Embedding Schramm spaces into Chanturiya classes. (English) Zbl 1457.42007
Necessary and sufficient conditions for the embedding of various spaces of functions of generalized bounded variation into the space \(H_p^{\omega}\) were studied by several mathematicians.
In the present paper a necessary and sufficient condition for the embedding of the Schramm space \(\Phi B V\) into the Chanturiya class \(V[\nu]\) is given. Namely, it is proved that if \(\Phi\) is a Schramm sequence and \(\nu\) is a modulus of variation, then \(\Phi B V\) embeds into \(V[\nu]\) if and only if \[ \limsup_{n\to \infty}\frac{n\Phi^{-1}_{n}(1)}{\nu(n)}<\infty . \] Applying this statement and Watermans theorem on \(HB V\) the author proves that if \(\Phi\) is a Schramm sequence that satisfies the condition \[ \sum_{n=1}^{\infty}\frac{\Phi^{-1}_{n}(1)}{ n} <\infty, \] then the Fourier series of any function \(f\) in \(\Phi B V\) converges everywhere. Furthermore, this convergence is uniform over any closed interval of continuity points of \(f\).
In the present paper a necessary and sufficient condition for the embedding of the Schramm space \(\Phi B V\) into the Chanturiya class \(V[\nu]\) is given. Namely, it is proved that if \(\Phi\) is a Schramm sequence and \(\nu\) is a modulus of variation, then \(\Phi B V\) embeds into \(V[\nu]\) if and only if \[ \limsup_{n\to \infty}\frac{n\Phi^{-1}_{n}(1)}{\nu(n)}<\infty . \] Applying this statement and Watermans theorem on \(HB V\) the author proves that if \(\Phi\) is a Schramm sequence that satisfies the condition \[ \sum_{n=1}^{\infty}\frac{\Phi^{-1}_{n}(1)}{ n} <\infty, \] then the Fourier series of any function \(f\) in \(\Phi B V\) converges everywhere. Furthermore, this convergence is uniform over any closed interval of continuity points of \(f\).
Reviewer: Rostom Getsadze (Uppsala)
MSC:
| 42A20 | Convergence and absolute convergence of Fourier and trigonometric series |
| 42A16 | Fourier coefficients, Fourier series of functions with special properties, special Fourier series |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 26A45 | Functions of bounded variation, generalizations |
Keywords:
Fourier series; uniform convergence; Fourier coefficients; generalized bounded variation; embeddingReferences:
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