Abstract
AbstractTruncating the Fourier transform averaged by means of a generalized Hausdorff operator, we approximate functions and the adjoint to that Hausdorff operator of the given function. We find estimates for the rate of approximation in various metrics in terms of the parameter of truncation and the components of the Hausdorff operator. Explicit rates of approximation of functions and comparison with approximate identities are given in the case of continuous functions from the class $\text {Lip }\alpha $ .
Highlights
Introduction e classicalHausdorff operator is defined, by means of a kernel φ, as (.) (Hφ f )(x) =φ(t) f Rt x t dt, and, as is shown first in [ ], such an operator is bounded in L (R) whenever φ ∈ L (R).In the last two decades, various problems related to Hausdorff operators have attracted a lot of attention. e number of publications is growing considerably; to add some of the most notable, we mention [, ]. ere are two survey papers: [ ] and [ ]
Πf (y)eix y d y does not hold for f ∈ L (R); in order to “repair” this, one can consider some transformation of the function f or of its Fourier transform
We will always assume that f ∈ L (R), so thatf is well defined, andf ∈ L∞(R)
Summary
Let us discuss some boundedness properties of the Hausdorff operator in Lebesgue spaces, in order for H∗( f ) (and the Hausdorff operator in ( . )) to be well defined. We will always assume that f ∈ L (R), so thatf is well defined, andf ∈ L∞(R). A sufficient condition for the operator H∗ to be bounded on Lp(R) is ( .). A sufficient condition (and necessary whenever φ ≥ a.e.) for the Hausdorff operator to be bounded on Lp(R) is that φ(t) a(t) p′ dt < ∞,. For further results on boundedness (and Pitt-type inequalities) of Hausdorff operators, we refer the reader to [ ]. Our main results concerning approximation of adjoint Hausdorff operators read as follows. E fact that the adjoint Hausdorff operator of a function is approximated may be unsatisfactory in principle, as one would rather approximate the function itself. Approximating a function instead of its adjoint Hausdorff operator is possible as a consequence of the following observation.
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