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A remark on extrapolation of rearrangement operators on dyadic $H^{s}$, 0 < s ≤ 1

100%

Geiss S. , Müller P. , Pillwein V.

Studia Mathematica

2005

tom 171

nr 2

196-205

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 $T_{s₀}$ is bounded on $H^{s₀}$, then for all 0 < s < 2 the operator $T_{s}$ is bounded on $H^{s}$.

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

88%

Georgieva I. , Hofreither C. , Koutschan C. , Pillwein V. , Thanatipanonda T.

Open Mathematics

2013

tom 11

nr 4

609-620

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.