Gauss–Kuzmin–Wirsing operator

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.

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.

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.

, unique up to scaling, which is the density of the measure invariant under the Gauss map.

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 for the continued fractions: if is the continued fraction representation of a number 0 < x < 1, then Because

This fact allows a short proof of the existence of Khinchin's constant.

Let us arrange the eigenvalues of the Gauss–Kuzmin–Wirsing operator according to an absolute value: It was conjectured in 1995 by Philippe Flajolet and Brigitte Vallée that In 2018, Giedrius Alkauskas gave a convincing argument that this conjecture can be refined to a much stronger statement:[2] here the function

The eigenvalues form a discrete spectrum, when the operator is limited to act on functions on the unit interval of the real number line.

More broadly, since the Gauss map is the shift operator on Baire space

(considered as a Banach space, with basis functions taken to be the indicator functions on the cylinders of the product topology).

In the later case, it has a continuous spectrum, with eigenvalues in the unit disk

A special case arises when one wishes to consider the Haar measure of the shift operator, that is, a function that is invariant under shifts.

[3] The Gauss map is in fact much more than ergodic: it is exponentially mixing,[4][5] but the proof is not elementary.

Compare this with base in topology, which is less restrictive as it allows non-disjoint unions.

Since the Lebesgue measure is outer regular, we can take an open set

, meaning the symmetric difference has arbitrarily small measure

Consider the set of all open intervals in the form

are rational, is a disjoint union of finitely many sets in

Since Lebesgue measure is outer regular, there exists an open set

The GKW operator is related to the Riemann zeta function.

Note that the zeta function can be written as which implies that by change-of-variable.

Consider the Taylor series expansions at x = 1 for a function f(x) and

The expansion is made about x = 1 because the GKW operator is poorly behaved at x = 0.

The Gauss–Kuzmin constant is easily computed to high precision by numerically diagonalizing the upper-left n by n portion.

play the analog of the Stieltjes constants, but for the falling factorial expansion.

They can be explicitly related to the Stieltjes constants by re-expressing the falling factorial as a polynomial with Stirling number coefficients, and then solving.

More generally, the Riemann zeta can be re-expressed as an expansion in terms of Sheffer sequences of polynomials.

This expansion of the Riemann zeta is investigated in the following references.