# Markov criterion

for best integral approximation

A theorem which in some cases enables one to give effectively the polynomial and the error of best integral approximation of a function $f$. It was established by A.A. Markov in 1898 (see [1]). Let $\{ \phi _ {k} ( x) \}$, $k = 1 \dots n$, be a system of linearly independent functions continuous on the interval $[ a , b ]$, and let the continuous function $\psi$ change sign at the points $x _ {1} < \dots < x _ {r}$ in $( a , b )$ and be such that

$$\int\limits _ { a } ^ { b } \phi _ {k} ( x) \mathop{\rm sgn} \psi ( x) d x = 0 ,\ \ k = 1 \dots n .$$

If the polynomial

$$P _ {n} ^ {*} ( x) = \ \sum _ { k= 1} ^ { n } c _ {k} ^ {*} \phi _ {k} ( x)$$

has the property that the difference $f - P _ {n} ^ {*}$ changes sign at the points $x _ {1} \dots x _ {r}$, and only at those points, then $P _ {n} ^ {*}$ is the polynomial of best integral approximation to $f$ and

$$\inf _ {\{ c _ {k} \} } \ \int\limits _ { a } ^ { b } \left | f ( x) - \sum _ { k= 1} ^ { n } c _ {k} \phi _ {k} ( x) \ \right | d x =$$

$$= \ \int\limits _ { a } ^ { b } \left | f ( x) - P _ {n} ^ {*} ( x) \right | d x = \left | \int\limits _ { a } ^ { b } f ( x) \mathop{\rm sgn} \psi ( x) d x \right | .$$

For the system $\{ 1 , \cos x \dots \cos n x \}$ on $[ 0 , \pi ]$, $\psi$ can be taken to be $\cos ( n + 1) x$; for the system $\{ \sin x \dots \sin n x \}$, $0 \leq x \leq \pi$, $\psi$ can be taken to be $\sin ( n + 1 ) x$; and for the system $\{ 1 , x \dots x ^ {n} \}$, $- 1 \leq x \leq 1$, one can take $\psi ( x) = \sin ( ( n + 2 ) \mathop{\rm arc} \cos x )$.

#### References

 [1] A.A. Markov, "Selected works" , Moscow-Leningrad (1948) (In Russian) [2] N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian) [3] I.K. Daugavet, "Introduction to the theory of approximation of functions" , Leningrad (1977) (In Russian)