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 Electronic Transactions on Numerical Analysis \\
 Volume 14, 2002 \\
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 \noindent
 Contents
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 \normalsize
 \begin{description}

 \item[1~~~~~~~~~]
Superlinear CG convergence for special right-hand sides.
{\em Bernhard Beckermann
 Arno B.~J. Kuijlaars.}

\medskip
\noindent
{\bf Abstract.}\newline
 Recently, we gave a theoretical explanation for superlinear convergence
 behavior observed while solving large symmetric systems of equations
 using the Conjugate Gradient method. Roughly speaking,
 one may observe superlinear convergence while solving a sequence of
 (symmetric positive definite) linear systems if the asymptotic
 eigenvalue distribution of the sequence of the corresponding
 matrices of coefficients is far from an equilibrium distribution.
 However, it is well known that the convergence of the Conjugate Gradient
 or other Krylov subspace methods does not only depend on the spectrum but
 also on the right-hand side of the underlying system and the starting
 vector.
 In this paper we present a family of examples based on the
 discretization via finite differences of the one dimensional
 Poisson problem
 where the asymptotic distribution equals an equilibrium distribution but
 one may as well observe superlinear convergence according to
 the particular
 choice of the right-hand sides.
 Our findings are related to some recent results concerning asymptotics
 of discrete orthogonal polynomials. An important tool in our
 investigations is a constrained energy problem in logarithmic
 potential theory, where an additional external field is used being
 related to our particular right-hand sides.

\medskip
\noindent {\bf Key Words.}\newline Superlinear convergence,
Conjugate gradients, Krylov subspace methods, Logarithmic
potential theory.

\medskip
\noindent {\bf AMS(MOS) Subject Classifications.}\newline 65F10,
65E05, 31A99, 41A10.

\medskip
\noindent {\bf Files.}\newline
vol.14.2002/pp1-19.dir/pp1-19.ps;\newline
vol.14.2002/pp1-19.dir/pp1-19.pdf;

\medskip
\noindent
{\bf Forward References.}\newline
\medskip

\item[20~~~~~]
L-curve curvature bounds
 via Lanczos bidiagonalization.
{\em D. Calvetti P. C. Hansen L. Reichel .}

\medskip
\noindent {\bf Abstract.}\newline The L-curve is often applied to
determine a suitable value of the regularization parameter when
solving ill-conditioned linear systems of equations with a
right-hand side contaminated by errors of unknown norm. The
location of the vertex of the L-curve typically yields a suitable
value of the regularization parameter. However, the computation of
 the L-curve and of its curvature is quite
costly for large problems; the determination of a point on the
L-curve requires that both the norm of the regularized approximate
solution and the norm of the corresponding residual vector be
available. Recently, the L-ribbon, which contains the L-curve in
its interior, has been shown to be suitable for the determination
of the regularization parameter for large-scale problems. In this
paper we describe how techniques similar to those employed for the
computation of the L-ribbon can be used to compute a
``curvature-ribbon,'' which contains the graph of the curvature of
the L-curve. Both curvature- and L-ribbon can be computed fairly
inexpensively by partial Lanczos bidiagonalization of the matrix
of the given linear system of equations. A suitable value of the
regularization parameter is then determined from these ribbons,
and we show that an associated approximate solution of the linear
system can be computed with little additional work.



\medskip
\noindent {\bf Key Words.}\newline Ill-posed problem,
regularization, L-curve, Gauss quadrature.



\medskip
\noindent {\bf AMS(MOS) Subject Classifications.}\newline
Ill-posed problem, regularization, L-curve, Gauss quadrature.


\medskip
\noindent {\bf Files.}\newline
vol.14.2002/pp20-35.dir/pp20-35.ps;\newline
vol.14.2002/pp20-35.dir/pp20-35.pdf;

\medskip
\noindent {\bf Forward References.}\newline
\medskip


\item[36~~~~~~~]
Uniform Approximation by Minimum Norm Interpolation.
{\em Franz-J\"urgen Delvos
 .}

\medskip
\noindent {\bf Abstract.}\newline Harmonic Hilbert spaces were
introduced as an extension of periodic Hilbert spaces introduced
by Babuka to the non-periodic case . In this paper we will
investigate approximation by minimum norm interpolation in
harmonic Hilbert spaces.


\medskip
\noindent {\bf Key Words.}\newline minimum norm interpolation,
harmonic Hilbert spaces, remainders.



\medskip
\noindent {\bf AMS(MOS) Subject Classifications.}\newline 41A05,
41A25, 41A65, 65D05, 65T99.


\medskip
\noindent {\bf Files.}\newline
vol.14.2002/pp36-44.dir/pp36-44.ps;\newline
vol.14.2002/pp36-44.dir/pp36-44.pdf;

\medskip
\noindent
{\bf Forward References.}\newline
\medskip


\item[45~~~~~~~]
An algorithm for nonharmonic signal analysis using Dirichlet
series on convex polygons. {\em Brigitte Forster.}

\medskip
\noindent {\bf Abstract.}\newline
 This article presents a new algorithm for nonharmonic signal analysis
 using Dirichlet series
 on a convex polygon as a generalization of Fourier series. Here
 denotes a quasipolynomial whose set of zeros generates
 a Riesz basis of the Smirnov space. The algorithm is based on a simple form of and on
 numerical properties of the dual basis of .



\medskip
\noindent
{\bf Key Words.}\newline
 nonharmonic Fourier series, Dirichlet series,
 signal analysis, time series analysis.



\medskip
\noindent {\bf AMS(MOS) Subject Classifications.}\newline 42C15,
30B50, 37M10.


\medskip
\noindent {\bf Files.}\newline
vol.14.2002/pp45-55.dir/pp45-55.ps;\newline
vol.14.2002/pp45-55.dir/pp45-55.pdf;

\medskip
\noindent
{\bf Forward References.}\newline
\medskip


\item[56~~~~~~~]
On Multiscale Denoising of Spherical Functions: Basic Theory and
Numerical Aspects. {\em W. Freeden T. Maier.}

\medskip
\noindent {\bf Abstract.}\newline The basic concepts of selective
multiscale reconstruction of functions on the sphere from
error--affected data is outlined for scalar functions. The
selective reconstruction mechanism is based on the premise that
multiscale approximation can be well--represented in terms of only
a relatively small number of expansion coefficients at various
resolution levels. A new proof, including non-bandlimited kernel
functions, of the pyramid scheme is presented to efficiently
remove the noise at different scales using a priori statistical
information, i.e. knowledge of the covariance function.



\medskip
\noindent {\bf Key Words.}\newline spherical wavelet theory,
scalar multiscale approximation, pyramid scheme, spectral and
multiscale variance-covariance model, hard and soft thresholding.



\medskip
\noindent {\bf AMS(MOS) Subject Classifications.}\newline 33C55,
42C40, 62-07, 65T60, 86A25.


\medskip
\noindent {\bf Files.}\newline
vol.14.2002/pp56-78.dir/pp56-78.ps;\newline
vol.14.2002/pp56-78.dir/pp56-78.pdf;

\medskip
\noindent
{\bf Forward References.}\newline
\medskip


\item[79~~~~~~~]
A Polynomial Collocation Method for Cauchy Singular Integral
Equations over the Interval. {\em P. Junghanns A. Rathsfeld.}

\medskip
\noindent {\bf Abstract.}\newline In this paper we consider a
polynomial collocation method for the numerical solution of a
singular integral equation over the interval. More precisely, the
operator of our integral equation is supposed to be of the form
with the Cauchy integral operator, with piecewise continuous
coefficients
 and with a regular integral operator and with a
Jacobi weight . To the equation
we apply a collocation method, where the collocation points are
the Chebyshev nodes of the second kind and where the trial space
is the space of polynomials multiplied by another Jacobi weight.
For the stability and convergence of this collocation in weighted
 spaces, we derive necessary and sufficient conditions.



\medskip
\noindent {\bf Key Words.}\newline Cauchy singular integral
equation; polynomial collocation method; stability.



\medskip
\noindent {\bf AMS(MOS) Subject Classifications.}\newline 45L10,
65R20, 65N38.


\medskip
\noindent {\bf Files.}\newline
vol.14.2002/pp79-126.dir/pp79-126.ps;\newline
vol.14.2002/pp79-126.dir/pp79-126.pdf;

\medskip
\noindent
{\bf Forward References.}\newline
\medskip


\item[127~~~~~]
Recent Trends on Analytic Properties of Matrix Orthonormal
Polynomials. {\em F. Marcell\'an
 H. O. Yakhlef .}

\medskip
\noindent {\bf Abstract.}\newline In this paper we give an
overview of recent results on analytic properties of matrix
orthonormal polynomials. We focus our attention on the
distribution of their zeros as well as on the asymptotic behavior
of such polynomials under some restrictions about the measure of
orthogonality.



\medskip
\noindent {\bf Key Words.}\newline
 matrix orthogonal polynomials, zeros, asymptotic behavior.



\medskip
\noindent {\bf AMS(MOS) Subject Classifications.}\newline 42C05,
15A15, 15A23.


\medskip
\noindent {\bf Files.}\newline
vol.14.2002/pp127-141.dir/pp127-141.ps;\newline
vol.14.2002/pp127-141.dir/pp127-141.pdf;

\medskip
\noindent
{\bf Forward References.}\newline
\medskip


\item[142~~~~~]
Polynomial Inequalities, Functional Spaces and Best Approximation
on the Real Semiaxis with Laguerre weights. {\em G. Mastroianni.}

\medskip
\noindent {\bf Abstract.}\newline This is a short survey on
polynomial approximation with Laguerre weights. Some new
polynomial inequalities are presented.



\medskip
\noindent {\bf Key Words.}\newline Orthogonal polynomials,
polynomial inequalities, Fourier series, Lagrange interpolation.



\medskip
\noindent {\bf AMS(MOS) Subject Classifications.}\newline 41A10,
26D05.


\medskip
\noindent {\bf Files.}\newline
vol.14.2002/pp142-151.dir/pp142-151.ps;\newline
vol.14.2002/pp142-151.dir/pp142-151.pdf;

\medskip
\noindent
{\bf Forward References.}\newline
\medskip




\item[152~~~~~]
Bounds for Vandermonde type determinants of
 orthogonal polynomials.
{\em Gerhard Schmeisser.}

\medskip
\noindent {\bf Abstract.}\newline Let be a system of monic
orthogonal polynomials. We establish upper and lower estimates for
determinants of the form For the proofs, we have to study the
monic orthogonal system
 obtained by inserting the polynomial
 as a weight into the inner product
defining . We also express the recurrence formula for
 in terms of Vandermonde type determinants.



\medskip
\noindent {\bf Key Words.}\newline Vandermonde type determinants,
orthogonal systems, polynomial weights, inequalities.



\medskip
\noindent {\bf AMS(MOS) Subject Classifications.}\newline 42C05,
15A15, 15A45, 30A10.


\medskip
\noindent {\bf Files.}\newline
vol.14.2002/pp152-164.dir/pp152-164.ps;\newline
vol.14.2002/pp152-164.dir/pp152-164.pdf;

\medskip
\noindent
{\bf Forward References.}\newline
\medskip


\item[165~~~~~]
Uncertainty principles revisited. {\em Kathi K. Selig.}

\medskip
\noindent {\bf Abstract.}\newline The Heisenberg uncertainty
principle and the uncertainty principle for self-adjoint operators
have been known and applied for decades. Both in quantum mechanics
and in time-frequency analysis they play an important role. In
this paper, the uncertainty principle is extended to symmetric
operators and to normal operators. Further, different function
spaces are studied in which we obtain a number of uncertainty
principles of same type using various operators.



\medskip
\noindent {\bf Key Words.}\newline uncertainty principle,
self-adjoint operators, symmetric operators, normal operators,
periodic functions, ultraspherical polynomials, sphere.



\medskip
\noindent {\bf AMS(MOS) Subject Classifications.}\newline 26D10,
42C25, 47A05, 47A30, 47B47.


\medskip
\noindent {\bf Files.}\newline
vol.14.2002/pp165-177.dir/pp165-177.ps;\newline
vol.14.2002/pp165-177.dir/pp165-177.pdf;

\medskip
\noindent
{\bf Forward References.}\newline
\medskip


\item[178~~~~~]
Comparing multilevel coarsening strategies.
{\em Frauke Sprengel.}

\medskip
\noindent {\bf Abstract.}\newline We compare several multilevel
coarsening strategies by using stable subspace splitting
techniques. The obtained condition numbers give an answer on how
well the coarsening strategies are suited for solving an
anisotropic elliptic boundary value problem.



\medskip
\noindent {\bf Key Words.}\newline Finite elements, multilevel
algorithms, semi-coarsening.



\medskip
\noindent {\bf AMS(MOS) Subject Classifications.}\newline 65N30,
65N55, 65N22.


\medskip
\noindent {\bf Files.}\newline
vol.14.2002/pp178-194.dir/pp178-194.ps;\newline
vol.14.2002/pp178-194.dir/pp178-194.pdf;

\medskip
\noindent
{\bf Forward References.}\newline
\medskip


\item[195~~~~~]
Asymptotics for Quadratic Hermite-Pad\'e Polynomials Associated
with the Exponential Function. {\em Herbert Stahl.}

\medskip
\noindent {\bf Abstract.}\newline The asymptotic behavior of
quadratic Hermite-Pad\'{e} polynomials $p_{n}$, $q_{n}$,
$r_{n}\in\mathcal{P}_{n}$ of type I and $\frak{p}_{n}$,
$\frak{q}_{n} $, $\frak{r}_{n}\in\mathcal{P}_{2n}$ of type II
associated with the exponential function are studied. In the
introduction the background of the definition of Hermite-Pad\'{e}
polynomials is reviewed. The quadratic Hermite-Pad\'{e}
polynomials $p_{n}$, $q_{n}$, $r_{n}\in\mathcal{P}_{n}$ of type I
are defined by the relation
\[
p_{n}(z)+q_{n}(z)e^{z}+r_{n}(z)e^{2z}\;=\;O(z^{3n+2})\quad\text{as}\quad
z\rightarrow0,
\]
and the polynomials $\frak{p}_{n}$, $\frak{q}_{n}$, $\frak{r}_{n}%
\in\mathcal{P}_{2n}$ of type II by the two relations
\begin{align*}
\frak{p}_{n}(z)e^{z}-\frak{q}_{n}(z)\;  &
=\;O(z^{3n+1})\quad\text{as}\quad
z\rightarrow0,\\
\frak{p}_{n}(z)e^{2z}-\frak{r}_{n}(z)\;  &
=\;O(z^{3n+1})\quad\text{as}\quad z\rightarrow0.
\end{align*}
Analytic descriptions are given for the arcs, on which the
contracted zeros of both sets of the polynomials $\left\{
p_{n},q_{n},r_{n}\right\}  $ and $\left\{
\frak{p}_{n},\frak{q}_{n},\frak{r}_{n}\right\}  $ cluster as
$n\rightarrow\infty$. Analytic expressions are also given for the
density functions of the asymptotic distributions of these zeros.

The description is based on an algebraic function of third degree
and a harmonic function defined on the Riemann surface, which is
associated with the algebraic function. The existence and basic
properties of the asymptotic distributions of the zeros and the
arcs on which these distributions\ live are proved, the asymptotic
relations themselves are only conjectured. Numerical calculations
are presented, which demonstrate the plausibility of these
conjectures.

\medskip
\noindent {\bf Key Words.}\newline Quadratic Hermite-Pad\'e
polynomials of type I and type II, the exponential function,
German and Latin polynomials, Hermite-Pad\'e approximants.



\medskip
\noindent {\bf AMS(MOS) Subject Classifications.}\newline 41A21,
30E10.


\medskip
\noindent {\bf Files.}\newline
vol.14.2002/pp195-222.dir/pp195-222.ps;\newline
vol.14.2002/pp195-222.dir/pp195-222.pdf;

\medskip
\noindent
{\bf Forward References.}\newline
\medskip


\item[223~~~~~]
On convergence and divergence of Fourier--Bessel series. {\em
Krzysztof Stempak.}

\medskip
\noindent {\bf Abstract.}\newline We furnish another proof, based
on an idea of Prestini, of a maximal inequality for the partial
sum operators of Fourier--Bessel expansions proved by Guadalupe,
P\'erez, Ruiz and Varona. Divergence results and mean convergence
are also discussed.



\medskip
\noindent {\bf Key Words.}\newline Fourier--Bessel expansions,
almost everywhere and norm convergence.



\medskip
\noindent {\bf AMS(MOS) Subject Classifications.}\newline 42C10.


\medskip
\noindent {\bf Files.}\newline
vol.14.2002/pp223-235.dir/pp223-235.ps;\newline
vol.14.2002/pp223-235.dir/pp223-235.pdf;

\medskip
\noindent
{\bf Forward References.}\newline
\medskip



\end{description}



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