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Weighted Approximation by a Certain Family of Summation Integral-Type Operators
AbstractIn this study, we investigate the weighted approximation properties of a general sequence of summation integral-type operators introduced by Srivastava and Gupta [1]. We estimate the rate of convergence of these operators for functions of polynomial growth in terms of weighted modulus of continuity on the interval [0, ∞)
Compactness Properties of Weighted Summation Operators on Trees
We investigate compactness properties of weighted summation operators
as mapping from into for some . Those operators are defined by where is a
tree with induced partial order (or ) for . Here and are given weights on . We introduce a metric
on such that compactness properties of imply two--sided
estimates for , the (dyadic) entropy numbers of
. The results are applied for concrete trees as e.g.
moderate increasing, biased or binary trees and for weights with
decreasing either polynomially or exponentially. We also
give some probabilistic applications for Gaussian summation schemes on trees
Compactness properties of weighted summation operators on trees - the critical case
The aim of this paper is to provide upper bounds for the entropy numbers of
summation operators on trees in a critical case. In a recent paper [10] we
elaborated a framework of weighted summation operators on general trees where
we related the entropy of the operator with those of the underlying tree
equipped with an appropriate metric. However, the results were left incomplete
in a critical case of the entropy behavior, because this case requires much
more involved techniques. In the present article we fill the gap left open in
[10]. To this end we develop a method, working in the context of general trees
and general weighted summation operators, which was recently proposed in [9]
for a particular critical operator on the binary tree. Those problems appeared
in natural way during the study of compactness properties of certain Volterra
integral operators in a critical case
Two-weight norm inequalities for potential type and maximal operators in a metric space
We characterize two-weight norm inequalities for potential type integral
operators in terms of Sawyer-type testing conditions. Our result is stated in a
space of homogeneous type with no additional geometric assumptions, such as
group structure or non-empty annulus property, which appeared in earlier works
on the subject. One of the new ingredients in the proof is the use of a finite
collection of adjacent dyadic systems recently constructed by the author and T.
Hyt\"onen. We further extend the previous Euclidean characterization of
two-weight norm inequalities for fractional maximal functions into spaces of
homogeneous type.Comment: 33 pages, v8 (some typos corrected; clarified the relationship
between the different constants present in the several steps of the proof of
the main result; Lemma 6.18 modified; examples of spaces and operators
included; fixed some technical details; Definition 2.14 and Lemma 2.15
modified; Lemma 6.17 corrected; measures allowed with point masses; some
imprecise arguments clarified
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