Luca Zampogni

Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces

In this paper we introduce a nonlinear version of the Kantorovich sampling type series in a nonuniform setting. By means of the above series we are able to reconstruct signals (functions) which are continuous or uniformly continuous. Moreover, we study the problem of the convergence in the setting of Orlicz spaces: this allows us to treat signals which are not necessarily continuous. Our theory applies to L^p-spaces, interpolation spaces, exponential spaces and many others. Several graphical examples are provided.

Approximation Results for a General Class of Kantorovich Type Operators

Abstract We introduce and study a family of integral operators in the Kantorovich sense acting on functions defined on locally compact topological groups. We obtain convergence results for the above operators with respect to the pointwise and uniform convergence and in the setting of Orlicz spaces with respect to the modular convergence. Moreover, we show how our theory applies to several classes of integral and discrete operators, as sampling, convolution and Mellin type operators in the Kantorovich sense, thus obtaining a simultaneous approach for discrete and integral operators. Further, we obtain general convergence results in particular cases of Orlicz spaces, as Lp−spaces, interpolation spaces and exponential spaces. Finally we construct some concrete examples of our operators and we show some graphical representations.

On Algebro-Geometric Solutions of the Camassa-Holm Hierarchy

Abstract We find global solutions of algebro geometric type for all the equations of a new commuting hierarchy containing the Camassa-Holm equation. This hierarchy is built in analogy to the classical K-dV and AKNS hierarchies. We use a zero curvature method to give recursion formulas. The time evolution of the solutions is completely determined, and the motion on a nonlinear subvariety Υ of a generalized Jacobian variety is obtained by solving an inverse problem for the Sturm-Liouville equation L(φ) = −φ″ + φ = λyφ. This is the natural setting for the expression of the solutions which depend linearly with respect to t and x, with coordinates on a curvilinear parallelogram contained in such a subvariety φ. φ is obtained as the restriction of the generalized Abel map I0 to the space Symmg(R) of unordered g-tuples of points on R, and the nonlinear parallelogram is the image through the restricted generalized Abel map of the moving poles Pi(x, t) (i = 1, . . . , g) of the Weyl m-functions m±(x, t, λ) of the dynamical Sturm-Liouville family of equations Lx(φ) = −φ″ + φ = λτx(y)φ, where τ is the translation flow. It turns out that the choice of a particular stationary initial condition ug(x, t0) completely determines the solution u(x, t) of all the equations in the hierarchy, as functions of the poles Pi(x, t) of the Weyl m-functions corresponding to the family Lx of Sturm-Liouville operators, with density function y(x, t) = uxx(x, t)/2 − 2u(x, t). For every t∊ℝ, the maps x ↦ y(x, t) lie in an isospectral class of the associated family of Sturm-Liouville equations Lx(φ), and are completely determined by assigning spectral parameters and initial conditions for the poles Pi(x, t).

The Sturm-Liouville Hierarchy of Evolution Equations and Limits of Algebro-Geometric Initial Data

The Sturm{Liouville hierarchy of evolution equations was introduced in (Adv. Nonlinear Stud. 11 (2011), 555{591) and includes the Korteweg{de Vries and the Camassa{ Holm hierarchies. We discuss some solutions of this hierarchy which are obtained as limits of algebro-geometric solutions. The initial data of our solutions are (generalized) reflectionless Sturm{Liouville potentials (Stoch. Dyn. 8 (2008), 413{449).