In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Fabry gap theorem is a result about the
analytic continuation of
complex power series whose non-zero terms are of orders that have a certain "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an
analytic function anywhere on the
boundary of its
disc of convergence.
The theorem may be deduced from the first main theorem of
Turán's method.
Statement of the theorem
Let 0 < ''p''
1 < ''p''
2 < ... be a
sequence of
integers such that the sequence ''p''
''n''/''n'' diverges to ∞. Let (''α''
''j'')
''j''∈N be a sequence of complex numbers such that the power series
:
has radius of convergence 1. Then the unit circle is a
natural boundary
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
for the series ''f''.
Converse
A converse to the theorem was established by
George Pólya. If lim inf ''p''
''n''/''n'' is finite then there exists a power series with exponent sequence ''p''
''n'', radius of convergence equal to 1, but for which the unit circle is not a natural boundary.
See also
*
Gap theorem (disambiguation) In mathematics, gap theorem may refer to:
* The Weierstrass gap theorem in algebraic geometry
* The Ostrowski–Hadamard gap theorem on lacunary function
* The Fabry gap theorem on lacunary functions
* The ''gap theorem'' of Fourier analysis, a ...
*
Lacunary function
*
Ostrowski–Hadamard gap theorem In mathematics, the Ostrowski–Hadamard gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a suitable "gap" between them. Such a power series is "badly behaved" in the sense ...
References
*
* {{cite journal , last=Erdős , first=Pál , authorlink=Paul Erdős , title=Note on the converse of Fabry's gap theorem , journal=
Transactions of the American Mathematical Society , volume=57 , pages=102–104 , year=1945 , issn=0002-9947 , jstor=1990169 , zbl=0060.20303 , doi=10.2307/1990169
Mathematical series
Theorems in complex analysis