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The minimality properties of Chebyshev polynomials and their lacunary series. (English) Zbl 1078.33007
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 $$1< p< \infty$$.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 42C15 General harmonic expansions, frames
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##### References:
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