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Session 4. Banach Spaces and Operator Theory with Applications
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Stable approximation, real interpolation and applications |
Eric Setterqvist, Department of Mathematics, Linköping University, Sweden
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The talk is based on joint work with Natan Kruglyak
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If we try to approximate the function \(f\) from \(L^{p}\) by the
ball of \(L^{\infty}\), we discover that the element of best
approximation does not depend on \(p<\infty\). This simple fact lies
at the core of the Marcinkiewicz interpolation theorem.
In this talk we present domains, different than the ball of
\(L^{\infty}\), with element of best approximation that is invariant
with respect to the \(L^{p}\)-norm. It turns out that domains with
this property are related to the classical
Hardy-Littlewood-Pólya majorization and \(K\)-monotonicity of
the couple \((L^{1},L^{\infty})\).
We will show some recent applications of this type of domains, and
their elements of best approximation, to smooth approximation of
Wiener process and buffer management problems in communication
theory. Algorithms for construction of the element of best
approximation will also be considered.
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