Convergence of Sequences of Convolution Operators
نویسندگان
چکیده
When (φn) is a sequence of positive functions in L1(R) such that the convolutions φn ? f converge in L1-norm to f for all f ∈ L1(R), then these convolutions may or may not converge almost everywhere too. There is always a subsequence (φnk ) such that at least φnk ? f converges almost everywhere to f for all f ∈ Lp(R), 1 < p < ∞. A special case of this is when the functions φn are the dilations DLnφ of a fixed function φ ∈ L1(R) with lim n→∞ Ln = ∞. In this case, the choice of Lnk ) sufficient for the almost everywhere convergence of Lnk φ ? f) for all f ∈ L2(R) cannot be made to be independent of the function φ.
منابع مشابه
On convergence of sample and population Hilbertian functional principal components
In this article we consider the sequences of sample and population covariance operators for a sequence of arrays of Hilbertian random elements. Then under the assumptions that sequences of the covariance operators norm are uniformly bounded and the sequences of the principal component scores are uniformly sumable, we prove that the convergence of the sequences of covariance operators would impl...
متن کاملApproximation of distributions by convolutions in the Hausdorff metric
The paper deals with finding criteria for the Hausdorff convergence of sequences of convolution operators on quasi-Banach spaces of periodic realvalued distributions (generalized functions). In particular, the criteria for convergence on the Hardy classes, on the class of regular Borel measures, and on the class of pseudomeasures are found. These assertions are special cases of the general resu...
متن کاملRates of approximation for general sampling-type operators in the setting of filter convergence
Keywords: Filter Modular space (Modular) filter convergence Rate of convergence Sampling operator Integral operator Urysohn operator Mellin convolution operator a b s t r a c t We investigate the order of approximation of a real-valued function f by means of suitable families of sampling type operators, which include both discrete and integral ones. We give a unified approach, by means of which...
متن کاملSubordination and Superordination Properties for Convolution Operator
In present paper a certain convolution operator of analytic functions is defined. Moreover, subordination and superordination- preserving properties for a class of analytic operators defined on the space of normalized analytic functions in the open unit disk is obtained. We also apply this to obtain sandwich results and generalizations of some known results.
متن کامل