A general approximation result for nonlinear integral operators and applications to signal processing^*

^*Dedicated to the memory of my Father, Professor Calogero Vinti, with all my respect, esteem and love, and in recognition of his fundamental work in approximation theory.

Abstract

Modular convergence theorems in Orlicz spaces for nets of nonlinear integral operators of the form

where G and H are topological groups and {h_w) is a family of homeomorphisms h_wH→h_w(H)CG are studied. The form of the above operators gives a unitary approach in order to obtain modular convergence theorems for several classical families of integral operators. In particular, in case of G = (R, +), H = (Z,+), h_w(k) = k/w, K_w(z,.) = K(wz,.), we obtain modular convergence theorems in a classical Orlicz space for the nonlinear version of the generalized sampling series of f of the form:

.

Keywords:

• Orlicz space
• Generalized sampling series
• Topological groups
• Modular convergence
• Jensen's inequality

AMS:

• 41A35
• 46E30
• 47A58
• 47B38
• 47G10

^*Dedicated to the memory of my Father, Professor Calogero Vinti, with all my respect, esteem and love, and in recognition of his fundamental work in approximation theory.

^† [email protected]

^*Dedicated to the memory of my Father, Professor Calogero Vinti, with all my respect, esteem and love, and in recognition of his fundamental work in approximation theory.

^† [email protected]

Notes

^*Dedicated to the memory of my Father, Professor Calogero Vinti, with all my respect, esteem and love, and in recognition of his fundamental work in approximation theory.

^† [email protected]