The aim of this paper is to introduce an abstract framework of certain approximation
processes using a cosine operator functions concept.
The mean-value first-order saddlepoint approximation
(MVFOSA)  is an alternative to FOSA.
Different from one-piecewise linear approximation
used in , a three-piecewise linear approximation
is presented to compute the initial seed in this paper.
The remaining part of this paper is organized as follows: we recall the basic notions of rough set, standard neutrosophic set and rough standard neutrosophic set on the crisp approximation
space, respectively, in Sections 2 and 3.
Figure 1 depicts a band-limited signal and its finite sum approximation
However, there are no contributions on the definitions of approximation
operators of general type-2 fuzzy sets in the Pawlak approximation
space and the generalized Pawlak approximation
In similar cases it is applied or piecewise approximation
, or a spline approximation
[4, 5, 6, 7].
The following approximation
theorem for a sequence of positive linear operators acting from [L.
In the first step of developing the approximation
algorithm a set of neural networks were tested based on preliminary results given by .
The next approximation
takes the rectangle to a trapezoid, reflecting the realization that the circumference of the bell at the mouth is larger than at the crown.
The more classical examples of linear positive operators throughout approximation
process are the Bernstein polynomials, which are defined by Bernstein  as following:
In comparison with the standard finite element method, the characteristics of mesh-independent and wave-based approximation
effectively make the meshfree methods attractive alternative numerical techniques for modeling the waveguide problems.