The minimality properties of Chebyshev polynomials and their lacunary series
Let \(T_n, U_n, V_n\) and \(W_n\) be four kinds of Chebyshey polynomials \[ \begin{alignedat}{2} T_n(x)&= \text{cos} (n\theta),&\qquad U_n(x)&={{\sin ((n+1)\theta)}\over {\sin\theta}},\\ V_n(x)&={{\cos ((n+{1\over 2})\theta)}\over{\cos\left({\theta\over 2}\right)}},&\quad W_n(x)&={{\sin \left(\left(n+{1\over 2}\right)\theta\right)}\over{\sin\left( {\theta\over 2}\right)}} \end{alignedat} \] where \(x=\cos\theta\), \(0\leq \theta\leq\pi\). Define weight functions \(w_p(x)\), \(-1\leq x\leq 1\), for these polynomials as follows: if \(1\leq p<\infty\), then \[ w_p(x)=\begin{cases} (1-x^2)^{-{1\over 2}} &\text{ for }T_n,\\ (1-x^2)^{{p-1}\over 2} &\text{ for } U_n,\\ (1+x)^{{p-1}\over 2} (1-x)^{-{1\over 2}} &\text{ for } V_n,\\ (1-x)^{{p-1}\over 2} (1+x)^{-{1\over 2}}&\text{ for } W_n \end{cases} \] and if \(p=\infty\) \[ w_\infty(x)=\lim_{p\to +\infty} (w_p(x))^{{1\over p}}. \] The author proves the following theorems that extend a number of known results on Chebyshev polynomials. Chebyshev's equioscillation theorem: Let \(p_n\) be a polynomial, \(\deg p=n\), and \(f\) be continuous. For four cases of \(w_{\infty}(x)\), the norm \[ \| f-p_n\|=\max_{-1\leq x\leq 1} |w_\infty(x) (f(x)-p_n(x))| \] is minimised if and only if \(w_\infty(f-p_n)\) attains its maximum magnitude with alternating signs on at least \(n+2\) consecutive points of \([-1, 1]\). The \(L_p\) minimality property: The monic polynomials which are corresponding \(T_n, U_n, V_n, W_n\) are the best \(L_p\) approximations to zero on \([-1, 1]\) with respect to their \(w_p(x)\), \(1\leq p\leq \infty\). A sufficient interpolation condition for weight \(L_1\) approximation: Let \(f\) be a continuous function on \([-1, 1]\). A polynomial \(p_n\), \(\deg p_n=n\), is the best \(L_1\) approximation of \(f\) with respect to \(w_1(x)\) if zeros of \(f-p_n\) coincide with zeros of the relevant \(n\)-th Chebyshev polynomial corresponding to \(w_1(x)\). The best \(L_p\) approximation by partial sums of lacunary series: Let \(S_n^r\) \((r=1, 2, 3, 4)\) denote the sum of the first \(n\) terms of the respective series: \[ \begin{aligned} \text{(i)}\quad S^1(x)\sim\sum_{k=1}^\infty a^k T_{b^k}(x), &\qquad \text{ (ii)} \quad S^2(x)\sim \sum_{k=1}^\infty a^k U_{b^k -1} (x),\\ \text{(iii)} \quad S^3(x)\sim \sum_{k=1}^\infty a^k V_{{1\over 2} (b^k -1)}(x), &\qquad \text{ (iv)} \;\;S^4(x)\sim \sum_{k=1}^\infty a^k W_{{1\over 2} (b^k-1)}(x). \end{aligned} \] Then \(S_n^r\) are the minimax approximation to \(S^r (r=1, 2, 3, 4)\) with respect to the corresponding \(w_\infty(x)\), given that \(a\) is real, \(b\) is an odd integer, and \(|a b|\leq 1\). The variants of the last theorem are also proved for \(p=1\) and \(1< p< \infty\).
best and near-best polynomial approximation
scientific article; zbMATH DE number 2161187