The Bohr-Pál theorem and the Sobolev space $W₂^{1/2}$

• Autorzy: Vladimir Lebedev
• Języki publikacji: EN

Abstrakty

EN

The well-known Bohr-Pál theorem asserts that for every continuous real-valued function f on the circle 𝕋 there exists a change of variable, i.e., a homeomorphism h of 𝕋 onto itself, such that the Fourier series of the superposition f ∘ h converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings f into the Sobolev space $W₂^{1/2}(𝕋)$. This refined version of the Bohr-Pál theorem does not extend to complex-valued functions. We show that if α < 1/2, then there exists a complex-valued f that satisfies the Lipschitz condition of order α and at the same time has the property that $f ∘ h ∉ W₂^{1/2}(𝕋)$ for every homeomorphism h of 𝕋.

Kategorie tematyczne

• 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2015
• Tom: 231
• Numer: 1
• Strony: 73-81

Daty

wydano

2015

Twórcy

autor

Vladimir Lebedev

• National Research University Higher School of Economics, 34 Tallinskaya St., Moscow, 123458, Russia

Identyfikatory

DOI

10.4064/sm8438-1-2016