9,303 research outputs found
Fock factorizations, and decompositions of the spaces over general Levy processes
We explicitly construct and study an isometry between the spaces of square
integrable functionals of an arbitrary Levy process and a vector-valued
Gaussian white noise. In particular, we obtain explicit formulas for this
isometry at the level of multiplicative functionals and at the level of
orthogonal decompositions, as well as find its kernel. We consider in detail
the central special case: the isometry between the spaces over a Poisson
process and the corresponding white noise. The key role in our considerations
is played by the notion of measure and Hilbert factorizations and related
notions of multiplicative and additive functionals and logarithm. The obtained
results allow us to introduce a canonical Fock structure (an analogue of the
Wiener--Ito decomposition) in the space over an arbitrary Levy process.
An application to the representation theory of current groups is considered. An
example of a non-Fock factorization is given.Comment: 35 pages; LaTeX; to appear in Russian Math. Survey
A category of kernels for equivariant factorizations, II: further implications
We leverage the results of the prequel in combination with a theorem of D.
Orlov to yield some results in Hodge theory of derived categories of
factorizations and derived categories of coherent sheaves on varieties. In
particular, we provide a conjectural geometric framework to further understand
M. Kontsevich's Homological Mirror Symmetry conjecture. We obtain new cases of
a conjecture of Orlov concerning the Rouquier dimension of the bounded derived
category of coherent sheaves on a smooth variety. Further, we introduce actions
of -graded commutative rings on triangulated categories and their associated
Noether-Lefschetz spectra as a new invariant of triangulated categories. They
are intended to encode information about algebraic classes in the cohomology of
an algebraic variety. We provide some examples to motivate the connection.Comment: v2: Updated references and addresses. Cleaved off a part. 54 pages.
v1: Expanded version of the latter half of arXiv:1105.3177. 92 pages.
Comments very welcome
Fej\'er-Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle
We give a complete characterization of the positive trigonometric polynomials
Q(\theta,\phi) on the bi-circle, which can be factored as
Q(\theta,\phi)=|p(e^{i\theta},e^{i\phi})|^2 where p(z,w) is a polynomial
nonzero for |z|=1 and |w|\leq 1. The conditions are in terms of recurrence
coefficients associated with the polynomials in lexicographical and reverse
lexicographical ordering orthogonal with respect to the weight
1/(4\pi^2Q(\theta,\phi)) on the bi-circle. We use this result to describe how
specific factorizations of weights on the bi-circle can be translated into
identities relating the recurrence coefficients for the corresponding
polynomials and vice versa. In particular, we characterize the Borel measures
on the bi-circle for which the coefficients multiplying the reverse polynomials
associated with the two operators: multiplication by z in lexicographical
ordering and multiplication by w in reverse lexicographical ordering vanish
after a particular point. This can be considered as a spectral type result
analogous to the characterization of the Bernstein-Szeg\H{o} measures on the
unit circle
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