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Harmonic interpolation based on Radon projections along the sides of regular polygons

Given information about a harmonic function in two variables, consisting of a finite number of values of its Radon projections, i.e., integrals along some chords of the unit circle, we study the problem of interpolating these data by a harmonic polynomial. With the help of symbolic summation techniques we show that this interpolation problem has a unique solution in the case when the chords form a regular polygon. Numerical experiments for this and more general cases are presented.

A remark on extrapolation of rearrangement operators on dyadic H s , 0 < s ≤ 1

Stefan GeissPaul F. X. MüllerVeronika Pillwein — 2005

Studia Mathematica

For an injective map τ acting on the dyadic subintervals of the unit interval [0,1) we define the rearrangement operator T s , 0 < s < 2, to be the linear extension of the map ( h I ) / ( | I | 1 / s ) ( h τ ( I ) ) ( | τ ( I ) | 1 / s ) , where h I denotes the L -normalized Haar function supported on the dyadic interval I. We prove the following extrapolation result: If there exists at least one 0 < s₀ < 2 such that T s is bounded on H s , then for all 0 < s < 2 the operator T s is bounded on H s .

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