In
mathematics, the Gauss–Kuzmin–Wirsing operator is the
transfer operator of the Gauss map that takes a positive number to the fractional part of its reciprocal. (This is not the same as the
Gauss map in
differential geometry.) It is named after
Carl Gauss,
Rodion Kuzmin, and
Eduard Wirsing. It occurs in the study of
continued fractions; it is also related to the
Riemann zeta function.
Relationship to the maps and continued fractions
The Gauss map

The Gauss function (map) ''h'' is :
:
where
denotes the
floor function
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least ...
.
It has an infinite number of
jump discontinuities at ''x'' = 1/''n'', for positive integers ''n''. It is hard to approximate it by a single smooth polynomial.
Operator on the maps
The Gauss–Kuzmin–Wirsing
operator
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...
acts on functions
as
:
Eigenvalues of the operator
The first
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, ...
of this operator is
:
which corresponds to an
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
of ''λ''
1=1. This eigenfunction gives the probability of the occurrence of a given integer in a continued fraction expansion, and is known as the
Gauss–Kuzmin distribution. This follows in part because the Gauss map acts as a truncating
shift operator
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function
to its translation . In time series analysis, the shift operator is called the lag operator.
Shift ...
for the
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integ ...
s: if
: