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.
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.
