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The Best Approximation of the Sinc Function by a Polynomial of Degree N with the Square Norm

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The Best Approximation of the Sinc Function by a Polynomial of Degree N with the Square Norm Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 307892, 12 pages doi:10.1155/2010/307892 Research Article The Best Approximation of the Sinc Function by a Polynomial of Degree n with the Square Norm Yuyang Qiu and Ling Zhu College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China Correspondence should be addressed to Yuyang Qiu, [email protected] Received 9 April 2010; Accepted 31 August 2010 Academic Editor: Wing-Sum Cheung Copyright q 2010 Y. Qiu and L. Zhu. This 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. The polynomial of degree n which is the best approximation of the sinc function on the interval 0, π/2 with the square norm is considered. By using Lagrange’s method of multipliers, we construct the polynomial explicitly. This method is also generalized to the continuous function on the closed interval a, b. Numerical examples are given to show the effectiveness. 1. Introduction Let sin cxsin x/x be the sinc function; the following result is known as Jordan inequality 1: 2 π ≤ sin cx < 1, 0 <x≤ , 1.1 π 2 where the left-handed equality holds if and only if x π/2. This inequality has been further refined by many scholars in the past few years 2–30. Ozban¨ 12 presented a new lower bound for the sinc function and obtained the following inequality: 2 1 4π − 3 π 2 π2 − 4x2 x − ≤ sin cx. 1.2 π π3 π3 2 The above inequality was generalized to an upper bound by Zhu 26: 2 1 12 − π2 π 2 sin cx ≤ π2 − 4x2 x − . 1.3 π π3 π3 2 2 Journal of Inequalities and Applications Later, Agarwal and his collaborators 2 proposed a more refined two-sided inequality: 4 −66 43π − 7π2 4 124 − 83π 14π2 412 − 4π 1 − x − x2 − x3 π2 π3 π4 4 −75 49π − 8π2 4 −142 95π − 16π2 412 − 4π ≤ sin cx ≤ 1 − x x2 − x3, π2 π3 π4 1.4 where the two-sided equalities hold if x tends to zero or x π/2. Note that the bounds of the sinc function sincx listed above are estimated by the given polynomials with the boundary constraints; the smaller the residual between the polynomial and the sinc function is, the more refined the estimation will be. Hence, our aim is to seek a polynomial of degree n, pn x , which is the best approximation of the sinc function with the square norm. In view of that, the sinc function is defined on 0,π/2 and two boundary constrained conditions are imposed. So we want to solve the following minimum problem: π/2 1/2 min sin cx − p x 2dx ∈P n pn x n 0 1.5 s.t. lim pnx lim sin cx, lim pnx lim sin cx, x → 0 x → 0 x → π/2 x → π/2 P where n is the set of the polynomial of degree n anditisdenotedby | ··· n ∈ Pn pn pnx a0 a1x anx ,ai R, i 1, 2,...,n 1.6 In this paper, an explicit representation for the approximating polynomial of sincx is presented by using Lagrange’s method of multipliers, and numerical examples are given to show the effectiveness. Moreover, this method can be generalized to the continuous function gx on the closed interval a, b. However, the residual error between the approximating polynomial pn x and g x is concussive, that is, it cannot keep positive or negative always. The rest of paper is organized as follows. In Section 2, we solve the problem 5 by Lagrange’s method of multipliers and this method is generalized to a continuous function on a, b in Section 3. Numerical examples are given in Section 4 to display the effectiveness of our estimations. 2. The Best Approximation of the Sinc Function by a Polynomial of Degree n on 0,π/2 Obviously, the constraints of 1.5 imply π 2 a 1,p . 2.1 0 n 2 π Journal of Inequalities and Applications 3 Note that π/2 π/2 n n − 2 2 − − i−1 i sin c x pn x dx sin cx 1 2sin cx 2 aix sin x 2 aix 0 0 i1 i1 ⎞ n n ij 2 2i⎠ 2 aiaj x ai x dx 1≤i<j≤n i1 π/2 n 2 − − i−1 sin cx 1 2sin cx 2 aix sin x dx 0 i1 n i1 2a a ij1 2ai π i j π i1 i 1 2 1≤i<j≤n i j 1 2 n a2 2i1 i π . i1 2i 1 2 2.2 Denote n 2 2aiaj π i j 1 a π 2i 1 i G a1,...,an h 2.3 1≤i<j≤n i j 1 2 i1 2i 1 2 with π/2 n n i1 − i−1 2ai π h 2 aix sin x dx , 2.4 0 i1 i1 i 1 2 ∈R where ai ,i 1, 2,...,n.So 1.5 is equivalent to solving the following minimum problem: min Ga1,...,an ai∈R 2.5 π π n 2 s.t.a ··· a − 1. 1 2 n 2 π This can be solved by using Lagrange’s method of multipliers. We construct the Lagrange function by π π n 2 La ,a ,...,a ,λ Ga ,...,a λ a ··· a − 1 2.6 1 2 n 1 n 1 2 n 2 π 4 Journal of Inequalities and Applications with Lagragian multiplier λ. Thus we need to equate to zero the partial derivatives of L with respect to each aj j 1, 2,...,n and λ,thatis, ∂L 0,j 1,...,n, ∂aj 2.7 π π n 2 a ··· a − 1 0. 1 2 n 2 π It gives a system of linear equations Au f, 2.8 where ⎛ ⎞ 2 π 3 2 π 4 2 π n2 π ⎜ ... ⎟ ⎜ 3 2 4 2 n 2 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 4 5 n3 2⎟ ⎜ 2 π 2 π 2 π π ⎟ ⎜ ... ⎟ ⎜ 4 2 5 2 n 3 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 5 6 n4 3⎟ ⎜ 2 π 2 π 2 π π ⎟ ⎜ ... ⎟ ⎜ 5 2 6 2 n 4 2 2 ⎟ ⎜ ⎟ A ⎜ ⎟, 2.9 ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ . ... ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 π n 2 2 π n 3 2 π 2n 1 π n⎟ ⎜ ... ⎟ ⎜n 2 2 n 3 2 2n 1 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ π π 2 π n ... 0 2 2 2 ⎛ ⎞ ∂h ⎜ ⎟ ⎜ ⎟ ⎜ ∂a1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ ⎜ ∂h ⎟ ⎜ ⎟ a1 ⎜ ⎟ ⎜ ⎟ ⎜ ∂a2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜a2⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ . ⎟ u ⎜ . ⎟,f −⎜ . ⎟. 2.10 ⎜ . ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝an⎠ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ λ ⎜ ∂h ⎟ ⎜ ⎟ ⎜ ∂an ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 2 1 − π Journal of Inequalities and Applications 5 To consider the consistence of the equations 2.8, we introduce the following lemma for the square matrix A of order n 1. Lemma 2.1. The square matrix A of order n 1 defined by 2.9 is nonsingular. Proof. We want to prove that detA / 0. Note that ⎛ ⎞ 2 2 π 1 2 π n−1 π −2 ⎜ ... ⎟ ⎜ 3 4 2 n 2 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 n−1 −2⎟ ⎜ 2 2 π 2 π π ⎟ ⎜ ... ⎟ ⎜ 4 5 2 n 3 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 π 1 2 π n−1 π −2⎟ ⎜ ⎟ ⎜ ... ⎟ ··· ⎜ 5 6 2 n 4 2 2 ⎟ π 3 4 n 2 1 ⎜ ⎟ detA det⎜ ⎟ ⎜ . ⎟ 2 ⎜ . ⎟ ⎜ . ... ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ − − ⎟ ⎜ 2 2 π 1 2 π n 1 π 2⎟ ⎜ ... ⎟ ⎜n 2 n 3 2 2n 1 2 2 ⎟ ⎜ ⎟ ⎝ ⎠ π 1 π n−1 1 ... 0 2 2 ⎛ ⎞ 2.11 2 2 2 ⎜ ... 1⎟ ⎜ 3 4 n 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 2 ⎟ ⎜ ... 1⎟ ⎜ 4 5 n 3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 2 ⎟ n5n/2n−1n/2−1 ⎜ ... 1⎟ π ⎜ 5 6 n 4 ⎟ det⎜ ⎟ 2 ⎜ ⎟ ⎜ ⎟ ⎜ . .⎟ ⎜ . ... .⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 2 ⎟ ⎜ ... 1⎟ ⎜n 2 n 3 2n 1 ⎟ ⎝ ⎠ 11... 10 π nn2−1 2H e det n , 2 eT 0 6 Journal of Inequalities and Applications where ⎛ ⎞ 1 1 1 ⎜ ... ⎟ ⎜ 3 4 n 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ ⎜ 1 1 1 ⎟ ⎜ ... ⎟ 1 ⎜ 4 5 n 3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜1⎟ ⎜ 1 1 1 ⎟ ⎜ ⎟ H ⎜ ⎟,e ⎜1⎟. 2.12 n ⎜ ... ⎟ ⎜ ⎟ ⎜ 5 6 n 4 ⎟ ⎝.⎠ ⎜ ⎟ . ⎜ ⎟ ⎜ . ⎟ 1 ⎜ . ⎟ ⎜ . ... ⎟ ⎜ ⎟ ⎝ ⎠ 1 1 1 ... n 2 n 3 2n 1 Since Hn is one n-order principal square submatrix of n 2 -order Hilbert matrix, together with Hilbert matrix being positive definite 31, volume 1, page 401 , then Hn is also positive −1 T −1 definite. Hence, Hn exists and it is positive definite, which implies e Hn e / 0. Moreover, ⎛ ⎞ 2Hn e 2Hn e ⎝ ⎠ det det 1 − . 2.13 eT 0 0 − eT H 1e 2 n 2Hn e So, det eT 0 / 0, that is, A is nonsingular. Because A is nonsingular, the solution of the equations 2.8 exists and is unique, as well as the best approximation of sin cx by a polynomial of degree n. Therefore, we obtain the following theorem. Theorem 2.2. Let 0 <x≤ π/2; suppose the matrix A and vector f are denoted by 2.9. Then the best approximation of sin cx by a polynomial of degree n on interval 0,π/2 with the square norm is given by ··· n−1 n pnx 1 a1x an−1x anx , 2.14 −1 where a1,...,an is the 1,2, ...n-th components of the vector A f. 3. The Best Approximation of the Continuous Function gx by a Polynomial of Degree n on a, b In this section, we generalize the above conclusion to the continuous function gx on interval a, b, that is, we want to consider the following minimum problem: b 1/2 min gx − p x 2dx 3.1 ∈P n pn x n a Journal of Inequalities and Applications 7 with the constraints pna ga,pnb gb, 3.2 where the polynomial pn x is rewritten as − ··· − n pnx a0 a1x a anx a 3.3 P − and n is defined by 1.6 .Ifwesett x a, problem 3.1 is equivalent to b−a 1/2 min gt a − p t 2dt 3.4 ∈P n pn t n 0 with − a0 ga, pnb a gb, 3.5 where ··· n pnt a0 a1t ant .
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