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Strong Inequalities for Hermite-Fejér Interpolations and Characterization of -Functionals

DOI: 10.1155/2014/781068

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The works of Smale and Zhou (2003, 2007), Cucker and Smale (2002), and Cucker and Zhou (2007) indicate that approximation operators serve as cores of many machine learning algorithms. In this paper we study the Hermite-Fejér interpolation operator which has this potential of applications. The interpolation is defined by zeros of the Jacobi polynomials with parameters , . Approximation rate is obtained for continuous functions. Asymptotic expression of the -functional associated with the interpolation operators is given. 1. Introduction Zhou and Jetter [1] used Bernstein-Durrmeyer operators for studying support vector machine classification algorithms. This work initiates the direction of applying more linear operators from approximation theory to learning theory. We will follow this direction and study Hermite-Fejér interpolation operator. It would be interesting to derive explicit learning rates by means of these operators for some specific learning algorithms. Let denote the Jacobi polynomial of order . Let be the zeros of . We assume that . For any continuous function on , the Hermite-Fejér interpolation is a polynomial of order that satisfies for any . Let be the norm on ( , ). Without introducing ambiguity we also use to denote the norm on (which is the totality of the continuous functions on with period , and in this case ). One has (see e.g., [2]) if and only if . Define Denote by the conjugate function of , and write . When , , one has (see [3, 4]) So when , , is saturation with and the saturation class is Note that for all , the associated classes are identical ([4, Theorem??6]). See [2, 5–7] for related works. Denote that and recursively with . Write . Define For any , one has [8] Let and . We use the following definition of -functional from [9]: We cite the following three Theorems from [9]. Theorem 1. Let , be fixed. Then there is a constant such that for all and all , one has in which the symbol does not rely on and . Theorem 2. Let , be fixed. Then the following relation holds: Theorem 3. Let , be fixed. Then there is a constant such that, for all and all , one has Here the symbol does not rely on and . The -functional for all , is characterized by the following Theorem 4. Let , and , be fixed. Then, for all and all , the following holds: in which Theorems 3 and 4 give Moreover, if , , then, due to (see (2.6) of [9]), we have Theorem 4 will be proved in Section 3. In Section 2 we will discuss some properties of Jacobi polynomials and make some remarks concerning the conjugate function . 2. Estimates for Jacobi Polynomials and Conjugate


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