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On generalization of Haar system and other function systems in spaces \(E_\phi\). (English. Russian original) Zbl 1392.42026

Russ. Math. 62, No. 1, 76-81 (2018); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2018, No. 1, 87-92 (2018).
Recall that a system \(\{f_n\}\) is called a representation system in an \(F\)-space \(E\) whenever any \(f\in E\) can be approximated by partial sums \(\sum_{n=1}^m c_n f_n\). The author considers a function with a finite number of discontinuity points, supported in \([0,1]\), \(|\psi(t)|=1\) and \(\int_0^1 \psi(t)dt=1\) and denoted \(\psi_{n,k}(t)=\psi(2^nt-k)\) for \(n=1,2,\cdots\) and \(k=0,1,\cdots, 2^n-1\). The paper presents some conditions on an Orlicz function \(\phi\) satisfying \(\lim_{t\to \infty} \frac{\phi(t)}{t}=0\) for a subsystem of \((\psi_{n,k})_{n,k}\) to become a representation system of \(E_\phi\), that is the closure of the bounded stepwise functions in the Orlicz space \(L_\phi\). Similar results were first proved by the author for the Haar system (see [V. I. Filippov, Math. Notes 51, No. 6, 1 (1992; Zbl 0798.42014); translation from Mat. Zametki 51, No. 6, 97–106 (1992)]).

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42C30 Completeness of sets of functions in nontrigonometric harmonic analysis

Citations:

Zbl 0798.42014
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Full Text: DOI

References:

[1] Musielak, J. Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034 (Springer-Verlag, Berlin, 1983). · Zbl 0557.46020 · doi:10.1007/BFb0072210
[2] Filippov, V. I. “Subsystems of the Haar System in the Spaces <Emphasis Type=”Italic“>Eϕ With xxx xxx = 0”, Math. Notes 51, No. 5-6, 593-599 (1992). · Zbl 0798.42014 · doi:10.1007/BF01263305
[3] Filippov, V. I. “Linear Continuous Functionals and Representation of Functions by Series in the Spaces <Emphasis Type=”Italic“>Eϕ”, Anal.Math. 27 (4), 239-260 (2001). · Zbl 1003.46020 · doi:10.1023/A:1014316127309
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[9] Ul’yanov, P. L. “Representation of Functions by Series, and the Classes ϕ(<Emphasis Type=”Italic“>L)”, Russ. Math. Surv. 27, No. 2, 1-54 (1972). · Zbl 0274.46027 · doi:10.1070/RM1972v027n02ABEH001370
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