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''in the complex domain''
 
''in the complex domain''
  
 
A differential equation of the form
 
A differential equation of the form
  
<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/e/e035/e035930/e0359301.png" /></td> </tr></table>
+
$$
 +
\sum _ {n = 0 } ^  \infty 
 +
a _ {n} ( z) y  ^ {(} n) ( z)  = \
 +
f ( z),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e0359302.png" /> is the unknown function of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e0359303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e0359304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e0359305.png" /> are given functions. Equations of infinite order which have been most thoroughly studied are those with constant coefficients:
+
where $  y( z) $
 +
is the unknown function of the complex variable $  z $
 +
and $  a _ {n} ( z) $,  
 +
$  f( z) $
 +
are given functions. Equations of infinite order which have been most thoroughly studied are those with constant coefficients:
  
<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/e/e035/e035930/e0359306.png" /></td> </tr></table>
+
$$
 +
Ly  \equiv \
 +
\sum _ {n = 0 } ^  \infty 
 +
a _ {n} y  ^ {(} n) ( z)  = \
 +
f ( z).
 +
$$
  
 
If the characteristic function
 
If the characteristic function
  
<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/e/e035/e035930/e0359307.png" /></td> </tr></table>
+
$$
 +
\phi ( \lambda )  = \
 +
\sum _ {n = 0 } ^  \infty 
 +
a _ {n} \lambda  ^ {n}
 +
$$
  
is an entire function of exponential type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e0359308.png" />, the left-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e0359309.png" /> makes sense for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593010.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593011.png" /> is an analytic function in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593014.png" />, one must assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593015.png" /> is an entire function. A difference from an equation of finite order consists already in the fact that the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593016.png" /> may have singular points even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593017.png" /> is an entire function. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593019.png" /> is an entire function, the domain of existence of a solution is convex [[#References|[1]]]. The general solution consists of a particular solution and the general solution of the corresponding homogeneous equation. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593020.png" /> be the roots of the characteristic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593021.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593022.png" /> be their respective multiplicities. The homogeneous equation has elementary particular solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593023.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593024.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593025.png" />). The solution of the homogeneous equation can be written as a series of elementary particular solutions, formed according to a definite rule. If the characteristic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593026.png" /> has regular growth (in a certain sense), it is possible to find a subsequence of the partial sums of this series converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593027.png" /> [[#References|[4]]]. In the general case the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593028.png" /> may be approximated, as accurately as one pleases, by finite linear combinations of elementary solutions [[#References|[5]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035930/e03593029.png" />, an equation of infinite order may have non-analytic solutions [[#References|[2]]]. Under certain conditions these solutions form a [[Quasi-analytic class|quasi-analytic class]] with weaker bounds on the growth of the derivatives than in the classical Denjoy–Carleman theorem (cf. [[Quasi-analytic class|Quasi-analytic class]]).
+
is an entire function of exponential type $  \sigma $,  
 +
the left-hand side $  Ly $
 +
makes sense for $  z = z _ {0} $
 +
if $  y( z) $
 +
is an analytic function in the disc $  | z - z _ {0} | < R $,  
 +
$  R > \sigma $.  
 +
If $  \sigma = \infty $,  
 +
one must assume that $  y( z) $
 +
is an entire function. A difference from an equation of finite order consists already in the fact that the solution $  y( z) $
 +
may have singular points even if $  f( z) $
 +
is an entire function. If $  \sigma = 0 $
 +
and $  f( z) $
 +
is an entire function, the domain of existence of a solution is convex [[#References|[1]]]. The general solution consists of a particular solution and the general solution of the corresponding homogeneous equation. Let $  \lambda _ {1} , \lambda _ {2} ,\dots $
 +
be the roots of the characteristic equation $  \phi ( \lambda ) = 0 $
 +
and let $  m _ {1} , m _ {2} ,\dots $
 +
be their respective multiplicities. The homogeneous equation has elementary particular solutions $  z  ^ {k} e ^ {\lambda _ {n} z } $(
 +
$  k = 0 \dots m _ {n} - 1 $;  
 +
$  n = 1, 2 ,\dots $).  
 +
The solution of the homogeneous equation can be written as a series of elementary particular solutions, formed according to a definite rule. If the characteristic function $  \phi ( \lambda ) $
 +
has regular growth (in a certain sense), it is possible to find a subsequence of the partial sums of this series converging to $  y( z) $[[#References|[4]]]. In the general case the function $  y( z) $
 +
may be approximated, as accurately as one pleases, by finite linear combinations of elementary solutions [[#References|[5]]]. If $  \sigma = 0 $,  
 +
an equation of infinite order may have non-analytic solutions [[#References|[2]]]. Under certain conditions these solutions form a [[Quasi-analytic class|quasi-analytic class]] with weaker bounds on the growth of the derivatives than in the classical Denjoy–Carleman theorem (cf. [[Quasi-analytic class|Quasi-analytic class]]).
  
 
Equations of infinite order have various applications. They are used in the study of sequences of Dirichlet polynomials, completeness of systems of analytic functions, uniqueness of analytic and harmonic functions, and in solvability questions of analytic problems such as the generalized quasi-analyticity problem, the generalized uniqueness problem of moments, etc.
 
Equations of infinite order have various applications. They are used in the study of sequences of Dirichlet polynomials, completeness of systems of analytic functions, uniqueness of analytic and harmonic functions, and in solvability questions of analytic problems such as the generalized quasi-analyticity problem, the generalized uniqueness problem of moments, etc.
Line 19: Line 69:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Pólya,  "Eine Verallgemeinung des Fabryschen Lückensatzes"  ''Nachr. Ges. Wiss. Göttingen''  (1927)  pp. 187–195</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Valiron,  "Sur les solutions des équations différentielles lineaires d'ordre infini et à coefficients constants"  ''Ann. Sci. École Norm. Sup. (3)'' , '''46''' :  1  (1929)  pp. 25–53</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.F. Leont'ev,  "Differential equations of infinite order and their applications" , ''Proc. 4-th All-Union Math. Congress (1961)'' , Leningrad  (1964)  pp. 648–660  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.F. Leont'ev,  "On representation of functions by sequences of Dirichlet polynomials"  ''Mat. Sb.'' , '''70 (112)''' :  1  (1966)  pp. 132–144  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.F. Krasichkov-Ternovskii,  "Invariant subspaces of analytic functions III. On the extension of spectral synthesis"  ''Math. USSR-Sb.'' , '''17''' :  3  (1972)  pp. 327–348  ''Mat. Sb.'' , '''88''' :  3  (1972)  pp. 331–352</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Pólya,  "Eine Verallgemeinung des Fabryschen Lückensatzes"  ''Nachr. Ges. Wiss. Göttingen''  (1927)  pp. 187–195</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Valiron,  "Sur les solutions des équations différentielles lineaires d'ordre infini et à coefficients constants"  ''Ann. Sci. École Norm. Sup. (3)'' , '''46''' :  1  (1929)  pp. 25–53</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.F. Leont'ev,  "Differential equations of infinite order and their applications" , ''Proc. 4-th All-Union Math. Congress (1961)'' , Leningrad  (1964)  pp. 648–660  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.F. Leont'ev,  "On representation of functions by sequences of Dirichlet polynomials"  ''Mat. Sb.'' , '''70 (112)''' :  1  (1966)  pp. 132–144  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.F. Krasichkov-Ternovskii,  "Invariant subspaces of analytic functions III. On the extension of spectral synthesis"  ''Math. USSR-Sb.'' , '''17''' :  3  (1972)  pp. 327–348  ''Mat. Sb.'' , '''88''' :  3  (1972)  pp. 331–352</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.I. Hirschmann,  "The convolution transform" , Princeton Univ. Press  (1955)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Ehrenpreis,  "Theory of infinite derivatives"  ''Amer. J. Math'' , '''81'''  (1959)  pp. 799–845</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Ehrenpreis,  "Fourier analysis in several complex variables" , Wiley (Interscience)  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.I. Hirschmann,  "The convolution transform" , Princeton Univ. Press  (1955)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Ehrenpreis,  "Theory of infinite derivatives"  ''Amer. J. Math'' , '''81'''  (1959)  pp. 799–845</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Ehrenpreis,  "Fourier analysis in several complex variables" , Wiley (Interscience)  (1970)</TD></TR></table>

Revision as of 19:37, 5 June 2020


in the complex domain

A differential equation of the form

$$ \sum _ {n = 0 } ^ \infty a _ {n} ( z) y ^ {(} n) ( z) = \ f ( z), $$

where $ y( z) $ is the unknown function of the complex variable $ z $ and $ a _ {n} ( z) $, $ f( z) $ are given functions. Equations of infinite order which have been most thoroughly studied are those with constant coefficients:

$$ Ly \equiv \ \sum _ {n = 0 } ^ \infty a _ {n} y ^ {(} n) ( z) = \ f ( z). $$

If the characteristic function

$$ \phi ( \lambda ) = \ \sum _ {n = 0 } ^ \infty a _ {n} \lambda ^ {n} $$

is an entire function of exponential type $ \sigma $, the left-hand side $ Ly $ makes sense for $ z = z _ {0} $ if $ y( z) $ is an analytic function in the disc $ | z - z _ {0} | < R $, $ R > \sigma $. If $ \sigma = \infty $, one must assume that $ y( z) $ is an entire function. A difference from an equation of finite order consists already in the fact that the solution $ y( z) $ may have singular points even if $ f( z) $ is an entire function. If $ \sigma = 0 $ and $ f( z) $ is an entire function, the domain of existence of a solution is convex [1]. The general solution consists of a particular solution and the general solution of the corresponding homogeneous equation. Let $ \lambda _ {1} , \lambda _ {2} ,\dots $ be the roots of the characteristic equation $ \phi ( \lambda ) = 0 $ and let $ m _ {1} , m _ {2} ,\dots $ be their respective multiplicities. The homogeneous equation has elementary particular solutions $ z ^ {k} e ^ {\lambda _ {n} z } $( $ k = 0 \dots m _ {n} - 1 $; $ n = 1, 2 ,\dots $). The solution of the homogeneous equation can be written as a series of elementary particular solutions, formed according to a definite rule. If the characteristic function $ \phi ( \lambda ) $ has regular growth (in a certain sense), it is possible to find a subsequence of the partial sums of this series converging to $ y( z) $[4]. In the general case the function $ y( z) $ may be approximated, as accurately as one pleases, by finite linear combinations of elementary solutions [5]. If $ \sigma = 0 $, an equation of infinite order may have non-analytic solutions [2]. Under certain conditions these solutions form a quasi-analytic class with weaker bounds on the growth of the derivatives than in the classical Denjoy–Carleman theorem (cf. Quasi-analytic class).

Equations of infinite order have various applications. They are used in the study of sequences of Dirichlet polynomials, completeness of systems of analytic functions, uniqueness of analytic and harmonic functions, and in solvability questions of analytic problems such as the generalized quasi-analyticity problem, the generalized uniqueness problem of moments, etc.

References

[1] G. Pólya, "Eine Verallgemeinung des Fabryschen Lückensatzes" Nachr. Ges. Wiss. Göttingen (1927) pp. 187–195
[2] G. Valiron, "Sur les solutions des équations différentielles lineaires d'ordre infini et à coefficients constants" Ann. Sci. École Norm. Sup. (3) , 46 : 1 (1929) pp. 25–53
[3] A.F. Leont'ev, "Differential equations of infinite order and their applications" , Proc. 4-th All-Union Math. Congress (1961) , Leningrad (1964) pp. 648–660 (In Russian)
[4] A.F. Leont'ev, "On representation of functions by sequences of Dirichlet polynomials" Mat. Sb. , 70 (112) : 1 (1966) pp. 132–144 (In Russian)
[5] I.F. Krasichkov-Ternovskii, "Invariant subspaces of analytic functions III. On the extension of spectral synthesis" Math. USSR-Sb. , 17 : 3 (1972) pp. 327–348 Mat. Sb. , 88 : 3 (1972) pp. 331–352

Comments

References

[a1] I.I. Hirschmann, "The convolution transform" , Princeton Univ. Press (1955)
[a2] L. Ehrenpreis, "Theory of infinite derivatives" Amer. J. Math , 81 (1959) pp. 799–845
[a3] L. Ehrenpreis, "Fourier analysis in several complex variables" , Wiley (Interscience) (1970)
How to Cite This Entry:
Equation of infinite order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equation_of_infinite_order&oldid=16246
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