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Register a new userPseudo-hyperbolic distance and n-best rational approximation in H^2 space
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Through reducing the problem to rational orthogonal system (Takenaka-Malmquist system), this note gives a proof for existence of n-best rational approximation to functions in the Hardy H2(D) space by using pseudohyperbolic distance.
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In this paper we give exact values of the best $n$-term approximation widths of diagonal operators between $ell_p(mathbb{N})$ and $ell_q(mathbb{N})$ with $0<p,qleq infty$. The result will be applied to obtain the asymptotic constants of best $n$-term approximation widths of embeddings of function spaces with mixed smoothness by trigonometric system.
In this paper we consider the compact plane sets known as Swiss cheese sets, which are a useful source of examples in the theory of uniform algebras and rational approximation. We introduce a notion of allocation map connected with Swiss cheeses, and we develop the theory of such maps. We use this theory to modify examples previously constructed in the literature to solve various problems, in order to obtain examples of Swiss cheese sets homeomorphic to the Sierpinski carpet which solve the same problems. In particular, this allows us to give examples of essential, regular uniform algebras on locally connected, compact plane sets. Our techniques also allow us to avoid certain technical difficulties in the literature.
The proximinality of certain subspaces of spaces of bounded affine functions is proved. The results presented here are some line
We consider composition operators $mathscr{C}_varphi$ on the Hardy space of Dirichlet series $mathscr{H}^2$, generated by Dirichlet series symbols $varphi$. We prove two different subordination principles for such operators. One concerns affine symbols only, and is based on an arithmetical condition on the coefficients of $varphi$. The other concerns general symbols, and is based on a geometrical condition on the boundary values of $varphi$. Both principles are strict, in the sense that they characterize the composition operators of maximal norm generated by symbols having given mapping properties. In particular, we generalize a result of J. H. Shapiro on the norm of composition operators on the classical Hardy space of the unit disc. Based on our techniques, we also improve the recently established upper and lower norm bounds in the special case that $varphi(s) = c + r2^{-s}$. A number of other examples are given.
Let $mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators $mathscr{C}_varphi$ on $mathscr{H}^2$ which are generated by symbols of the form $varphi(s) = c_0s + sum_{ngeq1} c_n n^{-s}$, in the case that $c_0 geq 1$. If only a subset $mathbb{P}$ of prime numbers features in the Dirichlet series of $varphi$, then the operator $mathscr{C}_varphi$ admits an associated orthogonal decomposition. Under sparseness assumptions on $mathbb{P}$ we use this to asymptotically estimate the approximation numbers of $mathscr{C}_varphi$. Furthermore, in the case that $varphi$ is supported on a single prime number, we affirmatively settle the problem of describing the compactness of $mathscr{C}_varphi$ in terms of the ordinary Nevanlinna counting function. We give detailed applications of our results to affine symbols and to angle maps.
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