The best approximation of the sinc function by a polynomial of degree \(n\) with the square norm
For the sinc function \(\text{sinc}(x)=\frac{\sin(x)}{x}\) the authors look for a polynomial \(p(x)\in{\mathcal P}_n\) (the set of polynomials of degree \(n\)) such that \((\int_0^{\pi/2} (\text{sinc}(x)-p(x))^2\,dx)^{1/2}\) is minimal. This problem is solved by using Lagrange's method of multipliers and the coefficients of \(p(x)\) are obtained explicitly. Also the general problem \[ \min_{p\in{\mathcal P}_n} \bigg(\int^b_a\big(g(x)-p(x)\big)^2\,dx\bigg)^{\frac{1}{2}}, \] where \(g\) is a continuous function on \([a,b]\), is solved by the same method. For \(n=2,3,4,5\) the residual error and the integration error for \(sinc(x)\) are given. The general problem is solved for \(g(x)=\cos x\), \(x\in[0,\pi]\) and \(g(x)=\sin x\), \(x\in[\frac{\pi}{2},\pi]\), by giving the polynomials \(p(x)\) in case \(n=3\) and \(n=4\).
scientific article; zbMATH DE number 5823159
scientific article