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)

Comments

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].

References

[a1]	J. Korevaar, R. Zeinstra, "Transformées de Laplace pour les courbes à pente bornée et un résultat correspondant du type Müntz–Szász" C.R. Acad. Sci. Paris , 301 (1985) pp. 695–698
[a2]	L.I. Ronkin, "Some questions of completeness and uniqueness for functions of several variables" Funct. Anal. Appl. , 7 (1973) pp. 37–45 Funkts. Anal. Prilozhen. , 7 (1973) pp. 45–55