Deviation of an approximating function

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

[1]	P.L. Chebyshev, "Complete collected works" , 2 , Moscow-Leningrad (1947) (In Russian)
[2]	I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian)
[3]	V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian)
[4]	N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
[5]	S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[a1]	E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982)
[a2]	A. Schönhage, "Approximationstheorie" , de Gruyter (1971)