# Müntz theorem

theorem on the completeness of a system of powers $\{ x ^ {\lambda _ {k} } \}$ on an interval $[ a , b ]$, $0 < a < b < \infty$

Let $0 < \lambda _ {1} < \lambda _ {2} < \dots$. In order that for any continuous function $f$ on $[ a , b ]$ and for any $\epsilon > 0$ there is a linear combination

$$P ( x) = \sum_{k=1} ^ { n } a _ {k} x ^ {\lambda _ {k} }$$

such that

$$\| f - P \| _ {C} = \ \max _ {a \leq x \leq b } \ | f ( x) - P ( x) | < \epsilon ,$$

it is necessary and sufficient that

$$\tag{* } \sum_{k=1} ^ \infty \frac{1}{\lambda _ {k} } = \infty .$$

In the case of an interval $[ 0 , b ]$ one adds the function which is identically equal to 1 to the system $\{ x ^ {\lambda _ {k} } \}$ and condition (*) is, as before, necessary and sufficient for the completeness of the enlarged system. The condition $a \geq 0$ is essential: the system $\{ x ^ {2k} \}_{k=0} ^ \infty$( which satisfies (*)) is not complete on $[ - 1 , 1 ]$( an odd function cannot be arbitrarily closely approximated by combinations of even powers).

Condition (*) is necessary and sufficient for the completeness of $\{ x ^ {\lambda _ {k} } \}$, $- 1 / p < \lambda _ {1} < \lambda _ {2} < {} \dots$, on $[ a , b ]$, $a \geq 0$, in the metric of $L _ {p}$, $p > 1$; that is, for each $f \in L _ {p} ( a , b )$ and any $\epsilon > 0$ there is a linear combination $P$ such that

$$\| f - P \| _ {L _ {p} } = \ \left | \int\limits _ { a } ^ { b } | f ( x) - P ( x) | ^ {p} \ d x \right | ^ {1/p} < \epsilon .$$

The theorem was proved by H. Müntz [1].

#### References

 [1] H. Müntz, "Ueber den Approximationssatz von Weierstrass" , Festschrift H.A. Schwarz , Schwarz–Festschrift , Berlin (1914) [2] N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)

There exists several extensions of the Müntz theorem. First, O. Szász showed that with exponents $\lambda _ {k} \in \mathbf C$, $\mathop{\rm Re} \lambda _ {k} > 0$,
$$\tag{a1 } \sum \mathop{\rm Re} \frac{1}{\lambda _ {k} } = \infty$$
is necessary and sufficient for completeness of the system $\{ x ^ {\lambda _ {k} } \}$ in $C [ a , b ]$ or $L _ {p} [ a , b ]$, $p > 1$, or, equivalently, completeness of $\{ e ^ {\lambda _ {k} z } \}$ in, say, $C _ {0} ( - \infty , 0 ]$. Later, J. Korevaar, A.F. Leont'ev, P. Malliavin, J.A. Siddigi, and others studied analogous completeness problems on curves $\gamma ( x) = x + i \eta ( x)$, $- \infty < x \leq 0$. Very recently it was shown that if $\eta$ is piecewise $C ^ {1}$, with $\mathop{\rm exp} | \eta ^ \prime | < \infty$, and $\{ \lambda _ {k} \}$ satisfies (a1) and is contained in a sufficiently small sector around the positive axis, then $\{ e ^ {\lambda _ {k} z } \}$ spans $C _ {0} [ \gamma ]$. See [a1], also for further references. Finally, attempts have been made to generalize the Müntz theorem to functions of several variables, see [a2].