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1
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Unconditionality of orthogonal spline systems in $L^{p}$

100%
Passenbrunner M.
Studia Mathematica
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2014
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tom 222
|
nr 1
51-86
EN
We prove that given any natural number k and any dense point sequence (tₙ), the corresponding orthonormal spline system is an unconditional basis in reflexive $L^{p}$.
2
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The Lebesgue constant for the periodic Franklin system

100%
Passenbrunner M.
Studia Mathematica
|
2011
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tom 205
|
nr 3
251-279
EN
We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots $t_{j}$ = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let $V_{n,ν}$ be the space of piecewise linear continuous functions on the torus with knots ${t_{j}: 0 ≤ j ≤ N-1}$. Finally, let $P_{n,ν}$ be the orthogonal projection operator from L²([0,1)) onto $V_{n,ν}$. The main result is $lim_{n→∞,ν=1} ||P_{n,ν}: L^{∞} → L^{∞}|| = sup_{n∈ℕ,0≤ν≤n} ||P_{n,ν}: L^{∞} → L^{∞}|| = 2 + (33-18√3)/13$. This shows in particular that the Lebesgue constant of the classical Franklin orthonormal system on the torus is 2 + (33-18√3)/13.
3
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Adaptive Deterministic Dyadic Grids on Spaces of Homogeneous Type

64%
Lechner R., Passenbrunner M.
Bulletin of the Polish Academy of Sciences. Mathematics
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2014
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tom 62
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nr 2
139-159
EN
In the context of spaces of homogeneous type, we develop a method to deterministically construct dyadic grids, specifically adapted to a given combinatorial situation. This method is used to estimate vector-valued operators rearranging martingale difference sequences such as the Haar system.
4
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On the distribution of random variables corresponding to Musielak-Orlicz norms

51%
Alonso-Gutiérrez D., Christensen S., Passenbrunner M., Prochno J.
Studia Mathematica
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2013
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tom 219
|
nr 3
269-287
EN
Given a normalized Orlicz function M we provide an easy formula for a distribution such that, if X is a random variable distributed accordingly and X₁,...,Xₙ are independent copies of X, then $1/C_{p} ||x||_M ≤ 𝔼 ||(x_{i}X_{i})ⁿ_{i=1}||_{p} ≤ C_{p}||x||_M$, where $C_{p}$ is a positive constant depending only on p. In case p = 2 we need the function t ↦ tM'(t) - M(t) to be 2-concave and as an application immediately obtain an embedding of the corresponding Orlicz spaces into L₁[0,1]. We also provide a general result replacing the $ℓ_{p}$-norm by an arbitrary N-norm. This complements some deep results obtained by Gordon, Litvak, Schütt, and Werner [Ann. Prob. 30 (2002)]. We also prove, in the spirit of that paper, a result which is of a simpler form and easier to apply. All results are true in the more general setting of Musielak-Orlicz spaces.
5
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Unconditionality of orthogonal spline systems in H¹

51%
Gevorkyan G., Kamont A., Keryan K., Passenbrunner M.
Studia Mathematica
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2015
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tom 226
|
nr 2
123-154
EN
We give a simple geometric characterization of knot sequences for which the corresponding orthonormal spline system of arbitrary order k is an unconditional basis in the atomic Hardy space H¹[0,1].
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