We consider the function systems of translates and dilates of one function in the spaces $L^p$ and $E_{\varphi}$. We study minimal conditions which the function sistem of dyadic translates and dilates of one fixed function forms a representation system in $L^p$ and $E_{\varphi}$, i.e., that any function $f \in L^p$ or $f\in E_{\varphi}$ can be at least one $L^p$ (or $E_{\varphi}$) convergent series with respect to this system. We give a criterion of the existence of linear continuous nontrivial functionals in the space $E_{\varphi}$ with a continuous measure. We also give a criterion that $\{x_n\}$ be a representation system in a space $E_{\varphi}$ with a special conditions on $\varphi$. We consider a Haar system pertubed in the sense of the $L^1(0,1)$-metric.
Biography
Graduated from Faculty of Mathematics and Mechanics of N. G. Chernyshevski Saratov State University (SSU) in 1982 (department of mathematical analysis). Ph. D. thesis was defended in 1987. A list of my works contains more than 30 titles.
In 1989 I was awarded the prize in the competition of young scientists of SSU.
Main publications:
Filippov V. I. On the completeness and other properties of some function systems in $L^p, 0<p<\infinity$ // J. Approx. Theory, 1998, 94, 42–53.
Filippov V. I. Linear continuous functionals and representation of functions by series in the spaces $E_{\varphi}$ // Anal. Math., 2001, 27(4), 239–260.
V. I. Filippov, “Integer expansion in systems of translates and dilates of a single function”, Izv. RAN. Ser. Mat., 84:4 (2020), 187–197; Izv. Math., 84:4 (2020), 796–806
V. I. Filippov, “Series of Fourier type with integer coefficients by systems of dilates and translates of one function in $L_p$, $p\geq 1$”, Izv. Vyssh. Uchebn. Zaved. Mat., 2019, no. 6, 58–64; Russian Math. (Iz. VUZ), 63:6 (2019), 51–57
2018
3.
V. I. Filippov, “On generalization of Haar system and other function systems in spaces $E_{\varphi}$”, Izv. Vyssh. Uchebn. Zaved. Mat., 2018, no. 1, 87–92; Russian Math. (Iz. VUZ), 62:1 (2018), 76–81
V. I. Filippov, “Representation systems obtained using translates and dilates of a single function in multidimensional spaces $E_{\varphi}$”, Izv. RAN. Ser. Mat., 76:6 (2012), 193–206; Izv. Math., 76:6 (2012), 1257–1270
V. I. Filippov, “On the Kolmogorov theorems on Fourier series and conjugate functions”, Izv. Vyssh. Uchebn. Zaved. Mat., 2012, no. 7, 21–34; Russian Math. (Iz. VUZ), 56:7 (2012), 18–29
V. I. Filippov, “Perturbation of the trigonometric system in $L_1(0,\pi)$”, Mat. Zametki, 80:3 (2006), 429–436; Math. Notes, 80:3 (2006), 410–416
2001
8.
V. I. Filippov, “Function systems obtained using translates and dilates of a single function in the paces $E_\varphi$ with $\lim_{t\to\infty}\frac{\varphi(t)}t=0$”, Izv. RAN. Ser. Mat., 65:2 (2001), 187–200; Izv. Math., 65:2 (2001), 389–402
V. I. Filippov, “Strong perturbations of the Haar system in the space $L_1(0,1)$”, Mat. Zametki, 66:4 (1999), 596–602; Math. Notes, 66:4 (1999), 489–494
V. I. Filippov, “Subsystems of the Haar system in spaces $E_{\varphi}$ with $\varliminf_{t\to\infty}\dfrac{\varphi(t)}{t}=0$”, Mat. Zametki, 51:6 (1992), 97–106; Math. Notes, 51:6 (1992), 593–599
V. I. Filippov, “Subsystems of the Faber–Schauder system in function space”, Izv. Vyssh. Uchebn. Zaved. Mat., 1991, no. 2, 78–85; Soviet Math. (Iz. VUZ), 35:2 (1991), 90–97
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