Müntz theorem
theorem on the completeness of a system of powers
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 |
Muentz theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Muentz_theorem&oldid=23427