9,664 research outputs found
On the diagonalization of the discrete Fourier transform
The discrete Fourier transform (DFT) is an important operator which acts on
the Hilbert space of complex valued functions on the ring Z/NZ. In the case
where N=p is an odd prime number, we exhibit a canonical basis of eigenvectors
for the DFT. The transition matrix from the standard basis to the canonical
basis defines a novel transform which we call the discrete oscillator transform
(DOT for short). Finally, we describe a fast algorithm for computing the
discrete oscillator transform in certain cases.Comment: Accepted for publication in the journal "Applied and Computational
Harmonic Analysis": Appl. Comput. Harmon. Anal. (2009),
doi:10.1016/j.acha.2008.11.003. Key words: Discrete Fourier Transform, Weil
Representation, Canonical Eigenvectors, Oscillator Transform, Fast Oscillator
Transfor
A Note on the Diagonalization of the Discrete Fourier Transform
Following the approach developed by S. Gurevich and R. Hadani, an analytical
formula of the canonical basis of the DFT is given for the case where
is a prime number and (mod 4).Comment: 12 pages, accepted by Applied and Computational Harmonic Analysi
Replica Fourier Transform: Properties and Applications
The Replica Fourier Transform is the generalization of the discrete Fourier
Transform to quantities defined on an ultrametric tree. It finds use in con-
junction of the replica method used to study thermodynamics properties of
disordered systems such as spin glasses. Its definition is presented in a
system- atic and simple form and its use illustrated with some representative
examples. In particular we give a detailed discussion of the diagonalization in
the Replica Fourier Space of the Hessian matrix of the Gaussian fluctuations
about the mean field saddle point of spin glass theory. The general results are
finally discussed for a generic spherical spin glass model, where the Hessian
can be computed analytically
Banach algebras of pseudodifferential operators and their almost diagonalization
We define new symbol classes for pseudodifferntial operators and investigate
their pseudodifferential calculus. The symbol classes are parametrized by
commutative convolution algebras. To every solid convolution algebra over a
lattice we associate a symbol class. Then every operator with such a symbol is
almost diagonal with respect to special wave packets (coherent states or Gabor
frames), and the rate of almost diagonalization is described precisely by the
underlying convolution algebra. Furthermore, the corresponding class of
pseudodifferential operators is a Banach algebra of bounded operators on . If a version of Wiener's lemma holds for the underlying convolution algebra,
then the algebra of pseudodifferential operators is closed under inversion. The
theory contains as a special case the fundamental results about Sj\"ostrand's
class and yields a new proof of a theorem of Beals about the H\"ormander class
of order 0.Comment: 28 page
A few remarks on time-frequency analysis of Gevrey, analytic and ultra-analytic functions
We give a brief survey of recent results concerning almost diagonalization of
pseudodifferential operators via Gabor frames. Moreover, we show new
connections between symbols with Gevrey, analytic or ultra-analityc regularity
and time-frequency analysis of the corresponding pseudodifferential operators.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1209.094
Linear approach to the orbiting spacecraft thermal problem
We develop a linear method for solving the nonlinear differential equations
of a lumped-parameter thermal model of a spacecraft moving in a closed orbit.
Our method, based on perturbation theory, is compared with heuristic
linearizations of the same equations. The essential feature of the linear
approach is that it provides a decomposition in thermal modes, like the
decomposition of mechanical vibrations in normal modes. The stationary periodic
solution of the linear equations can be alternately expressed as an explicit
integral or as a Fourier series. We apply our method to a minimal thermal model
of a satellite with ten isothermal parts (nodes) and we compare the method with
direct numerical integration of the nonlinear equations. We briefly study the
computational complexity of our method for general thermal models of orbiting
spacecraft and conclude that it is certainly useful for reduced models and
conceptual design but it can also be more efficient than the direct integration
of the equations for large models. The results of the Fourier series
computations for the ten-node satellite model show that the periodic solution
at the second perturbative order is sufficiently accurate.Comment: 20 pages, 11 figures, accepted in Journal of Thermophysics and Heat
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