91Ó°ÊÓ

The modulus of continuity of a function \(f: E \rightarrow \mathbb{R}\) is the function \(\omega(\delta)\) defined for \(\delta>0\) as follows: $$ \omega(\delta)=\sup _{\left|x_{1}-x_{2}\right|<\delta \atop x_{1}, x_{2} \in E}\left|f\left(x_{1}\right)-f\left(x_{2}\right)\right| $$ Thus, the least upper bound is taken over all pairs of points \(x_{1}, x_{2}\) of \(E\) whose distance apart is less than \(\delta .\) Show that a) the modulus of continuity is a nondecreasing nonnegative function having the limit \(^{7} \omega(+0)=\lim _{\delta \rightarrow+0} \omega(\delta)\) b) for every \(\varepsilon>0\) there exists \(\delta>0\) such that for any points \(x_{1}, x_{2} \in E\) the relation \(\left|x_{1}-x_{2}\right|<\delta\) implies \(\left|f\left(x_{1}\right)-f\left(x_{2}\right)\right|<\omega(+0)+\varepsilon\) c) if \(E\) is a closed interval, an open interval, or a half-open interval, the relation $$ \omega\left(\delta_{1}+\delta_{2}\right) \leq \omega\left(\delta_{1}\right)+\omega\left(\delta_{2}\right) $$ holds for the modulus of continuity of a function \(f: E \rightarrow \mathbb{R}\); d) the moduli of continuity of the functions \(x\) and \(\sin \left(x^{2}\right)\) on the whole real axis are respectively \(\omega(\delta)=\delta\) and the constant \(\omega(\delta)=2\) in the domain \(\delta>0\) e) a function \(f\) is uniformly continuous on \(E\) if and only if \(\omega(+0)=0\).

Let \(f\) and \(g\) be bounded functions defined on the same set \(X\). The quantity \(\Delta=\sup _{x \in X}|f(x)-g(x)|\) is called the distance between \(f\) and \(g\). It shows how well one function approximates the other on the given set \(X\). Let \(X\) be a closed interval \([a, b]\). Show that if \(f, g \in C[a, b]\), then \(\exists x_{0} \in[a, b]\), where \(\Delta=\left|f\left(x_{0}\right)-g\left(x_{0}\right)\right|\), and that such is not the case in general for arbitrary bounded functions.

Let \(P_{n}(x)\) be a polynomial of degree \(n\). We are going to approximate a bounded function \(f:[a, b] \rightarrow \mathbb{R}\) by polynomials. Let $$ \Delta\left(P_{n}\right)=\sup _{x \in[a, b]}\left|f(x)-P_{n}(x)\right| \text { and } E_{n}(f)=\inf _{P_{n}} \Delta\left(P_{n}\right) $$ where the infimum is taken over all polynomials of degree \(n .\) A polynomial \(P_{n}\) is called a polynomial of best approximation of \(f\) if \(\Delta\left(P_{n}\right)=E_{n}(f)\) Show that a) there exists a polynomial \(P_{0}(x) \equiv a_{0}\) of best approximation of degree zero; b) among the polynomials \(Q_{\lambda}(x)\) of the form \(\lambda P_{n}(x)\), where \(P_{n}\) is a fixed polynomial, there is a polynomial \(Q_{\lambda_{0}}\) such that $$ \Delta\left(Q_{\lambda_{0}}\right)=\min _{\lambda \in \mathbb{R}} \Delta\left(Q_{\lambda}\right) $$ c) if there exists a polynomial of best approximation of degree \(n\), there also exists a polynomial of best approximation of degree \(n+1\); d) for any bounded function on a closed interval and any \(n=0,1,2, \ldots\) there exists a polynomial of best approximation of degree \(n\).