Luca Zampogni - Academia.edu (original) (raw)

Papers by Luca Zampogni

Research paper thumbnail of Variation diminishing-type properties for multivariate sampling Kantorovich operators

Bollettino dell'Unione Matematica Italiana

In this paper we establish a variation-diminishing type estimate for the multivariate Kantorovich... more In this paper we establish a variation-diminishing type estimate for the multivariate Kantorovich sampling operators with respect to the concept of multidimensional variation introduced by Tonelli. A sharper estimate can be achieved when step functions with compact support (digital images) are considered. Several examples of kernels have been presented.

Research paper thumbnail of Some Remarks Concerning the Scattering Theory for the Sturm–Liouville Operator

Journal of Dynamics and Differential Equations

Research paper thumbnail of A General Approximation Approach for the Simultaneous Treatment of Integral and Discrete Operators

Advanced Nonlinear Studies

In this paper, we give a unitary approach for the simultaneous study of the convergence of discre... more In this paper, we give a unitary approach for the simultaneous study of the convergence of discrete and integral operators described by means of a family of linear continuous functionals acting on functions defined on locally compact Hausdorff topological groups. The general family of operators introduced and studied includes very well-known operators in the literature. We give results of uniform convergence and modular convergence in the general setting of Orlicz spaces. The latter result allows us to cover many other settings as the {L^{p}} -spaces, the interpolation spaces, the exponential spaces and many others.

Research paper thumbnail of Some examples of generalized reflectionless Schrödinger potentials

Discrete and Continuous Dynamical Systems - Series S

Research paper thumbnail of On algebro-geometric solutions of the Camassa-Holm hierarchy

Advanced Nonlinear Studies, 2007

We find global solutions of algebro geometric type for all the equations of a new commuting hiera... more 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 I

Research paper thumbnail of On the Lyapunov exponent of certain SL (2,R)-valued cocycles II

Differential Equations and Dynamical Systems, 2010

Research paper thumbnail of Chapter 2 Nonautonomous differential systems in two dimensions

Research paper thumbnail of On Algebro-Geometric Solutions of the Camassa-Holm Hierarchy

Advanced Nonlinear Studies, 2007

We find global solutions of algebro geometric type for all the equations of a new commuting hiera... more 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 I

Research paper thumbnail of Approximation Results for a General Class of Kantorovich Type Operators

Advanced Nonlinear Studies, 2014

We introduce and study a family of integral operators in the Kantorovich sense acting on function... more 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 L

Research paper thumbnail of * Insegnamenti I semestre **Insegnamenti II semestre II ses III ses I ses I ses CORSO DI LAUREA INTERFACOLTA

Research paper thumbnail of A unifying approach to convergence of linear sampling type operators in Orlicz spaces

Advances in Differential Equations

In this paper we give a unified approach to the study of the convergence of a general family of l... more In this paper we give a unified approach to the study of the convergence of a general family of linear sampling type operators which includes several operators very useful in signal reconstruction. We study both the pointwise and uniform convergence and the modular one in the general setting of Orlicz spaces. This creates the possibility of covering several settings such as L-p-spaces, the interpolation spaces, the exponential spaces and many others.

Research paper thumbnail of Remarks concerning the Lyapunov exponents of linear cocycles

We impose a condition of pointwise convergence on the Lyapunov exponents of a d-dimensional cocyc... more We impose a condition of pointwise convergence on the Lyapunov exponents of a d-dimensional cocycle over a compact metric minimal flow. This condition turns out to have significant consequences for the dynamics of the cocycle. We make use of such classical ODE techniques as the Lyapunov-Perron triangularization method, and the ergodic-theoretical techniques of Krylov and Bogoliubov.

Research paper thumbnail of Description of the Algebro Geometric S-L coefficients

Research paper thumbnail of M39 - Differential equations, dynamical systems and applic ations

Research paper thumbnail of A unified approach for the convergence of linear Kantorovich-type operators

Advanced Nonlinear Studies

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

Symmetry, Integrability and Geometry: Methods and Applications, 2014

The Sturm-Liouville hierarchy of evolution equations was introduced in [Adv. Nonlinear Stud. 11 (... more 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].

Research paper thumbnail of On the Lyapunov exponent of certain SL (2, ℝ)-valued cocycles II

Differential Equations and Dynamical …, 2010

We study the positivity of the Lyapunov exponent for a smooth SL (2, ℝ)-valued cocycle defined ov... more We study the positivity of the Lyapunov exponent for a smooth SL (2, ℝ)-valued cocycle defined over a flow from a class which includes the Kronecker flows and others as well. We also discuss the question of the density in the Hölder class of the set of SL (2, ℝ)- ...

Research paper thumbnail of On infinite order K-dV hierarchy

Research paper thumbnail of On infinite order KdV hierarchies

Using a zero-curvature method and an approximation process, the author finds solutions of the r-t... more Using a zero-curvature method and an approximation process, the author finds solutions of the r-th order KdV equation when the initial data u(x,0) determines a Schrödinger potential of algebro geometric-type. Then the author determines a hierarchy of commuting equations containg the KdV equation when the initial data u(x,0) determines a Schrödinger potential of reflectionless type.

Research paper thumbnail of Some topologies on the spaces of USCO maps and densely continuous forms

Acta Mathematica Hungarica, 2008

We study the completeness of three (metrizable) uniformities on the sets D(X, Y ) and U (X, Y ) o... more We study the completeness of three (metrizable) uniformities on the sets D(X, Y ) and U (X, Y ) of densely continuous forms and USCO maps from X to Y : the uniformity of uniform convergence on bounded sets, the Hausdor metric uniformity and the uniformity U B . We also prove that if X is a nondiscrete space, then the Hausdor metric on real-valued densely continuous forms D(X, R) (identied with their graphs) is not complete. The key to guarantee completeness of closed subsets of D(X, Y ) equipped with the Hausdor metric is dense equicontinuity introduced by Hammer and McCoy in [7].

Research paper thumbnail of Variation diminishing-type properties for multivariate sampling Kantorovich operators

Bollettino dell'Unione Matematica Italiana

In this paper we establish a variation-diminishing type estimate for the multivariate Kantorovich... more In this paper we establish a variation-diminishing type estimate for the multivariate Kantorovich sampling operators with respect to the concept of multidimensional variation introduced by Tonelli. A sharper estimate can be achieved when step functions with compact support (digital images) are considered. Several examples of kernels have been presented.

Research paper thumbnail of Some Remarks Concerning the Scattering Theory for the Sturm–Liouville Operator

Journal of Dynamics and Differential Equations

Research paper thumbnail of A General Approximation Approach for the Simultaneous Treatment of Integral and Discrete Operators

Advanced Nonlinear Studies

In this paper, we give a unitary approach for the simultaneous study of the convergence of discre... more In this paper, we give a unitary approach for the simultaneous study of the convergence of discrete and integral operators described by means of a family of linear continuous functionals acting on functions defined on locally compact Hausdorff topological groups. The general family of operators introduced and studied includes very well-known operators in the literature. We give results of uniform convergence and modular convergence in the general setting of Orlicz spaces. The latter result allows us to cover many other settings as the {L^{p}} -spaces, the interpolation spaces, the exponential spaces and many others.

Research paper thumbnail of Some examples of generalized reflectionless Schrödinger potentials

Discrete and Continuous Dynamical Systems - Series S

Research paper thumbnail of On algebro-geometric solutions of the Camassa-Holm hierarchy

Advanced Nonlinear Studies, 2007

We find global solutions of algebro geometric type for all the equations of a new commuting hiera... more 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 I

Research paper thumbnail of On the Lyapunov exponent of certain SL (2,R)-valued cocycles II

Differential Equations and Dynamical Systems, 2010

Research paper thumbnail of Chapter 2 Nonautonomous differential systems in two dimensions

Research paper thumbnail of On Algebro-Geometric Solutions of the Camassa-Holm Hierarchy

Advanced Nonlinear Studies, 2007

We find global solutions of algebro geometric type for all the equations of a new commuting hiera... more 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 I

Research paper thumbnail of Approximation Results for a General Class of Kantorovich Type Operators

Advanced Nonlinear Studies, 2014

We introduce and study a family of integral operators in the Kantorovich sense acting on function... more 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 L

Research paper thumbnail of * Insegnamenti I semestre **Insegnamenti II semestre II ses III ses I ses I ses CORSO DI LAUREA INTERFACOLTA

Research paper thumbnail of A unifying approach to convergence of linear sampling type operators in Orlicz spaces

Advances in Differential Equations

In this paper we give a unified approach to the study of the convergence of a general family of l... more In this paper we give a unified approach to the study of the convergence of a general family of linear sampling type operators which includes several operators very useful in signal reconstruction. We study both the pointwise and uniform convergence and the modular one in the general setting of Orlicz spaces. This creates the possibility of covering several settings such as L-p-spaces, the interpolation spaces, the exponential spaces and many others.

Research paper thumbnail of Remarks concerning the Lyapunov exponents of linear cocycles

We impose a condition of pointwise convergence on the Lyapunov exponents of a d-dimensional cocyc... more We impose a condition of pointwise convergence on the Lyapunov exponents of a d-dimensional cocycle over a compact metric minimal flow. This condition turns out to have significant consequences for the dynamics of the cocycle. We make use of such classical ODE techniques as the Lyapunov-Perron triangularization method, and the ergodic-theoretical techniques of Krylov and Bogoliubov.

Research paper thumbnail of Description of the Algebro Geometric S-L coefficients

Research paper thumbnail of M39 - Differential equations, dynamical systems and applic ations

Research paper thumbnail of A unified approach for the convergence of linear Kantorovich-type operators

Advanced Nonlinear Studies

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

Symmetry, Integrability and Geometry: Methods and Applications, 2014

The Sturm-Liouville hierarchy of evolution equations was introduced in [Adv. Nonlinear Stud. 11 (... more 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].

Research paper thumbnail of On the Lyapunov exponent of certain SL (2, ℝ)-valued cocycles II

Differential Equations and Dynamical …, 2010

We study the positivity of the Lyapunov exponent for a smooth SL (2, ℝ)-valued cocycle defined ov... more We study the positivity of the Lyapunov exponent for a smooth SL (2, ℝ)-valued cocycle defined over a flow from a class which includes the Kronecker flows and others as well. We also discuss the question of the density in the Hölder class of the set of SL (2, ℝ)- ...

Research paper thumbnail of On infinite order K-dV hierarchy

Research paper thumbnail of On infinite order KdV hierarchies

Using a zero-curvature method and an approximation process, the author finds solutions of the r-t... more Using a zero-curvature method and an approximation process, the author finds solutions of the r-th order KdV equation when the initial data u(x,0) determines a Schrödinger potential of algebro geometric-type. Then the author determines a hierarchy of commuting equations containg the KdV equation when the initial data u(x,0) determines a Schrödinger potential of reflectionless type.

Research paper thumbnail of Some topologies on the spaces of USCO maps and densely continuous forms

Acta Mathematica Hungarica, 2008

We study the completeness of three (metrizable) uniformities on the sets D(X, Y ) and U (X, Y ) o... more We study the completeness of three (metrizable) uniformities on the sets D(X, Y ) and U (X, Y ) of densely continuous forms and USCO maps from X to Y : the uniformity of uniform convergence on bounded sets, the Hausdor metric uniformity and the uniformity U B . We also prove that if X is a nondiscrete space, then the Hausdor metric on real-valued densely continuous forms D(X, R) (identied with their graphs) is not complete. The key to guarantee completeness of closed subsets of D(X, Y ) equipped with the Hausdor metric is dense equicontinuity introduced by Hammer and McCoy in [7].