Research Article Cardinal Basis Piecewise Hermite Interpolation on Fuzzy Data H. Vosoughi and S. Abbasbandy Department of Mathematics, Islamic Azad University, Science and Research Branch, Tehran, Iran Correspondence should be addressed to S. Abbasbandy; abbasbandy@yahoo.com Received 6 June 2016; Accepted 7 December 2016 Academic Editor: Katsuhiro Honda Copyright © 2016 H. Vosoughi and S. Abbasbandy. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A numerical method along with explicit construction to interpolation of fuzzy data through the extension principle results by widely 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. 1. Introduction Fuzzy interpolation problem was posed by Zadeh [1]. Lowen presented a solution to this problem, based on the fundamen- tal 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 not- a-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 [4–8] 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 frst 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 difculties, 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 inter- polation are widely used and when we take more knots, the error breaks down uniformly to zero. Using piecewise- polynomial 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. Tese 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]. Te paper is organized in fve sections. In Section 2, we have reviewed defnitions and preliminary results of several Hindawi Publishing Corporation Advances in Fuzzy Systems Volume 2016, Article ID 8127215, 8 pages http://dx.doi.org/10.1155/2016/8127215