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Operators of Hardy type. (English) Zbl 1127.47032

Among William Desmond Evans’s many outstanding contributions to analysis, his work on operators of Hardy type plays a prominent role. Such an operator \(T\) is of the form \(Tf(x)=\upsilon(x)\int_a^x u(t)f(t)\,dt\), where \(u\) and \(\upsilon\) are given functions on an interval \((a,b)\) that satisfy certain integrability conditions, and \(T\) is viewed as a map between Lebesgue spaces.
In this paper, the authors describe some of results concerning entropy and approximation numbers of operators of Hardy type in which Des Evans has played a major part, in the context of operators on intervals and on trees, and also mention some quite recent developments involving the Bernstein widths of \(T\) as well as its Kolmogorov widths and approximation numbers.

MSC:

47B38 Linear operators on function spaces (general)
47G10 Integral operators
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Biographic References:

Evans, William Desmond
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References:

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