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''theorem on the completeness of a system of powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m0655802.png" /> on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m0655803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m0655804.png" />''
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m0655805.png" />. In order that for any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m0655806.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m0655807.png" /> and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m0655808.png" /> there is a linear combination
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{{TEX|auto}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m0655809.png" /></td> </tr></table>
+
''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
 
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558010.png" /></td> </tr></table>
+
$$
 +
\| f - P \| _ {C}  = \
 +
\max _ {a \leq  x \leq  b } \
 +
| f ( x) - P ( x) |  < \epsilon ,
 +
$$
  
 
it is necessary and sufficient that
 
it is necessary and sufficient that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\sum_{k=1} ^  \infty 
 +
 
 +
\frac{1}{\lambda _ {k} }
 +
  = \infty .
 +
$$
  
In the case of an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558012.png" /> one adds the function which is identically equal to 1 to the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558013.png" /> and condition (*) is, as before, necessary and sufficient for the completeness of the enlarged system. The condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558014.png" /> is essential: the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558015.png" /> (which satisfies (*)) is not complete on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558016.png" /> (an odd function cannot be arbitrarily closely approximated by combinations of even powers).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558018.png" />, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558020.png" />, in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558022.png" />; that is, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558023.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558024.png" /> there is a linear combination <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558025.png" /> such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558026.png" /></td> </tr></table>
+
$$
 +
\| f - P \| _ {L _ {p}  }  = \
 +
\left | \int\limits _ { a } ^ { b }
 +
| f ( x) - P ( x) |  ^ {p} \
 +
d x \right |  ^ {1/p}  < \epsilon .
 +
$$
  
 
The theorem was proved by H. Müntz [[#References|[1]]].
 
The theorem was proved by H. Müntz [[#References|[1]]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Müntz,  "Ueber den Approximationssatz von Weierstrass" , ''Festschrift H.A. Schwarz'' , ''Schwarz–Festschrift'' , Berlin  (1914)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,  "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Müntz,  "Ueber den Approximationssatz von Weierstrass" , ''Festschrift H.A. Schwarz'' , ''Schwarz–Festschrift'' , Berlin  (1914)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,  "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
There exists several extensions of the Müntz theorem. First, O. Szász showed that with exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558028.png" />,
+
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 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
\sum  \mathop{\rm Re} 
 +
\frac{1}{\lambda _ {k} }
 +
  = \infty
 +
$$
  
is necessary and sufficient for completeness of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558031.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558033.png" />, or, equivalently, completeness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558034.png" /> in, say, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558035.png" />. Later, J. Korevaar, A.F. Leont'ev, P. Malliavin, J.A. Siddigi, and others studied analogous completeness problems on curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558037.png" />. Very recently it was shown that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558038.png" /> is piecewise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558039.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558040.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558041.png" /> satisfies (a1) and is contained in a sufficiently small sector around the positive axis, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558042.png" /> spans <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065580/m06558043.png" />. See [[#References|[a1]]], also for further references. Finally, attempts have been made to generalize the Müntz theorem to functions of several variables, see [[#References|[a2]]].
+
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 [[#References|[a1]]], also for further references. Finally, attempts have been made to generalize the Müntz theorem to functions of several variables, see [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR></table>

Latest revision as of 12:46, 6 January 2024


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
How to Cite This Entry:
Müntz theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=M%C3%BCntz_theorem&oldid=23426
This article was adapted from an original article by A.F. Leont'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article