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Published in:
2018 | OriginalPaper | Chapter
More on the density of analytic polynomials in abstract Hardy spaces
Authors : Alexei Karlovich, Eugene Shargorodsky
Published in: The Diversity and Beauty of Applied Operator Theory
Publisher: Springer International Publishing
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Let $$\left\{F_n\right\}$$ be the sequence of the Fejér kernels on the unit circle $$\mathbb{T}$$ .The First author recently proved that if X is a separable Banach function space on $$\mathbb{T}$$ such that the Hardy–Littlewood maximal operator M is bounded on its associate space $$X^\prime$$ , then $$\| f * F_n - f \|_X \to 0$$ for every $$f \in X\; \mathrm{as}\; n \to \infty$$ . This implies that the set of analytic polynomials $$\mathcal{P}_A$$ is dense in the abstract Hardy space $$H \left[X \right]$$ built upon a separable Banach function space X such that M is bounded on $$X^\prime$$ . In this note we show that there exists a separable weighted L1 space X such that the sequence $$f * F_n$$ does not always converge to $$f \in X$$ in the norm of X. On the other hand, we prove that the set $$\mathcal{P}_A$$ is dense in $$H \left[X \right]$$ under the assumption that X is merely separable.