2 research outputs found
Biorthogonal rational functions of type
In this work, a sequence of orthonormal rational functions that is also
biorthogonal to another sequence of rational functions arising from recurrence
relations of type is constructed. The biorthogonality is proved by a
procedure which we call Zhedanov method. A particular case of a sequence of
orthonormal rational functions having denominators of special form is
considered to motivate the general case. The particular case provides a
Christoffel type transformation of the generalized eigenvalue problem with a
reformulation different from the existing literature.Comment: 18 page
Riemann-Hilbert Characterisation of Rational Functions with a General Distribution of Poles on the Extended Real Line Orthogonal with Respect to Varying Exponential Weights: Multi-Point Pad\'e Approximants and Asymptotics
Given arbitrary poles, which are neither necessarily distinct nor
bounded, on the extended real line, a corresponding ordered base of rational
functions orthogonal with respect to varying exponential weights is
constructed: this gives rise to a -fold family of orthogonal rational
functions (ORFs). The ORF problem is characterised as a family of matrix
Riemann-Hilbert problems (RHPs) on the extended real line, and a corresponding
family of energy minimisation (variational) problems containing external
fields with singular points is formulated, and the existence, uniqueness, and
regularity properties of the associated family of equilibrium measures is
established. The family of equilibrium measures is used to derive a family
of model matrix RHPs on the extended real line that are amenable to
asymptotic analysis via the Deift-Zhou non-linear steepest-descent method: this
is used to derive uniform asymptotics, in a certain double-scaling limit, of
the ORFs and their leading coefficients, as well as related, important objects,
in the entire complex plane. A family of multi-point Pad\'e approximants
(MPAs) for the Markov-Stieltjes transform is also presented, and uniform
asymptotics, in a certain double-scaling limit, are obtained for the
corresponding MPAs and their associated errors in approximation (MPA error
terms) in the entire complex plane.Comment: 343 pages, 15 figure