85 research outputs found
How fast do radial basis function interpolants of analytic functions converge?
The question in the title is answered using tools of potential theory. Convergence and divergence rates of interpolants of analytic functions on the unit interval are analyzed. The starting point is a complex variable contour integral formula for the remainder in RBF interpolation. We study a generalized Runge phenomenon and explore how the location of centers and affects convergence. Special attention is given to Gaussian and inverse quadratic radial functions, but some of the results can be extended to other smooth basis functions. Among other things, we prove that, under mild conditions, inverse quadratic RBF interpolants of functions that are analytic inside the strip , where is the shape parameter, converge exponentially
Solution of a second order difference equation using the bilinear relations of Riemann
A recently proposed technique to solve a class of second order functional difference equations arising in electromagnetic diffraction theory is further investigated by applying it to a case of intermediate complexity. The proposed approach is conceptually simple and relies on first obtaining well-defined branched solutions to a pair of associated first order difference equations. The construction of these branched expressions leads to an equation system whose solution requires relationships akin to Riemannās bilinear relations for differentials of the first and third kinds; their derivation necessitates the application of Cauchyās theorem on Riemann surfaces of, in this particular instance, genera one and three. Branch-free solutions of the second order difference equation are then obtained by taking appropriate linear combinations of the branched solutions of the first order equations. Analysis and computation demonstrate that the resulting expressions have the desired analytical properties and recover known solutions in the appropriate limit. Ā© 2002 American Institute of Physics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71093/2/JMAPAQ-43-3-1598-1.pd
Geometrically stopped Markovian random growth processes and Pareto tails
Many empirical studies document power law behavior in size distributions of
economic interest such as cities, firms, income, and wealth. One mechanism for
generating such behavior combines independent and identically distributed
Gaussian additive shocks to log-size with a geometric age distribution. We
generalize this mechanism by allowing the shocks to be non-Gaussian (but
light-tailed) and dependent upon a Markov state variable. Our main results
provide sharp bounds on tail probabilities, a simple equation determining
Pareto exponents, and comparative statics. We present two applications: we show
that (i) the tails of the wealth distribution in a heterogeneous-agent dynamic
general equilibrium model with idiosyncratic investment risk are Paretian, and
(ii) a random growth model for the population dynamics of Japanese
municipalities is consistent with the observed Pareto exponent but only after
allowing for Markovian dynamics
Asymptotic expansion of stationary distribution for reflected Brownian motion in the quarter plane via analytic approach
Brownian motion in R 2 + with covariance matrix and drift in
the interior and reflection matrix R from the axes is considered. The
asymptotic expansion of the stationary distribution density along all paths in
R 2 + is found and its main term is identified depending on parameters
(, , R). For this purpose the analytic approach of Fayolle,
Iasnogorodski and Malyshev in [12] and [36], restricted essentially up to now
to discrete random walks in Z 2 + with jumps to the nearest-neighbors in the
interior is developed in this article for diffusion processes on R 2 + with
reflections on the axes
Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2)
A fundamental question, first raised by Langlands, is to know whether the
Rankin-Selberg product of two (not necessarily holomorphic) cusp forms f and g
is modular, i.e., if there exists an automorphic form f box g on GL(4)/Q whose
standard L-function equals L^*(s, f x g) after removing the ramified and
archimedean factors. The first main result of this paper is to answer it in the
affirmative, in fact with the base field Q replaced by any number field F. Our
proof uses a mixture of converse theorems, base change and descent, and it also
appeals to the local regularity properties of Eisenstein series and the scalar
products of their truncations.
One of the applications of this result is that the space of cusp forms on
SL(2) has multiplicity one. Concretely this means the following: If f, g are
newforms of holomorphic or Maass type with trivial character such that the
squares of the p-th coeficients of f and g are the same at almost all primes p,
then g must be a twist of f by a quadratic Dirichlet character.Comment: 67 pages, published version, abstract added in migratio
Nonlinear Fourier Spectrum Characterization of Time-Limited Signals
Addressing the optical communication systems employing the nonlinear Fourier transform (NFT) for the data modulation/demodulation, we provide an explicit proof for the properties of the signals emerging in the so-called -modulation method, the nonlinear signal modulation technique that provides explicit control over the signal extent. We present details of the procedure and related rigorous mathematical proofs addressing the case where the time-domain profile corresponding to the -modulated data has a limited duration, and when the bound states corresponding to specifically chosen discrete solitonic eigenvalues and norming constants, are also present. We also prove that the number of solitary modes that we can embed without violating the exact localisation of the time-domain profile, is actually infinite. Our theoretical findings are illustrated with numerical examples, where simple example waveforms are used for the -coefficient, demonstrating the validity of the developed approach. We also demonstrate the influence of the bound states on the noise tolerance of the b-modulated system
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