María José Muñoz-Bouzo

Projective system approach to the martingale characterization of the absence of arbitrage

Abstract The equivalence between the absence of arbitrage and the existence of an equivalent martingale measure fails when an infinite number of trading dates is considered. By enlarging the set of states of nature and the probability measure through a projective system of perfect measure spaces, we characterize the absence of arbitrage when the time set is countable.

Generalized sampling: from shift-invariant to U-invariant spaces

The aim of this article is to derive a sampling theory in U-invariant subspaces of a separable Hilbert space ℋ where U denotes a unitary operator defined on ℋ. To this end, we use some special dual frames for L2(0, 1), and the fact that any U-invariant subspace with stable generator is the image of L2(0, 1) by means of a bounded invertible operator. The used mathematical technique mimics some previous sampling work for shift-invariant subspaces of L2(ℝ). Thus, sampling frame expansions in U-invariant spaces are obtained. In order to generalize convolution systems and deal with the time-jitter error in this new setting we consider a continuous group of unitary operators which includes the operator U.

Finite Sampling in Multiple Generated

The relevance in a sampling theory of U-invariant subspaces of a Hilbert space H, where U denotes a unitary operator on H, is nowadays a recognized fact. Indeed, shift-invariant subspaces of L2(R) become a particular example; periodic extensions of finite signals also provide a remarkable example. As a consequence, the availability of an abstract U-sampling theory becomes a useful tool to handle these problems. In this paper, we derive a sampling theory for finite dimensional multiple generated U-invariant subspaces of a Hilbert space H. As the involved samples are identified as frame coefficients in a suitable euclidean space, the relevant mathematical technique is that of the finite frame theory. Since finite frames are nothing but spanning sets of vectors, the used technique naturally meets matrix analysis.

U

This paper is concerned with the characterization as frames of some sequences in -invariant spaces of a separable Hilbert space where denotes an unitary operator defined on ; besides, the dual frames having the same form are also found. This general setting includes, in particular, shift-invariant or modulation-invariant subspaces in , where these frames are intimately related to the generalized sampling problem. We also deal with some related perturbation problems. In doing so, we need the unitary operator to belong to a continuous group of unitary operators.

-Invariant Subspaces

The classical Kramer sampling theorem is, in the subject of self-adjoint bound- ary value problems, one of the richest sources to obtain sampling expansions. It has be- come very fruitful in connection with discrete Sturm-Liouville problems. In this paper, a discrete version of the analytic Kramer sampling theorem is proved. Orthogonal polyno- mials arising from indeterminate Hamburger moment problems as well as polynomials of the second kind associated with them provide examples of Kramer analytic kernels.

On Some Sampling-Related Frames in

The use of unitary invariant subspaces of a Hilbert space

U

\mathcal{H}

-Invariant Spaces

is nowadays a recognized fact in the treatment of sampling problems. Indeed, shift-invariant subspaces of

ON THE ANALYTIC FORM OF THE DISCRETE KRAMER SAMPLING THEOREM

L^2(\mathbb{R})

Modeling Sampling in Tensor Products of Unitary Invariant Subspaces

and also periodic extensions of finite signals are remarkable examples where this occurs. As a consequence, the availability of an abstract unitary sampling theory becomes a useful tool to handle these problems. In this paper we derive a sampling theory for tensor products of unitary invariant subspaces. This allows to merge the cases of finitely/infinitely generated unitary invariant subspaces formerly studied in the mathematical literature, it also allows to introduce the several variables case. As the involved samples are identified as frame coefficients in suitable tensor product spaces, the relevant mathematical technique is that of frame theory, involving both, finite/infinite dimensional cases.