Some applications of modular convergence in vector lattice setting

Abstract

The main purpose of this paper is to apply the theory of vector lattices and the related abstract modular convergence to the context of Mellin-type kernels and (non)linear vector lattice-valued operators, following the construction of an integral given in earlier papers.

A comparison between the sampling Kantorovich algorithm for digital image processing with some interpolation and quasi-interpolation methods

“…It is well-known that the operators, S w , w > 0, are approximation operators which are able to pointwise reconstruct, continuous and bounded signals, and to uniformly reconstruct, signals which are uniformly continuous and bounded, as w → +∞. Moreover, the operators S w can be used also to reconstruct not-necessarily continuous signals, e.g., signals belonging to the L p -spaces, 1 ≤ p < +∞ ( [2,3,6,7,9,10,26,32,33,41]).…”

Abstract Integration with Respect to Measures and Applications to Modular Convergence in Vector Lattice Setting

“…Taking into account that, by the Baire category theorem, the complement of a meager subset of Ω is dense in Ω, we get that the inequality in (20) holds for any ω ∈ Ω. From (20) it follows that ϕ satisfies the inequality in (17), taking…”

“…From ( 48), ( 49) and (50) it follows that the conditions in 6.7 on (U )-singularity are fulfilled. Note that it is possible to check that these conditions are satisfied also by other Mellin-type kernels, like for instance Mellin-Gauss-Weierstrass and Mellin-Poisson-Cauchy-type kernels (see, e.g., [15,20]).…”