Modular Convergence Theorems for Integral Operators in the Context of Filter Exhaustiveness and Applications

Vitali-type theorems for filter convergence related to vector lattice-valued modulars and applications to stochastic processes

“…In modular spaces the Vitali theorem is often used in the problem of approximating a realvalued function f by means of Urysohn-type or sampling operators (T n f ) n , (see also [8,9,11,12,13,32,1,2,3,16,17,4,34]). These operators are particularly useful in order to approximate a continuous or analog signal by means of discrete samples, and therefore they are widely applied for example in reconstructing images and videos.…”

LpSpaces in vector lattices and applications

“…exists uniformly with respect to E (see also [18,Proposition 2.14]), thus proving that ((f n + g n ) p ) n is a defining sequence for (f + g) p .…”

“…In this paper some fundamental properties of L p spaces in the vector lattice setting are investigated, continuing a research initiated by the authors in [12-14, 16, 21, 22] and developed later in [15]. The range of the involved functions is a vector lattice endowed with filter/ideal convergence (for a related literature, see also [3,[17][18][19][20]25,27]). Thanks to the triangle inequality, it is possible to view the space L p as a metric space endowed with a distance of the type d(f, g) = f −g p .…”

Convergence of sampling Kantorovich operators in modular spaces with applications

“…The notion of modular space L is very general and includes in itself several cases of well-know spaces, such as the L p -spaces, the Zygmund (or interpolation) spaces, the Sobolev spaces, and the exponential spaces (see, e.g., [22,46,49,64]).…”