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

Abstract

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.

Multivariate sampling Kantorovich operators: from the theory to the Digital Image Processing algorithm

“…A procedure that allow us to construct multivariate kernels by the product of univariate kernels satisfying the assumptions of the above theory can be showed (see e.g. [10,11,13,14]). Examples of one-dimensional kernels are the well-known Fejér's kernel, i.e., F (x) = , and many others, see e.g., [2,7,8].…”

“…The sampling Kantorovich operators S w (see also [9,14,15]) represent an L 1 -version of the generalized sampling operators and they revealed to be very suitable to reconstruct not necessarily continuous signals; thus applications to image reconstruction can be deduced.…”

Digital image processing algorithms for diagnosis in arterial diseases

“…Multivariate sampling Kantorovich operators [1,2,4,5,9,10] and subsequent wavelet analysis allow to emphasize the morphology of arterial vessel improving the visual diagnosis even without contrast medium introduction. The previous family of operators is defined in [5] by:…”

Approximation by Nonlinear Multivariate Sampling Kantorovich Type Operators and Applications to Image Processing

“…and A k := k 1 · k 2 · · · · · k n with k i := t k i +1 − t k i > 0, i = 1, 2, , n. The linear sampling Kantorovich type operators were first introduced in [1] in the univariate case and subsequently extended in [38] in the nonlinear univariate case. This kind of operator, such as those defined in (I), represents an averaged version in Kantorovich-sense of the generalized sampling operators introduced by Butzer and his school at Aachen in the 1980s (see, e.g., [4, 5, 8-11, 13-16, 28, 33, 36, 37]).…”