Nonlinear Integral Operators and Applications

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

“…Moreover, we denote by I ϕ : M (IR n ) → [0, +∞] the corresponding modular functional associated to L ϕ (IR n ) (see e.g. [4,9,12]), where M (IR n ) denotes the space of all Lebesgue-measurable functions. We can obtain what follows.…”

“…In order to weaken the above assumptions on the signal being reconstructed, the German mathematician P.L. Butzer introduced the generalized sampling operators, see e.g., [3,4]. These operators revealed to be very suitable to approximate (in some sense) continuous signals f by means of sampling series involving sample values f (k/w), k ∈ Z Z, w > 0.…”

Fatou type weighted pointwise convergence of nonlinear singular integral operators Depending on two parameters

“…In were published until that time, showed that the singularity of the operators was related to their linearity [11]. Afterwards, Swiderski and Wachnicki [12] investigated the pointwise convergence of the operators of the preceding type at Lebesgue points of the functions…”

“…Especially, effect of nonlinear integral operators in sampling theory must be emphasized here [11]. Further, signal and image processing are two major research fields around sampling theory.…”

A new approach to nonlinear singular integral operators depending on three parameters

“…After this important study, Swiderski and Wachnicki [16] investigated the pointwise convergence of the operators of type (1) at p Lebesgue points of functions f 2 L p . ; / .1 Ä p < 1/ : For further results concerning the convergence of several types of nonlinear singular integral operators in different function spaces, the studies [17,18] are strongly recommended.…”