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1
Content available remote Sharp inequalities for the Haar system and martingale transforms
Osękowski A.
Probability and Mathematical Statistics
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2016
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Vol. 36, Fasc. 1
147--164
EN
A classical result of Paley and Marcinkiewicz asserts that the Haar system on [0, 1] forms an unconditional basis in Lp provided 1 < p < ∞. The purpose of the paper is to study related weak-type inequalities, which can be regarded as a version of this property for p = 1. Probabilistic counterparts, leading to some sharp estimates for martingale transforms, are presented.
2
Content available remote Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2014
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Vol. 62, no 2
187--196
EN
Let (hk)k≥0 be the Haar system on [0,1]. We show that for any vectors ak from a separable Hilbert space H and any εk∈[−1,1], k=0,1,2,…, we have the sharp inequality ...[formula], where W([0,1]) is the weak-L∞ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ∥Y∥W(Ω)≤2∥X∥L∞(Ω), where X and Y stand for H-valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.
3
Content available Convolution algebras for topological groupoids with locally compact fibres
Buneci M. R.
Opuscula Mathematica
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2011
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Vol. 31, no. 2
159-172
EN
The aim of this paper is to introduce various convolution algebras associated with a topological groupoid with locally compact fibres. Instead of working with continuous functions on G, we consider functions having a uniformly continuity property on fibres. We assume that the groupoid is endowed with a system of measures (supported on its fibres) subject to the "left invariance" condition in the groupoid sense.
4
A Levy-Ciesielski expansion for quantum Brownian motion and the construction of quantum Brownian bridges
Applebaum D.
Journal of Applied Analysis
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2007
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Vol. 13, nr 2
275-290
EN
We introduce "probabilistic" and "stochastic Hilbertian structures". These seem to be a suitable context for developing a theory of "quantum Gaussian processes". The Schauder system is utilised to give a Levy-Ciesielski representation of quantum (bosonic) Brownian motion as operators in Fock space over a space of square summable sequences. Similar results hold for non-Fock, fermion, free and monotone Brownian motions. Quantum Brownian bridges are defined and a number of representations of these are given.
5
Content available remote Inequivalence of Wavelet Systems in L1 (Rd) and BV(Rd)
Bechler P.
Bulletin of the Polish Academy of Sciences. Mathematics
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2005
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Vol. 53, no 1
25-37
EN
Theorems stating sufficient conditions for the inequivalence of the d-variate Haar wavelet system and another wavelet system in the spaces L1 (Rd) and BV(Rd) are proved. These results are used to show that the Strömberg wavelet system and the system of continuous Daubechies wavelets with minimal supports are not equivalent to the Haar system in these spaces. A theorem stating that some systems of smooth Daubechies wavelets are not equivalent to the Haar system in L1 (Rd) is also shown.
6
Content available remote Refinement of some necessary conditions
Leindler L.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2004
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Vol. 44, nr spec.
145-154
EN
The traditional monotonicity requirement on the coefficients of orthogonal series is weakened to almost, or locally almost monotonicity assumptions in connection with the necessary conditions of convergence and absolute Cesaro summability of Haar series. Similar generalization is established regarding Fourier series and general orthogonal series.
7
Content available remote Square functions for wavelet type systems
Gevorkjan G., Wolnik B.
Bulletin of the Polish Academy of Sciences. Mathematics
|
1998
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Vol. 46, no 2
155-171
EN
We show the equivalence of the L[sub p] (0 < p [a is less than or equal to] 2) (quasi)-norms of square functions for the systems {2 [...], where f satisfies some decay condition. This implies the boundedness of the shift operator on the wavelet type unconditional basis on L[sub p], 1 < p < [infinity]. We prove also that such operator is unbounded on L[sub 1].
8
Content available remote Haar system as a Schauder basis in Lp(x,y),q(y) (I)
Pietrzyk B.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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1998
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[Z] 38
109-114
EN
In this paper we define Lp(x,y),q(y) (I). We investigate some condition of this space.
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