16 research outputs found
Integration of Oscillatory and Subanalytic Functions
We prove the stability under integration and under Fourier transform of a
concrete class of functions containing all globally subanalytic functions and
their complex exponentials. This paper extends the investigation started in
[J.-M. Lion, J.-P. Rolin: "Volumes, feuilles de Rolle de feuilletages
analytiques et th\'eor\`eme de Wilkie" Ann. Fac. Sci. Toulouse Math. (6) 7
(1998), no. 1, 93-112] and [R. Cluckers, D. J. Miller: "Stability under
integration of sums of products of real globally subanalytic functions and
their logarithms" Duke Math. J. 156 (2011), no. 2, 311-348] to an enriched
framework including oscillatory functions. It provides a new example of
fruitful interaction between analysis and singularity theory.Comment: Final version. Accepted for publication in Duke Math. Journal.
Changes in proofs: from Section 6 to the end, we now use the theory of
continuously uniformly distributed modulo 1 functions that provides a uniform
technical point of view in the proofs of limit statement
Some properties of the polynomially bounded o-minimal expansions of the real field and of some quasianalytic local rings
In this paper, we study the Weierstrass division theorem over the rings of smooth germs that are definable in an arbitrary polynomially bounded o-minimal expansion of the real field by giving some criteria for satisfying this theorem. Afterwards, we study some topological properties of some quasianalytic subrings of the ring of smooth germs for the -adic topology by showing that these rings are separable metric spaces. Also, we cite a criterion for their completeness with respect to the -adic topology
Periods, Power Series, and Integrated Algebraic Numbers
Periods are defined as integrals of semialgebraic functions defined over the
rationals. Periods form a countable ring not much is known about. Examples are
given by taking the antiderivative of a power series which is algebraic over
the polynomial ring over the rationals and evaluate it at a rational number. We
follow this path and close these algebraic power series under taking iterated
antiderivatives and nearby algebraic and geometric operations. We obtain a
system of rings of power series whose coefficients form a countable real closed
field. Using techniques from o-minimality we are able to show that every period
belongs to this field. In the setting of o-minimality we define exponential
integrated algebraic numbers and show that the Euler constant is an exponential
integrated algebraic number. Hence they are a good candiate for a natural
number system extending the period ring and containing important mathematical
constants.Comment: Final versio