Multiplicative systems on zero-dimensional locally compact Abelian groups

Author: Nika Areshidze
Keywords: locally compact Abelian group, Haar measure, Q_p field, Vilenkin group, multiplicative systems, generalized derivative.
Annotation:

In general, the derivative of a function on locally compact Abelian group can be defined in many ways. In 1973, Butzer and Wagner (\cite{butzer1}) introduced and investigated the strong derivative $D^{(r)}(f)\in L^{p}(G),\, r\in\mathbb{N},$ of $f\in L^p(G)$ function (see def. \ref{diff1}). They proved that $D_{X}^{(1)}(\psi_k)=k\psi_k,\, k\in\mathbb{N}_{+}.$ Onneweer (\cite{onneweer1}) later generalized the Butzer-Wagner derivative for Vilenkin groups (see. def. \ref{diff2}). Zelin proposed a new definition of $P$ - derivative and integral, which is simpler than one of Butzer-Wagner or Onneweer. In this master's thesis the existence of $f^{[r]}$ in $L^{p(\cdot)}(G_m)$ is proved. Additionally, analogs of Jackson's and Bernstein's inequalities with respect to Vilenkin system in variable Lebesgue spaces are established. We prove the equivalence of the norms of the operator $Q(f)$ and $f$ in $L^{p(\cdot)}(G_m)$ which is then used to estimate the best approximation of $f^{[r]}$ by the best approximation of $f$. We also define $K$-functional and we estimate best approximation in variable Lebesgue spaces in terms of the $K$.

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პრეზენტაცია [ka]
სამაგისტრო ნაშრომი [ka]