensional case. In particular, we show that if $p<2$ and $f$ is a continuous function on the torus $\mathbb T^d$, $d\ge2$, such that $f\circ h\in A_p(\mathbb T^d)$ for every homeomorphism $h\colon \mathbb T^d\to\mathbb T^d$, then $f$ is constant.

Keywords: homeomorphisms of the circle, Fourier series.

DOI: https://doi.org/10.4213/im2745

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English version:
Izvestiya: Mathematics, 2010, 74:2, 347–378

Bibliographic databases:

Document Type: Article
UDC: 517.51
MSC: 42A16, 42B05
Received: 12.11.2007

Citation: V. V. Lebedev, “On the topological stability of continuous functions in certain spaces related to Fourier series”, Izv. RAN. Ser. Mat., 74:2 (2010), 131–164; Izv. Math., 74:2 (2010), 347–378

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