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Hilbert space factorization and Fourier type of operators

100%
Hinrichs A.
Studia Mathematica
|
2001
|
tom 145
|
nr 3
199-212
EN
It is an open question whether every Fourier type 2 operator factors through a Hilbert space. We show that at least the natural gradations of Fourier type 2 norms and Hilbert space factorization norms are not uniformly equivalent. A corresponding result is also obtained for a number of other sequences of ideal norms instead of the Fourier type 2 gradation including the Walsh function analogue of Fourier type. Our main tools are ideal norms and random matrices.
2
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On the type constants with respect to systems of characters of a compact abelian group

100%
Hinrichs A.
Studia Mathematica
|
1996
|
tom 118
|
nr 3
231-243
EN
We prove that there exists an absolute constant c such that for any positive integer n and any system Φ of $2^n$ characters of a compact abelian group, $2^{-n/2} t_Φ(T) ≤ c n^{-1/2} t_n(T)$, where T is an arbitrary operator between Banach spaces, $t_Φ(T)$ is the type norm of T with respect to Φ and $t_n(T)$ is the usual Rademacher type-2 norm computed with n vectors. For the system of the first $2^n$ Walsh functions this is even true with c=1. This result combined with known properties of such type norms provides easy access to quantitative versions of the fact that a nontrivial type of a Banach space implies finite cotype and nontrivial type with respect to the Walsh system or the trigonometric system.
3
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On the randomized complexity of Banach space valued integration

63%
Heinrich S., Hinrichs A.
Studia Mathematica
|
2014
|
tom 223
|
nr 3
205-215
EN
We study the complexity of Banach space valued integration in the randomized setting. We are concerned with r times continuously differentiable functions on the d-dimensional unit cube Q, with values in a Banach space X, and investigate the relation of the optimal convergence rate to the geometry of X. It turns out that the nth minimal errors are bounded by $cn^{-r/d - 1 + 1/p}$ if and only if X is of equal norm type p.
4
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On the non-equivalence of rearranged Walsh and trigonometric systems in $L_{p}$

63%
Hinrichs A., Wenzel J.
Studia Mathematica
|
2003
|
tom 159
|
nr 3
435-451
EN
We consider the question of whether the trigonometric system can be equivalent to some rearrangement of the Walsh system in $L_{p}$ for some p ≠ 2. We show that this question is closely related to a combinatorial problem. This enables us to prove non-equivalence for a number of rearrangements. Previously this was known for the Walsh-Paley order only.
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