Sampling of Paley-Wiener functions on stratified groups

We consider a generalization of entire functions of spherical exponential type on stratified groups. An analog of the Paley-Wiener theorem is given. We also show that every spectral entire function on a stratified group is uniquely determined by its values on some discrete subgroups. The main result of the article is reconstruction formula of spectral entire functions from their values on discrete subgroups using Lagrangian splines.     

Approximation of a function of two variables by a product of functions of one variable on a given domain

We are concerned with the problem of uniform approximation of a continuous function of two variables by a product of continuous functions of one variable on some domain D. This problem have been examined so far only on a rectangular domain D = U × V, where U and V are compact sets. An algorithm to give a solution of this problem in the discrete case is available. We put forward an algorithm which in certain cases allows one to construct an approximate solution of the problem on a given domain (not necessarily rectangular). This approximate solution is built in the form of interpolating natural splines, which in turn are constructed by means of discrete approximation. Depending on the degree of the splines, the problem can be solved in classes of functions with appropriate degree of smoothness.     

Calculus on the Sierpinski gasket I: polynomials, exponentials and power series

We study the analog of power series expansions on the Sierpinski gasket, for analysis based on the Kigami Laplacian. The analog of polynomials are multiharmonic functions, which have previously been studied in connection with Taylor approximations and splines. Here the main technical result is an estimate of the size of the monomials analogous to xⁿ/n!. We propose a definition of entire analytic functions as functions represented by power series whose coefficients satisfy exponential growth conditions that are stronger than what is required to guarantee uniform convergence. We present a characterization of these functions in terms of exponential growth conditions on powers of the Laplacian of the function. These entire analytic functions enjoy properties, such as rearrangement and unique determination by infinite jets, that one would expect. However, not all exponential functions (eigenfunctions of the Laplacian) are entire analytic, and also many other natural candidates, such as the heat kernel, do not belong to this class. Nevertheless, we are able to use spectral decimation to study exponentials, and in particular to create exponentially decaying functions for negative eigenvalues.     

Band limited functions on quantum graphs

The notion of band limited functions is introduced on a quantum graph. The main results of the paper are a uniqueness theorem and a reconstruction algorithm of such functions from discrete sets of values. It turns out that some of our band limited functions can have compact supports and their frequencies can be localized on the ``time" side. It opens an opportunity to consider signals of a variable band width and to develop a sampling theory with variable rate of sampling.

     

Augmented Lagrangian method for probabilistic optimization

We analyze nonlinear stochastic optimization problems with probabilistic constraints described by continuously differentiable non-convex functions. We describe the tangent and the normal cone to the level sets of the underlying probability function and provide new insight into their structure. Furthermore, we formulate fist order and second order conditions of optimality for these problems based on the notion of p-efficient points. We develop an augmented Lagrangian method for the case of discrete distribution functions. The method is based on progressive inner approximation of the level set of the probability function by generation of p-efficient points. Numerical experience is provided.     

Combinatorial Laplacians of matroid complexes

We combinatorially interpret the spectra of discrete Laplace operators from the boundary maps in the simplicial complex of independent sets of a matroid. The interpretation follows from a surprising orthogonal decomposition of the simplicial chain groups. This decomposition is in general finer than the spectral decomposition. As a consequence, the spectra are integral. One corollary to our combinatorial interpretation may be paraphrased as stating that one can ``hear" the characteristic polynomial of a matroid.

     

Delaunay configurations and multivariate splines: A generalization of a result of B. N. Delaunay

In the 1920s, B. N. Delaunay proved that the dual graph of the Voronoi diagram of a discrete set of points in a Euclidean space gives rise to a collection of simplices, whose circumspheres contain no points from this set in their interior. Such Delaunay simplices tessellate the convex hull of these points. An equivalent formulation of this property is that the characteristic functions of the Delaunay simplices form a partition of unity. In the paper this result is generalized to the so-called Delaunay configurations. These are defined by considering all simplices for which the interiors of their circumspheres contain a fixed number of points from the given set, in contrast to the Delaunay simplices, whose circumspheres are empty. It is proved that every family of Delaunay configurations generates a partition of unity, formed by the so-called simplex splines. These are compactly supported piecewise polynomial functions which are multivariate analogs of the well-known univariate B-splines. It is also shown that the linear span of the simplex splines contains all algebraic polynomials of degree not exceeding the degree of the splines.

     

Variational Splines and Paley–Wiener Spaces on Combinatorial Graphs

Notions of interpolating variational splines and Paley–Wiener spaces are introduced on a combinatorial graph G. Both of these definitions explore existence of a combinatorial Laplace operator on G. The existence and uniqueness of interpolating variational splines on a graph is shown. As an application of variational splines, the paper presents a reconstruction algorithm of Paley–Wiener functions on graphs from their uniqueness sets.      

On the Interpolation by Discrete Splines with Equidistant Nodes

In this paper we consider equidistant discrete splines S(j), j , which may grow as O(|j|^s) as |j|→∞. Such splines are relevant for the purposes of digital signal processing. We give the definition of the discrete B-splines and describe their properties. Discrete splines are defined as linear combinations of shifts of the B-splines. We present a solution to the problem of discrete spline cardinal interpolation of the sequences of power growth and prove that the solution is unique within the class of discrete splines of a given order.     

Approximation of Surfaces by Fairness Bicubic Splines

In this paper we present an approximation method of surfaces by a new type of splines, which we call fairness bicubic splines, from a given Lagrangian data set. An approximating problem of surface is obtained by minimizing a quadratic functional in a parametric space of bicubic splines. The existence and uniqueness of this problem are shown as long as a convergence result of the method is established. We analyze some numerical and graphical examples in order to prove the validity of our method.     

A Globally Convergent Lagrangian Barrier Algorithm for Optimization with General Inequality Constraints and Simple Bounds

We consider the global and local convergence properties of a class of Lagrangian barrier methods for solving nonlinear programming problems. In such methods, simple bound constraints may be treated separately from more general constraints. The objective and general constraint functions are combined in a Lagrangian barrier function. A sequence of such functions are approximately minimized within the domain defined by the simple bounds. Global convergence of the sequence of generated iterates to a first-order stationary point for the original problem is established. Furthermore, possible numerical difficulties associated with barrier function methods are avoided as it is shown that a potentially troublesome penalty parameter is bounded away from zero. This paper is a companion to previous work of ours on augmented Lagrangian methods.

     

Approximation by Discrete GB-Splines

This paper addresses the definition and the study of discrete generalized splines. Discrete generalized splines are continuous piecewise defined functions which meet some smoothness conditions for the first and second divided differences at the knots. They provide a generalization both of smooth generalized splines and of the classical discrete cubic splines. Completely general configurations for steps in divided differences are considered. Direct algorithms are proposed for constructing discrete generalized splines and discrete generalized B-splines (discrete GB-splines for short). Explicit formulae and recurrence relations are obtained for discrete GB-splines. Properties of discrete GB-splines and their series are studied. It is shown that discrete GB-splines form weak Chebyshev systems and that series of discrete GB-splines have a variation diminishing property.     

Superposition operators between weighted Banach spaces of analytic functions of controlled growth

We characterize the entire functions which transform a weighted Banach space of holomorphic functions on the disc of type $H^{\infty }$ into another such space by superposition. We also show that all the superposition operators induced by such entire functions map bounded sets into bounded sets and are continuous. Superposition operators that map bounded sets into relatively compact sets are also considered.     

Variational and Geometric Structures of Discrete Dirac Mechanics

In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew’s triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange–Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange–d’Alembert–Pontryagin and Hamilton–d’Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.     

Convergence of discrete and penalized least squares spherical splines

We study the convergence of discrete and penalized least squares spherical splines in spaces with stable local bases. We derive a bound for error in the approximation of a sufficiently smooth function by the discrete and penalized least squares splines. The error bound for the discrete least squares splines is explicitly dependent on the mesh size of the underlying triangulation. The error bound for the penalized least squares splines additionally depends on the penalty parameter.     

Minimal Lagrangian submanifolds in ℂP<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with diagonal metric

We propose a method for constructing minimal Lagrangian submanifolds in ℂP ⁿ using the Baker-Akhiezer functions of algebraic spectral curves.     

The modified KdV equation on a finite interval

We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann–Hilbert problem in the complex k-plane. This Riemann–Hilbert problem has explicit (x,t)-dependence and it involves certain functions of k referred to as “spectral functions”. Some of these functions are defined in terms of the initial condition q(x,0)=q₀(x), while the remaining spectral functions are defined in terms of two sets of boundary values. We show that the spectral functions satisfy an algebraic “global relation” that characterize the boundary values in spectral terms. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A local Lagrange interpolation method based on cubic splines on Freudenthal partitions

A trivariate Lagrange interpolation method based on cubic splines is described. The splines are defined over a special refinement of the Freudenthal partition of a cube partition. The interpolating splines are uniquely determined by data values, but no derivatives are needed. The interpolation method is local and stable, provides optimal order approximation, and has linear complexity.

     

Nonsmooth analysis on smooth manifolds

We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function.

A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.