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Best L1 Approximation of Heaviside-Type Functions in a Weak-Chebyshev Space

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Best L1 Approximation of Heaviside-Type Functions in a Weak-Chebyshev Space Best L1 approximation of Heaviside-type functions in Chebyshev and weak-Chebyshev spaces Laurent Gajny,1 Olivier Gibaru1,,2 Eric Nyiri1, Shu-Cherng Fang3 Abstract In this article, we study the problem of best L1 approximation of Heaviside-type functions in Chebyshev and weak-Chebyshev spaces. We extend the Hobby-Rice theorem [2] into an appro- priate framework and prove the unicity of best L1 approximation of Heaviside-type functions in an even-dimensional Chebyshev space under the condition that the dimension of the subspace composed of the even functions is half the dimension of the whole space. We also apply the results to compute best L1 approximations of Heaviside-type functions by polynomials and Her- mite polynomial splines with fixed knots. Keywords : Best approximation, L1 norm, Heaviside function, polynomials, polynomial splines, Chebyshev space, weak-Chebyshev space. Introduction Let [a, b] be a real interval with a < b and ν be a positive measure defined on a σ-field of subsets of [a, b]. The space L1([a, b],ν) of ν-integrable functions with the so-called L1 norm : b k · k1 : f ∈ L1([a, b],ν) 7−→ kfk1 = |f(x)| dν(x) Za forms a Banach space. Let f ∈ L1([a, b],ν) and Y be a subset of L1([a, b],ν) such that f∈ / Y where Y is the closure of Y . The problem of the best L1 approximation of the function f in Y consists in finding a function g∗ ∈ Y such that : ∗ arXiv:1407.0228v3 [math.FA] 26 Oct 2015 kf − g k1 ≤ kf − gk1, ∀g ∈ Y. (1) If Y is a compact set or a finite dimensional subspace of L1([a, b],ν), then such a function exists for any f ∈ L1([a, b],ν). In this article, we focus on the finite-dimensional case. An important characterization theorem when ν is a non-atomic measure (See for example [3, 7] and a development in [15]) claims that finding a best L1 approximation of a function f remains to define a sign-changes function s (i.e. a function for which values alternate between -1 and 1 at certain abscissae) such that : b s(x)g(x) dν(x) = 0, ∀g ∈ Y. Za 1Arts et M´etiers ParisTech, LSIS - UMR CNRS 7296, 8 Boulevard Louis XIV, 59046 Lille Cedex, France 2INRIA Lille-Nord-Europe, NON-A research team, 40, avenue Halley 59650 Villeneuve d’Ascq 3North Carolina State University, Edward P. Fitts Department of Industrial and Systems Engineering, NC 27695- 7906, United States of America. 1 The Hobby-Rice theorem [2] enables to show the existence of such functions. This major result helps characterize and compute solutions of best L1 approximation of a continuous function in Chebyshev and weak-Chebyshev spaces. It has been widely studied in the litterature [4, 8, 12, 16, 20, 21]. We recall that an n-dimensional Chebyshev space (resp. weak-Chebyshev space) on [a, b] is a subspace of C0[a, b] composed of functions which have at most n − 1 distinct zeros on [a, b] (resp. strong changes of sign) [5, 19]. Classical examples of such spaces are respectively polynomial functions and polynomial spline functions with fixed knots. To the best of our knowledge, there are few references in the litterature that deal with the challenging problem of best L1 approximation of discontinuous functions, especially Heaviside-type functions. Definition 1. We say that f is a Heaviside-type function on [a, b] with a jump at δ ∈ (a, b) if f ∈ C0([a, b]\{δ}) and if the limits of the function |f| on both sides of δ exist and are finite. We denote by Jδ[a, b] the vector space of such functions. Existing papers focusing on one or two-sided best L1 approximation deal with specific spaces. Bustamante et al. have shown that best polynomial one-sided L1 approximation of the Heaviside function can be obtained by Hermite interpolation at zeros of some Jacobi polynomials [1]. Moskona et al. have studied the problem of best two-sided Lp (p ≥ 1) approximation of the Heaviside function using trigonometrical polynomials [9]. Saff and Tashev have done a similar work using polygonal lines [18]. We propose in this article a more general framework for best (two-sided) L1 approximation of Heaviside-type functions in Chebyshev and weak-Chebyshev spaces which includes the two previous cases. This problem is presented in Section 1. We evidence the encountered difficulties that lead us to extend the classical Hobby-Rice theorem. This extension is presented in details in Section 2. In the third section, we strongly use this result to characterize best L1 approximations of Heaviside-type functions in a Chebyshev space. We give sufficient conditions on the Chebyshev space to obtain a unique L1 best approximation for every Heaviside-type function. We apply these results to polynomial approximation. In Section 4, we study the problem of best L1 approximation of Heaviside-type functions in a weak-Chebyshev space. In particular, this theory is applied to the Hermite polynomial spline case in Section 4. 1 Best L1 approximation In this section, we recall an important characterization theorem of best L1 approximation (See for example [3, 7]) and the Hobby-Rice theorem. We evidence the reason why the Hobby-Rice theorem fails to give a more precise characterization of best L1 approximations of Heaviside-type functions. For convenience, we define the zero set Z(f)= {x ∈ [a, b], f(x) = 0} and the sign function : 1 if x> 0, sign : x ∈ R 7−→ sign(x)= 0 if x = 0, −1 if x< 0. ∗ Theorem 1.1. Let Y be a subspace of L1([a, b],ν) and f ∈ L1([a, b],ν)\Y . Then g is a best L1 approximation of f in Y if and only if : b sign(f(x) − g∗(x))g(x) dν(x) ≤ |g(x)| dν(x), for all g ∈ Y. (2) Z Z ∗ a Z(f−g ) ∗ ∗ In particular, if ν(Z(f − g )) = 0, then g is a best L1 approximation of f in Y if and only if : b sign(f(x) − g∗(x))g(x) dν(x) = 0, (3) Za 2 for all g ∈ Y . The second part of the latter theorem claims that finding a best L1 approximation of a function f may remain to find a sign-changes function s, that alternates between −1 and 1 at certain abscissae, such that: b s(x)g(x) dν(x) = 0, ∀g ∈ Y. (4) Za If there exists a function g∗ ∈ Y which interpolates f at these abscissae and no others, then g∗ is a best L1 approximation of f in Y . Thus, the non-linear problem of best L1 approximation remains to a simple Lagrange interpolation problem. Actually, based on a result by Phelps (Lemma 2 in [13]), it can be shown that if ν is a non-atomic ∗ measure then g is a best L1 approximation of f in Y if and only if there exists a sign changes function satisfying (4) and which coincides ν-almost everywhere with sgn(f −g∗) on [a, b]\Z(f −g∗) (see also Theorem 2.3 in [15]). From now, we consider in this article that we are in this case. Thus, either if a best L1 approximation coincides or not with its reference function on a set of non-zero measure, the determination of a sign-changes function verifying (4) is fundamental. Indeed, it characterizes a best L1 approximation in both cases. The Hobby-Rice theorem [2] enables to show the existence of such a sign-changes function with the number of sign-changes lower or equal to the dimension of Y . Theorem 1.2 (Hobby, Rice, 1965). Let Y be an n-dimensional subspace of L1([a, b],ν) where ν is a finite, non-atomic measure. Then there exists a sign-changes function : r+1 i s(x)= (−1) 1[αi−1,αi[(x), (5) Xi=1 such that for all g ∈ Y : b s(x)g(x) dν(x) = 0, (6) Za where r ≤ n and : a = α0 < α1 < α2 < · · · < αr < αr+1 = b. Remark. A simple proof of this theorem can be found in [14] using the Borsuk antipodality theorem (See for example [10], page 21). One cannot guarantee in general that a sign function with r ≤ n changes of sign as stated in the Hobby-Rice theorem enables us to define a function g∗ that coincides with the function to approximate f at these abscissae and no others. In Figure 1, we can observe a best L1 approximation of a continuous function, x 7→ sin(3x), in a 4-dimensional space - the space of cubic functions on the interval [−1, 1] - having five intersections with the above function. The following corollary partially explains this observation. Corollary 1. Let Y be an n-dimensional subspace of L1([a, b],ν) where ν is a finite, non-atomic measure and m be an integer such that m ≥ n. Then there exist a sign-changes function : rm+1 i s(x)= (−1) 1[αi−1,αi[(x), (7) Xi=1 3 1.5 1 0.5 0 -0.5 -1 -1.5α α α α α α α 0 1 2 3 4 5 6 Figure 1: Best L1 cubic approximation (solid line) of x 7→ sin(3x) (dashed line) on [−1, 1]. such that for all g ∈ Y : b s(x)g(x) dν(x) = 0, (8) Za where rm ≤ m : a = α0 < α1 < α2 < · · · < αrm < αrm+1 = b. Proof. Complete Y in a m-dimensional subspace of L1([a, b],ν).
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