On the orthogonal democratic systems in the $L^p$ spaces

K. S. Kazarian; A. San Antolin

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arXiv · 1812.11905

On the orthogonal democratic systems in the $L^p$ spaces

K. S. Kazarian·A. San Antolin

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Abstract

The concept of bidemocratic pair for a Banach space was introduced in \cite{KS:18}. We construct a family of orthonormal systems $\mathfrak{F}_{l},$ $l\in (0,\infty)$ of functions defined on $[-1,1]$ such that the pair $(\mathfrak{F}_{l},\mathfrak{F}_{l})$ is bidemocratic for $L^{p}[-1,1]$ and for $L^{p'}[-1,1]$ if $l\in (0, \frac{p}{2(p-2)}]$, where $p>2$ and $p'= \frac{p}{p-1}$. The system $\mathfrak{F}_{l}$ is not democratic for $L^{p'}[-1,1]$ when $l\in (\frac{p}{2(p-2)}, \frac{p}{p-2}). $ When $l> \frac{p}{2(p-2)}$ the pair $(\mathfrak{F}_{l},\mathfrak{F}_{l})$ is not bidemocratic neither for $L^{p}[-1,1]$ nor for $L^{p'}[-1,1]$.

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BibTeX RIS

K. S. Kazarian, A. San Antolin. 2018-12-31. On the orthogonal democratic systems in the $L^p$ spaces. https://arxiv.org/abs/1812.11905

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