3,405 research outputs found
Spiraling of approximations and spherical averages of Siegel transforms
We consider the question of how approximations satisfying Dirichlet's theorem
spiral around vectors in . We give pointwise almost everywhere
results (using only the Birkhoff ergodic theorem on the space of lattices). In
addition, we show that for unimodular lattice, on average, the
directions of approximates spiral in a uniformly distributed fashion on the
dimensional unit sphere. For this second result, we adapt a very recent
proof of Marklof and Str\"ombergsson \cite{MS3} to show a spherical average
result for Siegel transforms on
. Our
techniques are elementary. Results like this date back to the work of
Eskin-Margulis-Mozes \cite{EMM} and Kleinbock-Margulis \cite{KM} and have
wide-ranging applications. We also explicitly construct examples in which the
directions are not uniformly distributed.Comment: 20 pages, 1 figure. Noteworthy changes from the previous version: New
title. New result added (Theorem 1.1). Strengthening of Theorem 1.
Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I
Classical Schur analysis is intimately connected to the theory of orthogonal
polynomials on the circle [Simon, 2005]. We investigate here the connection
between multipoint Schur analysis and orthogonal rational functions.
Specifically, we study the convergence of the Wall rational functions via the
development of a rational analogue to the Szeg\H o theory, in the case where
the interpolation points may accumulate on the unit circle. This leads us to
generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields
asymptotics of a novel type.Comment: a preliminary version, 39 pages; some changes in the Introduction,
Section 5 (Szeg\H o type asymptotics) is extende
On the finiteness and periodicity of the --adic Jacobi--Perron algorithm
Multidimensional continued fractions (MCFs) were introduced by Jacobi and
Perron in order to obtain periodic representations for algebraic irrationals,
as it is for continued fractions and quadratic irrationals. Since continued
fractions have been also studied in the field of --adic numbers , also MCFs have been recently introduced in together to a
--adic Jacobi--Perron algorithm. In this paper, we address th study of two
main features of this algorithm, i.e., finiteness and periodicity. In
particular, regarding the finiteness of the --adic Jacobi--Perron algorithm
our results are obtained by exploiting properties of some auxiliary integer
sequences. Moreover, it is known that a finite --adic MCF represents
--linearly dependent numbers. We see that the viceversa is not
always true and we prove that in this case infinite partial quotients of the
MCF have --adic valuations equal to . Finally, we show that a periodic
MCF of dimension converges to algebraic irrationals of degree less or equal
than and for the case we are able to give some more detailed
results
Inverse problems for periodic generalized Jacobi matrices
Some inverse problems for semi-infinite periodic generalized Jacobi matrices
are considered. In particular, a generalization of the Abel criterion is
presented. The approach is based on the fact that the solvability of the
Pell-Abel equation is equivalent to the existence of a certainly normalized
-unitary -matrix polynomial (the monodromy matrix).Comment: 11 pages (some typos are corrected
The linear pencil approach to rational interpolation
It is possible to generalize the fruitful interaction between (real or
complex) Jacobi matrices, orthogonal polynomials and Pade approximants at
infinity by considering rational interpolants, (bi-)orthogonal rational
functions and linear pencils zB-A of two tridiagonal matrices A, B, following
Spiridonov and Zhedanov.
In the present paper, beside revisiting the underlying generalized Favard
theorem, we suggest a new criterion for the resolvent set of this linear pencil
in terms of the underlying associated rational functions. This enables us to
generalize several convergence results for Pade approximants in terms of
complex Jacobi matrices to the more general case of convergence of rational
interpolants in terms of the linear pencil. We also study generalizations of
the Darboux transformations and the link to biorthogonal rational functions.
Finally, for a Markov function and for pairwise conjugate interpolation points
tending to infinity, we compute explicitly the spectrum and the numerical range
of the underlying linear pencil.Comment: 22 page
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