The Best Approximation of the Sinc Function by a Polynomial of Degree with the Square Norm
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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 cx sin x/x be the sinc function; the following result is known as Jordan inequality 1: π 2 ≤ sin cx < 1, 0 < x ≤ , π 2
1.1
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: 4π − 3 1 π 2 2 3 π 2 − 4x2 x − ≤ sin cx. π π 2 π3
1.2
The above inequality was generalized to an upper bound by Zhu 26: sin cx ≤
12 − π 2 2 1 π 2 3 π 2 − 4x2 x − . π π 2 π3
1.3
2
Journal of Inequalities and Applications
Later, Agarwal and his collaborators 2 proposed a more refined two-sided inequality: 4 −66 43π − 7π 2 4 124 − 83π 14π 2 2 412 − 4π 3 1− x− x − x π2 π3 π4 4 −75 49π − 8π 2 4 −142 95π − 16π 2 2 412 − 4π 3 ≤ sin cx ≤ 1 − x x − x , 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 min
pn x∈Pn
s.t.
2 sin cx − pn x dx
1.5
0
lim pn x lim sin cx,
x → 0
1/2
lim pn x lim sin cx,
x→0
x → π/2
x → π/2
where Pn is the set of the polynomial of degree n and it is denoted by
Pn pn | pn x a0 a1 x · · · an xn , 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 effe
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