View source for Deviation of an approximating function

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The distance $\rho(g,f)$ between the approximating function $g\in K$ and a given function $f\in\mathfrak M$. In one and the same class $\mathfrak M$ different metrics $\rho$ may be considered, e.g. the uniform metric

$$\rho(g,f)=\max_{a\leq x\leq b}|g(x)-f(x)|,$$

an integral metric

$$\rho(g,f)=\left(\int\limits_a^b|g(x)-f(x)|^pdx\right)^{1/p},\quad p\geq1,$$

and other metrics. As the class $K$ of approximating functions one may consider algebraic polynomials, trigonometric polynomials and also partial sums of orthogonal expansions of $f$ in an orthogonal system, linear averages of these partial sums as well as a number of other sets.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,   "Complete collected works" , '''2''' , Moscow-Leningrad  (1947)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.P. Natanson,   "Constructive function theory" , '''1–3''' , F. Ungar  (1964–1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.L. Goncharov,   "The theory of interpolation and approximation of functions" , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,   "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.M. Nikol'skii,   "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,   "Introduction to approximation theory" , Chelsea, reprint  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Schönhage,   "Approximationstheorie" , de Gruyter  (1971)</TD></TR></table>