78,040 research outputs found
Generalized Fourier-Mukai Transforms
The paper sets out a generalized framework for Fourier-Mukai transforms and
illustrates their use via vector bundle transforms. A Fourier-Mukai transform
is, roughly, an isomorphism of derived categories of (sheaves) on smooth
varieties X and Y. We show that these can only exist if the first Chern class
of the varieties vanishes and, in the case of vector bundle transforms, will
exist if and only if there is a bi-universal bundle on XxY which is "strongly
simple" in a suitable sense. Some applications are given to abelian varieties
extending the work of Mukai.Comment: 13 pages, AMSLaTeX 1.
Bessel Integrals and Fundamental Solutions for a Generalized Tricomi Operator
Partial Fourier transforms are used to find explicit formulas for two
remarkable fundamental solutions for a generalized Tricomi operator. These
fundamental solutions reflect clearly the mixed type of the operator. In order
to prove these results, we establish explicit formulas for Fourier transforms
of some type of Bessel functions
A generalized Fourier inversion theorem
In this work we define operator-valued Fourier transforms for suitable
integrable elements with respect to the Plancherel weight of a (not necessarily
Abelian) locally compact group. Our main result is a generalized version of the
Fourier inversion Theorem for strictly-unconditionally integrable Fourier
transforms. Our results generalize and improve those previously obtained by Ruy
Exel in the case of Abelian groups.Comment: 15 pages; some typos correcte
Efficient Quantum Transforms
Quantum mechanics requires the operation of quantum computers to be unitary,
and thus makes it important to have general techniques for developing fast
quantum algorithms for computing unitary transforms. A quantum routine for
computing a generalized Kronecker product is given. Applications include
re-development of the networks for computing the Walsh-Hadamard and the quantum
Fourier transform. New networks for two wavelet transforms are given. Quantum
computation of Fourier transforms for non-Abelian groups is defined. A slightly
relaxed definition is shown to simplify the analysis and the networks that
computes the transforms. Efficient networks for computing such transforms for a
class of metacyclic groups are introduced. A novel network for computing a
Fourier transform for a group used in quantum error-correction is also given.Comment: 30 pages, LaTeX2e, 7 figures include
Fourier transforms of hypercomplex signals
An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal. Subsequently, attention is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained
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