Cardinal Basis Piecewise Hermite Interpolation on Fuzzy Data

A numericalmethod alongwith explicit construction to interpolation of fuzzy data through the extension principle results bywidely used fuzzy-valued piecewise Hermite polynomial in general case based on the cardinal basis functions, which satisfy a vanishing property on the successive intervals, has been introduced here. We have provided a numerical method in full detail using the linear space notions for calculating the presented method. In order to illustrate the method in computational examples, we take recourse to three prime cases: linear, cubic, and quintic.


Introduction
Fuzzy interpolation problem was posed by Zadeh [1].Lowen presented a solution to this problem, based on the fundamental polynomial interpolation theorem of Lagrange (see, e.g., [2]).Computational and numerical methods for calculating the fuzzy Lagrange interpolate were proposed by Kaleva [3].He introduced an interpolating fuzzy spline of order .Important special cases were  = 2, the piecewise linear interpolant, and  = 4, a fuzzy cubic spline.Moreover, Kaleva obtained an interpolating fuzzy cubic spline with the nota-knot condition.Interpolating of fuzzy data was developed to simple Hermite or osculatory interpolation, (3) cubic splines, fuzzy splines, complete splines, and natural splines, respectively, in [4][5][6][7][8] by Abbasbandy et al.Later, Lodwick and Santos presented the Lagrange fuzzy interpolating function that loses smoothness at the knots at every -cut; also every -cut ( ̸ = 1) of fuzzy spline with the not--knot boundary conditions of order  has discontinuous first derivatives on the knots and based on these interpolants some fuzzy surfaces were constructed [9].Zeinali et al. [10] presented a method of interpolation of fuzzy data by Hermite and piecewise cubic Hermite that was simpler and consistent and also inherited smoothness properties of the generator interpolation.However, probably due to the switching points difficulties, the method was expressed in a very special case and none of three remaining important cases was not investigated and this is a fundamental reason for the method weakness.
In total, low order versions of piecewise Hermite interpolation are widely used and when we take more knots, the error breaks down uniformly to zero.Using piecewisepolynomial interpolants instead of high order polynomial interpolants on the same material and spaced knots is a useful way to diminish the wiggling and to improve the interpolation.These facts, as well as cardinal basis functions perspective, motivated us in [11] to patch cubic Hermite polynomials together to construct piecewise cubic fuzzy Hermite polynomial and provide an explicit formula in a succinct algorithm to calculate the fuzzy interpolant in cubic case as a new replacement method for [4,10].Now, in this paper, in light of our previous work, we want to introduce a wide general class of fuzzy-valued interpolation polynomials by extending the same approach in [11] applying a very special case of which general class of fuzzy polynomials could be an alternative to fuzzy osculatory interpolation in [4] and so its lowest order case ( = 1), namely, the piecewise linear polynomial, is an analogy of fuzzy linear spline in [3].Meanwhile, when  = 2 with exactly the same data, we will simply produce the second lower order form of mentioned general class that was introduced in [11] and the interpolation of fuzzy data in [10].
The paper is organized in five sections.In Section 2, we have reviewed definitions and preliminary results of several 2 Advances in Fuzzy Systems basic concepts and findings; next, we construct piecewise fuzzy Hermite polynomial in detail based on cardinal basis functions and prove some new properties of the introduced general interpolant (Section 3).In Section 4, we have produced three initial, linear, cubic [11], and quintic cases and shown the relationship between some of the mentioned cases and the newly presented interpolants in [3,4,10].Furthermore, to illustrate the method, some computational examples are provided.Finally, the conclusions of this interpolation are in Section 5.

Preliminaries
To begin, let us introduce some brief account of notions used throughout the paper.We shall denote the set of fuzzy numbers by R F the family of all nonempty convex, normal, upper semicontinuous, and compactly supported fuzzy subsets defined on the real axis R.
The linear space of all polynomials of degree at most  will be designated by   .Full Hermite interpolation problem defines a unique polynomial, called   (), which solves the following problem.
(iii) The sign of all other elements of B is not negative on .
The derivative has two (−2)th order zeros at   and  +1 and its two other zeros are  −1 ,  +2 .Then, it has at least 2 − 1 zeros on the interval [ −1 ,  +2 ], which is a contradiction.The cases  = 0 and  =  − 1 are treated similarly.
A similar proof via definition of basis functions and ( 6) follows the claim (iii).

Piecewise-Polynomial Linear, Cubic, and Quintic Fuzzy Hermite Interpolation
We consider  = 1 and compute the piecewise fuzzy linear interpolant as the initial case of the presented method based on ( 12) and for a given set of fuzzy data {(  ,   ) |   ∈ R F , 0 ≤  ≤ }, as follows: where  0 =   , 0 ≤  ≤ , and subject to conditions (6), The obtained () is the same as fuzzy spline of order  = 2 that had been introduced in [3] because the basic splines and the cardinal basis functions in two interpolants are equal.
In Figure 1, the dashed line is the 0.5-cut set of piecewise cubic fuzzy interpolation (),  ∈ [1,4] and the solid lines represent the support and the core of ().When  = 2, we get the piecewise cubic fuzzy Hermite polynomial interpolant in [11] where   =  ()  ,  = 0, 1, 0 ≤  ≤ .An outstanding feature of this study is that, by simply applying the second case of the presented general method and exactly the same data, we have produced an alternative to simple fuzzy Hermite polynomial interpolation in [4].Heretofore, the mentioned cubic case (23) was independently introduced in [10] but only in very weak conditions and without using the extension principle.