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A moment sequence in the q-world

100%
Kula A.
Banach Center Publications
|
2007
|
tom 78
|
nr 1
201-210
EN
The aim of the paper is to present some initial results about a possible generalization of moment sequences to a so-called q-calculus. A characterization of such a q-analogue in terms of appropriate positivity conditions is also investigated. Using the result due to Maserick and Szafraniec, we adapt a classical description of Hausdorff moment sequences in terms of positive definiteness and complete monotonicity to the q-situation. This makes a link between q-positive definiteness and q-complete monotonicity.
2
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A q-analogue of complete monotonicity

100%
Kula A.
Colloquium Mathematicum
|
2008
|
tom 111
|
nr 2
169-181
EN
The aim of this paper is to give a q-analogue for complete monotonicity. We apply a classical characterization of Hausdorff moment sequences in terms of positive definiteness and complete monotonicity, adapted to the q-situation. The method due to Maserick and Szafraniec that does not need moments turns out to be useful. A definition of a q-moment sequence appears as a by-product.
3
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A limit theorem for the q-convolution

100%
Kula A.
Banach Center Publications
|
2011
|
tom 96
|
nr 1
245-255
EN
The q-convolution is a measure-preserving transformation which originates from non-commutative probability, but can also be treated as a one-parameter deformation of the classical convolution. We show that its commutative aspect is further certified by the fact that the q-convolution satisfies all of the conditions of the generalized convolution (in the sense of Urbanik). The last condition of Urbanik's definition, the law of large numbers, is the crucial part to be proved and the non-commutative probability techniques are used.
4
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Convolutions related to q-deformed commutativity

100%
Kula A.
Banach Center Publications
|
2010
|
tom 89
|
nr 1
189-200
EN
Two important examples of q-deformed commutativity relations are: aa* - qa*a = 1, studied in particular by M. Bożejko and R. Speicher, and ab = qba, studied by T. H. Koornwinder and S. Majid. The second case includes the q-normality of operators, defined by S. Ôta (aa* = qa*a). These two frameworks give rise to different convolutions. In particular, in the second scheme, G. Carnovale and T. H. Koornwinder studied their q-convolution. In the present paper we consider another convolution of measures based on the so-called (p,q)-commutativity, a generalization of ab = qba. We investigate and compare properties of both convolutions (associativity, commutativity and positivity) and corresponding Fourier transforms.
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