On direct and converse theorems of the approximation theory in the metric of a sign sensitive weight
Notations and Definitions. A pair \(p_+(x)\) and \(p_-(x)\) of continuous functions on \(\Delta= [a,b]\subset (-\infty,\infty)\) is called a continuous sign sensitive weight \((p_-(x), p_+(x))\) denoted by \(p(x)\). \(p(x)\) is regular if \(p_+(x)p_-(x)> 0\) and monotonic if \(p_+(x)\) and \(p_-(x)\) are monotonic. As usual \(x^+= x\) if \(x>0\) and \(x^+= 0\) otherwise. Similarly for \(x^-\). The following notations are also used with continuous functions \(f\) in connection with sign sensitive weights. \(|(f, p)(x)|= f^+(x) p_+(x)+ f^-(x) p_-(x)\), \(|f|_p=|f|_{p,\Delta}=|f|_{(p_-,p_+)}= \sup\{|(f,p)|: x\in\Delta\}\). For \(2\pi\)-periodic functions and weights the supremum is taken on any interval of length \(2\pi\). Also \(|f|= |f|_\Delta= \max\{|f(x)|: x\in\Delta\}\). \(v(f)= v(f,\Delta)\) is the total variation of \(f\) on \(\Delta\) and \(\Omega(f)= \Omega(f,\Delta)\) is its oscillation on \(\Delta\). The moduli of continuity used in various theorems are: \[ \omega(f,\delta)= \sup\{|(f_h- f)(x)|: x,x+ h\in\Delta, |h|\leq\delta\},\;f_h(x)= f(x+ h), \] \[ \omega(\delta, f,p)= \sup\{|(f_h- f,p)(x)|: x,x+h\in \Delta, |h|\leq\delta\}, \] \[ \omega_+(\delta, f)_p= \sup\biggl\{\{\sup|(f_h(y)- f(y),p(x))|: y,y+ h\in[x, b]\}: x\in\Delta, |h|\leq\delta\biggr\}. \] Similarly \(\omega_-(\delta, f)_p\) is defined by changing \([x,b]\) to \([a,x]\). If in the definition of \(\omega(\delta, f,p)\) instead \(|h|\leq\delta\) one takes \(0\leq h\leq\delta\), \(-\delta\leq h\leq 0\), the definitions of \(\omega_+(\delta, f,p)\) and \(\omega_-(\delta, f,p)\) are obtained. In case of \(2\pi\)-periodic functions and weights in the definition of the moduli of continuity instead of \(x,x+ h[0,2\pi]\) one can take also \(x,x+ h\in[-\infty,\infty]\). The results can be different but the notations remain the same. In the first section of the paper, the author compares the four basic known properties of the usual uniform modulus of continuity \(\omega(f,\delta)\) to the other generalized forms which all coincide with it in the case \(p_+(x)\equiv p_-(x)\equiv 1\). Section two deals with approximation by polynomials. Here the main definitions are: \(E_n(f,p)= E_n(f,p_-,p_+)= \inf|T- f|_p\) is the best approximation of a \(2\pi\)-periodic function \(f(x)\) by trigonometric polynomials \(T(x)\) of degree not exceeding \(n\). In case of algebraic polynomials the best approximation is denoted by \(E_n(f, p,\Delta)\). The following three theorems are the main results of section 2. Theorem 1. For arbitrary continuous \(2\pi\)-periodic functions \(f(x)\) and weight \(p(x)= (p_-(x),p_+(x))\), \(n= 1,2,\dots\), the inequality \[ E_n(f, p)\leq 14\omega\Biggl({1\over n+1}, f,p\Biggr)+ 31\omega\Biggl(f,{1\over n+1}\Biggr) \omega\Biggl(p,{1\over n+1}\Biggr). \] Theorem 2. Constants \(C\), \(C_1\) which satisfy the inequality \[ E_n(f, p,[a, b])\leq C_1\omega\Biggl(C{b- a\over n}, f,p\Biggr),\quad n=1,2,\dots \] for all continuous \(f(x)\) on \([a,b]\) and some weight \(p(x)\) on that interval, exist if and only if the weight \(p(x)\) is monotonic. Theorem 3. For an arbitrary \(2\pi\)-periodic weight \(p(x)= (p_-(x),p_+(x))\) and an arbitrary trigonometric polynomial \(T(x)\) of degree \(n\), \(n=0,1,2,\dots\), the inequality \[ |T'|_p\leq n\max\{|T|_p, |-T|_p\}+ B|T|, \] where \(B= \min\{n\Omega(p), A\omega(p, \pi/n)\}\), \(\Omega(p)= \max\{\Omega(p_+, [0,2\pi])\), \(\Omega(p_-,[0, 2\pi])\}\), \[ A= 2\sum^n_{m= 1} mR_m\leq (2/\pi^2)n\ln n+ 2n, \] \(R_m= {1\over 4n}\sin^{-2}(x_m/2)\), \(x_m= {2m-1\over 2n}\pi\) \((m=1,2,\dots, 2n)\). Section 3 deals with best rational approximations \(R_n(f, p)=\inf|r-f|_{p,\Delta}\), \(R_n(f)= \inf|r- f|_\Delta\), where \(r\) runs through the set of rational functions with real coefficients of degree not exceeding \(n\), \(n=0,1,2,\dots\)\ . Estimates from above of the best rational approximation can be obtained from the inequality \(R_n(f, p)\leq E_n(f, p)\). The author thus treats in detail an analogue to the inverse problem of estimating the variation of a continuous function \(f(x)\) on an interval \(\Delta\) by its best uniform rational approximation \(R_n(f)\) in the spirit of \textit{E. P. Dolzhenko} [Mat. Sb., Nov. Ser. 56(98), 403-432 (1962; Zbl 0115.05701)], namely \(v(f,\Delta)\leq 12\sum^\infty_{n= 0} R_n(f)\). The author explains that a direct analogous result is impossible even for a monotonic weight. However for a wide class of weight functions which satisfy a certain condition, which the author calls the \(D\)-condition, he is able to obtain an estimate of the variation of the product of the function \(f(x)\) and its two weight functions \(p_-(x)\), \(p_+(x)\) defining a given weight. The details are too lengthy to be included in this review.
scientific article; zbMATH DE number 817240
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