norm convergence
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2021 ◽  
pp. 2150037
Author(s):  
Inyoung Park ◽  
Jian Zhao ◽  
Kehe Zhu

For positive [Formula: see text] and real [Formula: see text] let [Formula: see text] denote the weighted Bergman spaces of the unit ball [Formula: see text] introduced in [R. Zhao and K. Zhu, Theory of Bergman Spaces on the Unit Ball in [Formula: see text], Mémoires de la Société Mathématique de France, Vol. 115 (2008)]. It is well known that, at least in the case [Formula: see text], all functions in [Formula: see text] can be approximated in norm by their Taylor polynomials if and only if [Formula: see text]. In this paper we show that, for [Formula: see text] with [Formula: see text], we always have [Formula: see text] as [Formula: see text], where [Formula: see text] and [Formula: see text] is the [Formula: see text]th Taylor polynomial of [Formula: see text]. We also show that for every [Formula: see text] in the Hardy space [Formula: see text], [Formula: see text], we always have [Formula: see text] as [Formula: see text], where [Formula: see text]. This generalizes and improves a result in [J. McNeal and J. Xiong, Norm convergence of partial sums of [Formula: see text] functions, Internat. J. Math. 29 (2018) 1850065, 10 pp.].


2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Carlo Bardaro ◽  
Ilaria Mantellini

<p style='text-indent:20px;'>In this paper we study boundedness properties of certain semi-discrete sampling series in Mellin–Lebesgue spaces. Also we examine some examples which illustrate the theory developed. These results pave the way to the norm-convergence of these operators</p>


Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1005
Author(s):  
Antonio Boccuto ◽  
Bipan Hazarika ◽  
Hemanta Kalita

We introduce Kuelbs–Steadman-type spaces ( K S p spaces) for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate the main properties and embeddings of L q -type spaces into K S p spaces, considering both the norm associated with the norm convergence of the involved integrals and that related to the weak convergence of the integrals.


Author(s):  
Valentin Zagrebnov

The paper improves approximation theory based on the Trotter–Kato product formulae. For self-adjoint \(C_0\)-semigroups we develop a lifting of the strongly convergent Chernoff approximation (or product) formula to convergence in the operator-norm topology. This allows to obtain optimal estimate for the rate of operator-norm convergence of Trotter–Kato product formulae for Kato functions from the class \(K_2\).


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