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A Levy-Ciesielski expansion for quantum Brownian motion and the construction of quantum Brownian bridges

Applebaum D.

Journal of Applied Analysis

2007

Vol. 13, nr 2

275-290

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.

Inequivalence of Wavelet Systems in L1 (Rd) and BV(Rd)

Bechler P.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 1

25-37

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.

Square functions for wavelet type systems

Gevorkjan G., Wolnik B.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 2

155-171

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].