6 research outputs found
Aspects of 2D-Adaptive Fourier Decompositions
As a new type of series expansion, the so-called one-dimensional adaptive
Fourier decomposition (AFD) and its variations (1D-AFDs) have effective
applications in signal analysis and system identification. The 1D-AFDs have
considerable influence to the rational approximation of one complex variable
and phase retrieving problems, etc. In a recent paper, Qian developed 2D-AFDs
for treating square images as the essential boundary of the 2-torus embedded
into the space of two complex variables. This paper studies the numerical
aspects of multi-dimensional AFDs, and in particular 2D-AFDs, which mainly
include (i) Numerical algorithms of several types of 2D-AFDs in relation to
image representation; (ii) Perform experiments for the algorithms with
comparisons between 5 types of image reconstruction methods in the Fourier
category; and (iii) New and sharper estimations for convergence rates of
orthogonal greedy algorithm and pre-orthogonal greedy algorithm. The comparison
shows that the 2D-AFD methods achieve optimal results among the others.Comment: 12 pages, 51 figure
Rational Approximation in the Bergman Spaces
It is known that adaptive Fourier decomposition (AFD) offers efficient
rational approxima- tions to functions in the classical Hardy H2 spaces with
significant applications. This study aims at rational approximation in Bergman,
and more widely, in weighted Bergman spaces, the functions of which have more
singularity than those in the Hardy spaces. Due to lack of an effective inner
function theory, direct adaptation of the Hardy-space AFD is not performable.
We, however, show that a pre-orthogonal method, being equivalent to AFD in the
classical cases, is available for all weighted Bergman spaces. The theory in
the Bergman spaces has equal force as AFD in the Hardy spaces. The methodology
of approximation is via constructing the rational orthogonal systems of the
Bergman type spaces, called Bergman space rational orthog- onal (BRO) system,
that have the same role as the Takennaka-Malmquist (TM) system in the Hardy
spaces. Subsequently, we prove a certain type direct sum decomposition of the
Bergman spaces that reveals the orthogonal complement relation between the span
of the BRO system and the zero-based invariant spaces. We provide a sequence of
examples with different and ex- plicit singularities at the boundary along with
a study on the inclusion relations of the weighted Bergman spaces. We finally
present illustrative examples for effectiveness of the approximation
Two-Dimensional Adaptive Fourier Decomposition
One-dimensional adaptive Fourier decomposition, abbreviated as 1-D AFD, or
AFD, is an adaptive representation of a physically realizable signal into a
linear combination of parameterized Szeg\"o and higher order Szeg\"o kernels of
the context. In the present paper we study multi-dimensional AFDs based on
multivariate complex Hardy spaces theory. We proceed with two approaches of
which one uses Product-TM Systems; and the other uses Product-Szeg\"o
Dictionaries. With the Product-TM Systems approach we prove that at each
selection of a pair of parameters the maximal energy may be attained, and,
accordingly, we prove the convergence. With the Product-Szeg\"o dictionary
approach we show that Pure Greedy Algorithm is applicable. We next introduce a
new type of greedy algorithm, called Pre-Orthogonal Greedy Algorithm (P-OGA).
We prove its convergence and convergence rate estimation, allowing a weak type
version of P-OGA as well. The convergence rate estimation of the proposed P-OGA
evidences its advantage over Orthogonal Greedy Algorithm (OGA). In the last
part we analyze P-OGA in depth and introduce the concept P-OGA-Induced Complete
Dictionary, abbreviated as Complete Dictionary . We show that with the Complete
Dictionary P-OGA is applicable to the Hardy space on -torus.Comment: 24 page
Sparse Approximation to the Dirac-{\delta} Distribution
The Dirac-{\delta} distribution may be realized through sequences of
convlutions, the latter being also regarded as approximation to the identity.
The present study proposes the so called pre-orthogonal adaptive Fourier
decomposition (POAFD) method to realize fast approximation to the identity. The
type of sparse representation method has potential applications in signal and
image analysis, as well as in system identification.Comment: 17 page
A Sparse Representation of Random Signals
Studies of sparse representation of deterministic signals have been well
developed. Amongst there exists one called adaptive Fourier decomposition (AFD)
established through adaptive selections of the parameters defining a
Takenaka-Malmquist system in one-complex variable. The AFD type algorithms give
rise to sparse representations of signals of finite energy. The multivariate
generalization of AFD is one called pre-orthogonal AFD (POAFD), the latter
being established with the context Hilbert space possessing a dictionary. The
purpose of the present study is to generalize both AFD and POAFD to random
signals. We work on two types of random signals. One is those expressible as
the sum of a deterministic signal with an error term such as a white noise; and
the other is, in general, as mixture of several classes of random signals
obeying certain distributive law. In the first part of the paper we develop an
AFD type sparse representation for one-dimensional random signals by making use
analysis of one complex variable. In the second part, without complex analysis,
we treat multivariate random signals in the context of stochastic Hilbert space
with a dictionary. Like in the deterministic signal case the established random
sparse representations are powerful tools in practical signal analysis.Comment: 28 page
Noncommutative analysis, Multivariable spectral theory for operators in Hilbert space, Probability, and Unitary Representations
Over the decades, Functional Analysis has been enriched and inspired on
account of demands from neighboring fields, within mathematics, harmonic
analysis (wavelets and signal processing), numerical analysis (finite element
methods, discretization), PDEs (diffusion equations, scattering theory),
representation theory; iterated function systems (fractals, Julia sets, chaotic
dynamical systems), ergodic theory, operator algebras, and many more. And
neighboring areas, probability/statistics (for example stochastic processes,
Ito and Malliavin calculus), physics (representation of Lie groups, quantum
field theory), and spectral theory for Schr\"odinger operators.
We have strived for a more accessible book, and yet aimed squarely at
applications; -- we have been serious about motivation: Rather than beginning
with the four big theorems in Functional Analysis, our point of departure is an
initial choice of topics from applications. And we have aimed for flexibility
of use; acknowledging that students and instructors will invariably have a host
of diverse goals in teaching beginning analysis courses. And students come to
the course with a varied background. Indeed, over the years we found that
students have come to the Functional Analysis sequence from other and different
areas of math, and even from other departments; and so we have presented the
material in a way that minimizes the need for prerequisites. We also found that
well motivated students are easily able to fill in what is needed from measure
theory, or from a facility with the four big theorems of Functional Analysis.
And we found that the approach "learn-by-using" has a comparative advantage.Comment: 442 pages, 62 figure