6,945 research outputs found
Differentiation by integration using orthogonal polynomials, a survey
This survey paper discusses the history of approximation formulas for n-th
order derivatives by integrals involving orthogonal polynomials. There is a
large but rather disconnected corpus of literature on such formulas. We give
some results in greater generality than in the literature. Notably we unify the
continuous and discrete case. We make many side remarks, for instance on
wavelets, Mantica's Fourier-Bessel functions and Greville's minimum R_alpha
formulas in connection with discrete smoothing.Comment: 35 pages, 3 figures; minor corrections, subsection 3.11 added;
accepted by J. Approx. Theor
The fractional orthogonal derivative
This paper builds on the notion of the so-called orthogonal derivative, where
an n-th order derivative is approximated by an integral involving an orthogonal
polynomial of degree n. This notion was reviewed in great detail in a paper in
J. Approx. Theory (2012) by the author and Koornwinder. Here an approximation
of the Weyl or Riemann-Liouville fractional derivative is considered by
replacing the n-th derivative by its approximation in the formula for the
fractional derivative. In the case of, for instance, Jacobi polynomials an
explicit formula for the kernel of this approximate fractional derivative can
be given. Next we consider the fractional derivative as a filter and compute
the transfer function in the continuous case for the Jacobi polynomials and in
the discrete case for the Hahn polynomials. The transfer function in the Jacobi
case is a confluent hypergeometric function. A different approach is discussed
which starts with this explicit transfer function and then obtains the
approximate fractional derivative by taking the inverse Fourier transform. The
theory is finally illustrated with an application of a fractional
differentiating filter. In particular, graphs are presented of the absolute
value of the modulus of the transfer function. These make clear that for a good
insight in the behavior of a fractional differentiating filter one has to look
for the modulus of its transfer function in a log-log plot, rather than for
plots in the time domain.Comment: 32 pages, 7 figures. The section between formula (4.15) and (4.20) is
correcte
Asymptotic approximations to the nodes and weights of Gauss-Hermite and Gauss-Laguerre quadratures
Asymptotic approximations to the zeros of Hermite and Laguerre polynomials
are given, together with methods for obtaining the coefficients in the
expansions. These approximations can be used as a standalone method of
computation of Gaussian quadratures for high enough degrees, with Gaussian
weights computed from asymptotic approximations for the orthogonal polynomials.
We provide numerical evidence showing that for degrees greater than the
asymptotic methods are enough for a double precision accuracy computation
(- digits) of the nodes and weights of the Gauss--Hermite and
Gauss--Laguerre quadratures.Comment: Submitted to Studies in Applied Mathematic
Computing a partial Schur factorization of nonlinear eigenvalue problems using the infinite Arnoldi method
The partial Schur factorization can be used to represent several eigenpairs
of a matrix in a numerically robust way. Different adaptions of the Arnoldi
method are often used to compute partial Schur factorizations. We propose here
a technique to compute a partial Schur factorization of a nonlinear eigenvalue
problem (NEP). The technique is inspired by the algorithm in [8], now called
the infinite Arnoldi method. The infinite Arnoldi method is a method designed
for NEPs, and can be interpreted as Arnoldi's method applied to a linear
infinite-dimensional operator, whose reciprocal eigenvalues are the solutions
to the NEP. As a first result we show that the invariant pairs of the operator
are equivalent to invariant pairs of the NEP. We characterize the structure of
the invariant pairs of the operator and show how one can carry out a
modification of the infinite Arnoldi method by respecting the structure. This
also allows us to naturally add the feature known as locking. We nest this
algorithm with an outer iteration, where the infinite Arnoldi method for a
particular type of structured functions is appropriately restarted. The
restarting exploits the structure and is inspired by the well-known implicitly
restarted Arnoldi method for standard eigenvalue problems. The final algorithm
is applied to examples from a benchmark collection, showing that both
processing time and memory consumption can be considerably reduced with the
restarting technique
Spectral methods for bivariate Markov processes with diffusion and discrete components and a variant of the Wright-Fisher model
The aim of this paper is to study differential and spectral properties of the
infinitesimal operator of two dimensional Markov processes with diffusion and
discrete components. The infinitesimal operator is now a second-order
differential operator with matrix-valued coefficients, from which we can derive
backward and forward equations, a spectral representation of the probability
density, study recurrence of the process and the corresponding invariant
distribution. All these results are applied to an example coming from group
representation theory which can be viewed as a variant of the Wright-Fisher
model involving only mutation effects.Comment: 6 figure
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