Müntz theorem

theorem on the completeness of a system of powers on an interval ,

Let . In order that for any continuous function on and for any there is a linear combination

such that

it is necessary and sufficient that

(*)

In the case of an interval one adds the function which is identically equal to 1 to the system and condition (*) is, as before, necessary and sufficient for the completeness of the enlarged system. The condition is essential: the system (which satisfies (*)) is not complete on (an odd function cannot be arbitrarily closely approximated by combinations of even powers).

Condition (*) is necessary and sufficient for the completeness of , , on , , in the metric of , ; that is, for each and any there is a linear combination such that

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 , ,

(a1)

is necessary and sufficient for completeness of the system in or , , or, equivalently, completeness of in, say, . Later, J. Korevaar, A.F. Leont'ev, P. Malliavin, J.A. Siddigi, and others studied analogous completeness problems on curves , . Very recently it was shown that if is piecewise , with , and satisfies (a1) and is contained in a sufficiently small sector around the positive axis, then spans . 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