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Elementary real an complex analysis / Georgii E. Shilov
Abstract
Traducción de: Matematichesskii analiz funktsii odnovo peremennovo Incluye índice
... The infinite sum ∑ ∞ k=−∞ c k ϕ k is called the Fourier series expansion of f , where {c k } are the Fourier coefficients. Using Equation (29) and taking the inner product of both sides of Equation (30) by ϕ n , for any n ∈ Z, yields c n ...
... .} in (X, ρ) contains a Cauchy subsequence {x n1 , x n2 , . . .} (see Definition A3 and also see [29]). ...
Functional analysis is a well-developed field in the discipline of Mathematics, which provides unifying frameworks for solving many problems in applied sciences and engineering. In particular, several important topics (e.g., spectrum estimation, linear prediction, and wavelet analysis) in signal processing had been initiated and developed through collaborative efforts of engineers and mathematicians who used results from Hilbert spaces, Hardy spaces, weak topology, and other topics of functional analysis to establish essential analytical structures for many subfields in signal processing. This paper presents a concise tutorial for understanding the theoretical concepts of the essential elements in functional analysis, which form a mathematical framework and backbone for central topics in signal processing, specifically statistical and adaptive signal processing. The applications of these concepts for formulating and analyzing signal processing problems may often be difficult for researchers in applied sciences and engineering, who are not adequately familiar with the terminology and concepts of functional analysis. Moreover, these concepts are not often explained in sufficient details in the signal processing literature; on the other hand, they are well-studied in textbooks on functional analysis, yet without emphasizing the perspectives of signal processing applications. Therefore, the process of assimilating the ensemble of pertinent information on functional analysis and explaining their relevance to signal processing applications should have significant importance and utility to the professional communities of applied sciences and engineering. The information, presented in this paper, is intended to provide an adequate mathematical background with a unifying concept for apparently diverse topics in signal processing. The main objectives of this paper from the above perspectives are summarized below: (1) Assimilation of the essential information from different sources of functional analysis literature, which are relevant to developing the theory and applications of signal processing. (2) Description of the underlying concepts in a way that is accessible to non-specialists in functional analysis (e.g., those with bachelor-level or first-year graduate-level training in signal processing and mathematics). (3) Signal-processing-based interpretation of functional-analytic concepts and their concise presentation in a tutorial format.
... Let us define a metric d(z 1 , z 2 ) = |z 1 − z 2 | on set Z. Then (Z, d) is a metric space 1 . Further, consider Z as a subset of the real line R and consider the same metric d on R. A well-known result from analysis states that (R, d) is a complete space [17]. Since the maximum possible value of p n is ...
... closed subspace of R. Since (Z, d) is a closed subspace of the complete space (R, d), it is a complete space [17]. ...
A resource allocation mechanism based on matching and bargaining is presented. There are many resource providers and many resource seekers. The resource valuations of the agents are private and are uniformly distributed over a common support. The regulator establishes two bid levels. Each participating agent is randomly paired with a counterpart and the allocations arise out of these bilateral encounters based on agents’ individual bid choices. We show a Bayes Nash equilibrium in dominant strategy when the regulator fixes the strategy for one set of agents. The mechanism has parameters to tune the allocations.
... According to those expressions, the transformed medium will be isotropic only when the underlying coordinate transformation is conformal, i.e., a transformation for which the Cauchy-Riemann equations are satisfied [9]: ...
... The transformation shows no reflection at a given boundary if, and only if, it is continuous with the external coordinate system at that boundary [10]. However, this condition can only be exactly met by the identity transformation due to the uniqueness theorem [9] as discussed in [1], [17]. It is possible, however, to minimize reflections by approximating the identity transformation close to the boundaries at the price of acceptable levels of anisotropy. ...
In this paper, we present new strategies to reduce anisotropy in transformation
optics designs and compare them to other techniques. Perturbation functions are used to
modify the original transformation to achieve a quasi-conformal map, resulting in a medium
with isotropic properties. All strategies investigated have no effect on the original
boundary conditions of the transformation, such that neither the initial design is affected
nor are reflections at the boundaries introduced. The results show that there exists a clear
compromise between the residual anisotropy and the required refractive index contrast in
the optimized transformation, but the former can be made as small as desired when the
degree of freedom provided by perturbation functions is increased, although it is never
exactly zero
... Given that all trajectories of h J,b starting at any x ∈ D (J, b) lie in a compact subset W of D (J, b), the uniqueness of its flow follows from the Lipschitz continuity of h J,b in W since a locally Lipschitz function on S(τ ) is Lipschitz on every compact subset of S(τ ), also refer to Theorem 3.3 in [82]. Moreover, this unique flow is continuous and piecewise smooth since it is the integral of the continuous and piecewise smooth vector field h J,b [94], which completes the proof. ...
We introduce the use of hierarchical clustering for relaxed, deterministic
coordination and control of multiple robots. Traditionally an unsupervised
learning method, hierarchical clustering offers a formalism for identifying and
representing spatially cohesive and segregated robot groups at different
resolutions by relating the continuous space of configurations to the
combinatorial space of trees. We formalize and exploit this relation,
developing computationally effective reactive algorithms for navigating through
the combinatorial space in concert with geometric realizations for a particular
choice of hierarchical clustering method. These constructions yield
computationally effective vector field planners for both hierarchically
invariant as well as transitional navigation in the configuration space. We
apply these methods to the centralized coordination and control of $n$
perfectly sensed and actuated Euclidean spheres in a $d$-dimensional ambient
space (for arbitrary $n$ and $d$). Given a desired configuration supporting a
desired hierarchy, we construct a hybrid controller which is quadratic in $n$
and algebraic in $d$ and prove that its execution brings all but a measure zero
set of initial configurations to the desired goal with the guarantee of no
collisions along the way.
... An entire function is a function with no singularities in the finite s plane, but which may have one at infinity [64,123]. If the object is on or near a ground plane or embedded in a lossy dielectric volume, two branch points of order 1 are introduced in the s-plane, with a branch cut between them [37]. ...
The goal of this research is to develop algorithms that recognize targets by exploiting properties in the late-time resonance induced by ultra-wide band radar signals. A new variant of the Matrix Pencil Method algorithm is developed that identifies complex resonant frequencies present in the scattered signal. Kalman filters are developed to represent the dynamics of the signals scattered from several target types. The Multiple Model Adaptive Estimation algorithm uses the Kalman filters to recognize targets. The target recognition algorithm is shown to be successful in the presence of noise. The performance of the new algorithms is compared to that of previously published algorithms.
We introduce a monotone class theory of Prospect Theory's value functions, which shows that they can be replaced almost surely by a topological lifting comprised of a class of compact isomorphic maps that embed weakly co-monotonic probability measures, attached to state space, in outcome space. Thus, agents solve a signal extraction problem to obtain estimates of empirical probability weights for prospects under risk and uncertainty. By virtue of the topological lifting, we prove an almost sure isomorphism theorem between compact stochastic choice operators, and well defined outcomes which, under Brouwer-Schauder theory, guarantees fixed point convergence in convex choice sets. Along the way we introduce a risk operator in the Hoffman-Jorgensen class of lifting operators, and value function [averaging] operators with respect to Radon measure. In that set up, suitable binary operations on gain-loss space show that our risk operator is isometric for gains and skewed for losses. The point spectrum from this operator constitutes the range of admissible observations for loss aversion index in a well designed experiment.
The calculation of fractional or integer order derivatives and integrals has been demonstrated to be simple and fast in the frequency domain. It is also the most sensible method if one wishes to calculate derivatives or integrals of periodic signals. In this paper, error analysis is carried out for the numerical algorithm for Weyl fractional derivatives. To derive an upper bound for the numerical error, some knowledge of the smoothness of the signal must be known in advance or it must be estimated. The derived error analysis is tested with sampled functions with known regularity and with real vibration measurements from rotating machines. Compared to previous publications which deal with error analysis of integer order numerical derivatives in the frequency domain using L^2 errors, the result of this paper is in terms of maximum absolute error and it is based on a novel result on the signal's regularity. The general conclusion using either error estimates is the same: the error of numerical Weyl derivatives is bounded by some constant times the sequence length raised to a negative power. The exponent depends on the smoothness of the signal. This contrasts with using difference quotients in numerical differentiation, in which case the error is bounded by a constant times the sequence length raised to a some fixed negative power and the order of the method defines that exponent.
The main result of this paper is, that if we suppose that a function is absolutely continuous and uniformly H\"older continuous and that it's finite difference function does not oscillate infinitely often on a bounded interval, then the decay rate of its Fourier coefficients can be estimated exactly. This rate of decay predicts the same uniform H\"older continuity but the two other conditions are not necessary. Several examples from literature and by the author show that none of the assumptions can be relaxed without weakening the decay for some functions. The uniform H\"older continuity of chirps and the decay of their Fourier coefficients and Fourier transforms are studied thoroughly.
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