Expand all Collapse all | Results 1 - 23 of 23 |
1. CJM 2010 (vol 62 pp. 737)
Approximation by Dilated Averages and K-Functionals For a positive finite measure $d\mu(\mathbf{u})$ on $\mathbb{R}^d$
normalized to satisfy $\int_{\mathbb{R}^d}d\mu(\mathbf{u})=1$, the dilated average of
$f( \mathbf{x})$ is given by \[ A_tf(\mathbf{x})=\int_{\mathbb{R}^d}f(\mathbf{x}-t\mathbf{u})d\mu(\mathbf{u}). \] It
will be shown that under some mild assumptions on $d\mu(\mathbf{u})$ one has
the equivalence \[ \|A_tf-f\|_B\approx \inf \{
(\|f-g\|_B+t^2 \|P(D)g\|_B): P(D)g\in B\}\quad\text{for }t>0, \]
where $\varphi(t)\approx \psi(t)$ means
$c^{-1}\le\varphi(t)/\psi(t)\le c$, $B$ is a Banach space of functions
for which translations are continuous isometries and $P(D)$ is an
elliptic differential operator induced by $\mu$. Many applications are
given, notable among which is the averaging operator with $d\mu(\mathbf{u})=
\frac{1}{m(S)}\chi_S(\mathbf{u})d\mathbf{u}$, where $S$ is a bounded convex set
in $\mathbb{R}^d$ with an interior point, $m(S)$ is the Lebesgue measure of
$S$, and $\chi_S(\mathbf{u})$ is the characteristic function of $S$. The rate
of approximation by averages on the boundary of a convex set under
more restrictive conditions is also shown to be equivalent to an
appropriate $K$-functional.
Keywords:rate of approximation, K-functionals, strong converse inequality Categories:41A27, 41A35, 41A63 |
2. CJM 2009 (vol 61 pp. 1341)
Simultaneous Polynomial Approximations of the Lerch Function We construct bivariate polynomial approximations of the Lerch
function that for certain specialisations of the variables and
parameters turn out to be Hermite--Pad\'e approximants either of
the polylogarithms or of Hurwitz zeta functions. In the former
case, we recover known results, while in the latter the results
are new and generalise some recent works of Beukers and Pr\'evost.
Finally, we make a detailed comparison of our work with Beukers'.
Such constructions are useful in the arithmetical study of the
values of the Riemann zeta function at integer points and of the
Kubota--Leopold $p$-adic zeta function.
Categories:41A10, 41A21, 11J72 |
3. CJM 2009 (vol 61 pp. 373)
An Infinite Order Whittaker Function In this paper we construct a flat smooth section of an induced space
$I(s,\eta)$ of $SL_2(\mathbb{R})$ so that the attached Whittaker function
is not of finite order.
An asymptotic method of classical analysis is used.
Categories:11F70, 22E45, 41A60, 11M99, 30D15, 33C15 |
4. CJM 2009 (vol 61 pp. 299)
\v{C}eby\v{s}ev Sets in Hyperspaces over $\mathrm{R}^n$ A set in a metric space is called a \emph{\v{C}eby\v{s}ev set} if
it has a unique ``nearest neighbour'' to each point of the space. In
this paper we generalize this notion, defining a set to be
\emph{\v{C}eby\v{s}ev relative to} another set if every point in the
second set has a unique ``nearest neighbour'' in the first. We are
interested in \v{C}eby\v{s}ev sets in some hyperspaces over $\R$,
endowed with the Hausdorff metric, mainly the hyperspaces of compact
sets, compact convex sets, and strictly convex compact sets.
We present some new classes of \v{C}eby\v{s}ev and relatively
\v{C}eby\v{s}ev sets in various hyperspaces. In particular, we show
that certain nested families of sets are \v{C}eby\v{s}ev. As these
families are characterized purely in terms of containment, without
reference to the semi-linear structure of the underlying metric space,
their properties differ markedly from those of known \v{C}eby\v{s}ev
sets.
Keywords:convex body, strictly convex set, \v{C}eby\v{s}ev set, relative \v{C}eby\v{s}ev set, nested family, strongly nested family, family of translates Categories:41A52, 52A20 |
5. CJM 2007 (vol 59 pp. 730)
Large Sieve Inequalities via Subharmonic Methods and the Mahler Measure of the Fekete Polynomials We investigate large sieve inequalities such as
\[
\frac{1}{m}\sum_{j=1}^{m}\psi ( \log | P( e^{i\tau
_{j}}) | ) \leq \frac{C}{2\pi }\int_{0}^{2\pi }\psi \left(
\log [ e| P( e^{i\tau }) | ] \right) \,d\tau
,
\]
where $\psi $ is convex and increasing, $P$ is a polynomial or an
exponential of a potential, and the constant $C$ depends on the degree of $P$,
and the distribution of the points $0\leq \tau _{1}<\tau _{2}<\dots<\tau
_{m}\leq 2\pi $. The method allows greater generality and is in some ways
simpler than earlier ones. We apply our results to estimate the Mahler
measure of Fekete polynomials.
Category:41A17 |
6. CJM 2007 (vol 59 pp. 3)
Holomorphic Generation of Continuous Inverse Algebras We study complex commutative Banach algebras
(and, more generally, continuous
inverse algebras) in which the holomorphic functions of a fixed $n$-tuple
of elements are dense. In particular, we characterize the compact subsets
of~$\C^n$ which appear as joint spectra of such $n$-tuples. The
characterization is compared with several established notions of holomorphic
convexity by means of approximation
conditions.
Keywords:holomorphic functional calculus, commutative continuous inverse algebra, holomorphic convexity, Stein compacta, meromorphic convexity, holomorphic approximation Categories:46H30, 32A38, 32E30, 41A20, 46J15 |
7. CJM 2006 (vol 58 pp. 1026)
Karamata Renewed and Local Limit Results Connections between behaviour of real analytic functions (with no
negative Maclaurin series coefficients and radius of convergence one)
on the open unit interval, and to a lesser extent on arcs of the unit
circle, are explored, beginning with Karamata's approach. We develop
conditions under which the asymptotics of the coefficients are related
to the values of the function near $1$; specifically, $a(n)\sim
f(1-1/n)/ \alpha n$ (for some positive constant $\alpha$), where
$f(t)=\sum a(n)t^n$. In particular, if $F=\sum c(n) t^n$ where $c(n)
\geq 0$ and $\sum c(n)=1$, then $f$ defined as $(1-F)^{-1}$ (the
renewal or Green's function for $F$) satisfies this condition if $F'$
does (and a minor additional condition is satisfied). In come cases,
we can show that the absolute sum of the differences of consecutive
Maclaurin coefficients converges. We also investigate situations in
which less precise asymptotics are available.
Categories:30B10, 30E15, 41A60, 60J35, 05A16 |
8. CJM 2006 (vol 58 pp. 249)
Convergence of Fourier--PadÃ© Approximants for Stieltjes Functions We prove convergence of diagonal multipoint Pad\'e approximants of
Stieltjes-type functions when a certain moment problem is
determinate. This is used for the study of the convergence of
Fourier--Pad\'e and nonlinear Fourier--Pad\'e approximants for such
type of functions.
Keywords:rational approximation, multipoint PadÃ© approximants, Fourier--PadÃ© approximants, moment problem Categories:41A20, 41A21, 44A60 |
9. CJM 2005 (vol 57 pp. 1224)
Convex Polynomial Approximation in the Uniform Norm: Conclusion Estimating the degree of approximation in the uniform norm, of a
convex function on a finite interval, by convex algebraic
polynomials, has received wide attention over the last twenty
years. However, while much progress has been made especially in
recent years by, among others, the authors of this article,
separately and jointly, there have been left some interesting open
questions. In this paper we give final answers to all those open
problems. We are able to say, for each $r$th differentiable convex
function, whether or not its degree of convex polynomial
approximation in the uniform norm may be estimated by a
Jackson-type estimate involving the weighted Ditzian-Totik $k$th
modulus of smoothness, and how the constants in this estimate
behave. It turns out that for some pairs $(k,r)$ we have such
estimate with constants depending only on these parameters. For
other pairs the estimate is valid, but only with constants that
depend on the function being approximated, while there are pairs
for which the Jackson-type estimate is, in general, invalid.
Categories:41A10, 41A25, 41A29 |
10. CJM 2003 (vol 55 pp. 1019)
More Eventual Positivity for Analytic Functions Eventual positivity problems for real convergent Maclaurin series lead
to density questions for sets of harmonic functions. These are solved
for large classes of series, and in so doing, asymptotic estimates are
obtained for the values of the series near the radius of convergence
and for the coefficients of convolution powers.
Categories:30B10, 30D15, 30C50, 13A99, 41A58, 42A16 |
11. CJM 2003 (vol 55 pp. 576)
Automorphic Orthogonal and Extremal Polynomials It is well known that many polynomials which solve extremal problems
on a single interval as the Chebyshev or the Bernstein-Szeg\"o
polynomials can be represented by trigonometric functions and their
inverses. On two intervals one has elliptic instead of trigonometric
functions. In this paper we show that the counterparts of the Chebyshev
and Bernstein-Szeg\"o polynomials for several intervals can be represented
with the help of automorphic functions, so-called Schottky-Burnside
functions. Based on this representation and using the Schottky-Burnside
automorphic functions as a tool several extremal properties of such
polynomials as orthogonality properties, extremal properties with
respect to the maximum norm, behaviour of zeros and recurrence
coefficients {\it etc.} are derived.
Categories:42C05, 30F35, 31A15, 41A21, 41A50 |
12. CJM 2002 (vol 54 pp. 945)
Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications |
Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications Given a homogeneous elliptic partial differential operator $L$ with constant
complex coefficients and a class of functions (jet-distributions) which
are defined on a (relatively) closed subset of a domain $\Omega$ in $\mathbf{R}^n$ and
which belong locally to a Banach space $V$, we consider the problem of
approximating in the norm of $V$ the functions in this class by ``analytic''
and ``meromorphic'' solutions of the equation $Lu=0$. We establish new Roth,
Arakelyan (including tangential) and Carleman type theorems for a large class
of Banach spaces $V$ and operators $L$. Important applications to boundary
value problems of solutions of homogeneous elliptic partial differential
equations are obtained, including the solution of a generalized Dirichlet
problem.
Keywords:approximation on closed sets, elliptic operator, strongly elliptic operator, $L$-meromorphic and $L$-analytic functions, localization operator, Banach space of distributions, Dirichlet problem Categories:30D40, 30E10, 31B35, 35Jxx, 35J67, 41A30 |
13. CJM 2001 (vol 53 pp. 489)
Bivariate Polynomials of Least Deviation from Zero Bivariate polynomials with a fixed leading term $x^m y^n$, which
deviate least from zero in the uniform or $L^2$-norm on the unit disk
$D$ (resp. a triangle) are given explicitly. A similar problem in
$L^p$, $1 \le p \le \infty$, is studied on $D$ in the set of products
of linear polynomials.
Categories:41A10, 41A50, 41A63 |
14. CJM 2000 (vol 52 pp. 815)
On the Maximum and Minimum Modulus of Rational Functions We show that if $m$, $n\geq 0$, $\lambda >1$, and $R$ is a rational function
with numerator, denominator of degree $\leq m$, $n$, respectively, then there
exists a set $\mathcal{S}\subset [0,1] $ of linear measure $\geq
\frac{1}{4}\exp (-\frac{13}{\log \lambda })$ such that for $r\in
\mathcal{S}$,
\[
\max_{|z| =r}| R(z)| / \min_{|z| =r} | R(z) |\leq \lambda ^{m+n}.
\]
Here, one may not replace $\frac{1}{4}\exp ( -\frac{13}{\log \lambda })$
by $\exp (-\frac{2-\varepsilon }{\log \lambda })$, for any $\varepsilon >0$.
As our motivating application, we prove a convergence result for diagonal
Pad\'{e} approximants for functions meromorphic in the unit ball.
Categories:30E10, 30C15, 31A15, 41A21 |
15. CJM 1999 (vol 51 pp. 546)
Strong Converse Inequalities for Averages in Weighted $L^p$ Spaces on $[-1,1]$ Averages in weighted spaces $L^p_\phi[-1,1]$ defined by additions
on $[-1,1]$ will be shown to satisfy strong converse inequalities
of type A and B with appropriate $K$-functionals. Results for
higher levels of smoothness are achieved by combinations of
averages. This yields, in particular, strong converse inequalities
of type D between $K$-functionals and suitable difference operators.
Keywords:averages, $K$-functionals, weighted spaces, strong converse inequalities Categories:41A25, 41A63 |
16. CJM 1998 (vol 50 pp. 1273)
Mean convergence of Lagrange interpolation for exponential weights on $[-1,1]$ We obtain necessary and sufficient conditions for mean convergence of
Lagrange interpolation at zeros of orthogonal polynomials for weights on
$[-1,1]$, such as
\[
w(x)=\exp \bigl(-(1-x^{2})^{-\alpha }\bigr),\quad \alpha >0
\]
or
\[
w(x)=\exp \bigl(-\exp _{k}(1-x^{2})^{-\alpha }\bigr),\quad k\geq 1,
\ \alpha >0,
\]
where $\exp_{k}=\exp \Bigl(\exp \bigl(\cdots\exp (\ )\cdots\bigr)\Bigr)$
denotes the $k$-th iterated exponential.
Categories:41A05, 42C99 |
17. CJM 1998 (vol 50 pp. 412)
Asymptotic transformations of $q$-series For the $q$-series $\sum_{n=0}^\infty a^nq^{bn^2+cn}/(q)_n$
we construct a companion $q$-series such that the asymptotic
expansions of their logarithms as $q\to 1^{\scriptscriptstyle -}$
differ only in the dominant few terms. The asymptotic expansion
of their quotient then has a simple closed form; this gives rise
to a new $q$-hypergeometric identity. We give an asymptotic
expansion of a general class of $q$-series containing some of
Ramanujan's mock theta functions and Selberg's identities.
Categories:11B65, 33D10, 34E05, 41A60 |
18. CJM 1998 (vol 50 pp. 152)
Inequalities for rational functions with prescribed poles This paper considers the rational system ${\cal P}_n
(a_1,a_2,\ldots,a_n):= \bigl\{ {P(x) \over \prod_{k=1}^n (x-a_k)},
P\in {\cal P}_n\bigr\}$ with nonreal elements in
$\{a_k\}_{k=1}^{n}\subset\Bbb{C}\setminus [-1,1]$ paired by complex
conjugation. It gives a sharp (to constant) Markov-type inequality
for real rational functions in ${\cal P}_n (a_1,a_2,\ldots,a_n)$.
The corresponding Markov-type inequality for high derivatives
is established, as well as Nikolskii-type inequalities. Some
sharp Markov- and Bernstein-type inequalities with curved majorants
for rational functions in ${\cal P}_n(a_1,a_2,\ldots,a_n)$ are
obtained, which generalize some results for the classical
polynomials. A sharp Schur-type inequality is also proved and
plays a key role in the proofs of our main results.
Keywords:Markov-type inequality, Bernstein-type inequality, Nikolskii-type inequality, Schur-type inequality, rational functions with prescribed poles, curved majorants, Chebyshev polynomials Categories:41A17, 26D07, 26C15 |
19. CJM 1997 (vol 49 pp. 1242)
$1$-complemented subspaces of spaces with $1$-unconditional bases We prove that if $X$ is a complex strictly monotone sequence
space with $1$-un\-con\-di\-tion\-al basis, $Y \subseteq X$ has no bands
isometric to $\ell_2^2$ and $Y$ is the range of norm-one projection from
$X$, then $Y$ is a closed linear span a family of mutually
disjoint vectors in $X$.
We completely characterize $1$-complemented subspaces and norm-one
projections in complex spaces $\ell_p(\ell_q)$ for $1 \leq p, q <
\infty$.
Finally we give a full description of the subspaces that are spanned
by a family of disjointly supported vectors and which are
$1$-complemented in (real or complex) Orlicz or Lorentz sequence
spaces. In particular if an Orlicz or
Lorentz space $X$ is not isomorphic to $\ell_p$ for some $1 \leq p <
\infty$ then the only subspaces
of $X$ which are $1$-complemented and disjointly supported are the
closed linear spans of block bases with constant
coefficients.
Categories:46B20, 46B45, 41A65 |
20. CJM 1997 (vol 49 pp. 1034)
Ray sequences of best rational approximants for $|x|^\alpha$ The convergence behavior of best uniform rational
approximations $r^\ast_{mn}$ with numerator degree~$m$
and denominator degree~$n$ to the function $|x|^\alpha$,
$\alpha>0$, on $[-1,1]$ is investigated. It is assumed
that the indices $(m,n)$ progress along a ray sequence in
the lower triangle of the Walsh table, {\it i.e.} the
sequence of indices $\{ (m,n)\}$ satisfies
$$
{m\over n}\rightarrow c\in [1, \infty)\quad\hbox{as } m+
n\rightarrow\infty.
$$
In addition to the convergence behavior, the asymptotic
distribution of poles and zeros of the approximants and the
distribution of the extreme points of the error function
$|x|^\alpha - r^\ast_{mn} (x)$ on $[-1,1]$ will be studied.
The results will be compared with those for paradiagonal
sequences $(m=n+2[\alpha/2])$ and for sequences of best
polynomial approximants.
Keywords:Walsh table, rational approximation, best approximation,, distribution of poles and zeros. Categories:41A25, 41A44 |
21. CJM 1997 (vol 49 pp. 944)
Approximation by multiple refinable functions We consider the shift-invariant space,
$\bbbs(\Phi)$, generated by a set $\Phi=\{\phi_1,\ldots,\phi_r\}$
of compactly supported distributions on $\RR$ when the vector of
distributions $\phi:=(\phi_1,\ldots,\phi_r)^T$ satisfies a system
of refinement equations expressed in matrix form as
$$
\phi=\sum_{\alpha\in\ZZ}a(\alpha)\phi(2\,\cdot - \,\alpha)
$$
where $a$ is a finitely supported sequence of $r\times r$ matrices
of complex numbers. Such {\it multiple refinable functions} occur
naturally in the study of multiple wavelets.
The purpose of the present paper is to characterize the {\it accuracy}
of $\Phi$, the order of the polynomial space contained in
$\bbbs(\Phi)$, strictly in terms of the refinement mask $a$. The
accuracy determines the $L_p$-approximation order of $\bbbs(\Phi)$ when
the functions in $\Phi$ belong to $L_p(\RR)$ (see Jia~[10]).
The characterization is achieved in terms of the eigenvalues and
eigenvectors of the subdivision operator associated with the mask $a$.
In particular, they extend and improve the results of Heil, Strang
and Strela~[7], and of Plonka~[16]. In addition, a
counterexample is given to the statement of Strang and Strela~[20]
that the eigenvalues of the subdivision operator determine the
accuracy. The results do not require the linear independence of
the shifts of $\phi$.
Keywords:Refinement equations, refinable functions, approximation, order, accuracy, shift-invariant spaces, subdivision Categories:39B12, 41A25, 65F15 |
22. CJM 1997 (vol 49 pp. 568)
A counterexample in $L^p$ approximation by harmonic functions For ${n \over {n-2}}\leq p<\infty$ we show that the
conditions $C_{2,q}(G\setminus \dox)=C_{2,q}(G \setminus
X)$ for all open sets $G$, $C_{2,q}$ denoting Bessel capacity, are not
sufficient to characterize the compact
sets $X$ with the property that each function harmonic on $\dox$
and in $L^p(X)$ is the limit in the $L^p$ norm of a sequence
of functions which are harmonic on neighbourhoods of $X$.
Categories:41A30, 31C15 |
23. CJM 1997 (vol 49 pp. 74)
Constrained approximation in Sobolev spaces Positive, copositive, onesided and intertwining (co-onesided) polynomial
and spline approximations of functions $f\in\Wp^k\mll$ are considered.
Both uniform and pointwise estimates, which are exact in some sense, are
obtained.
Keywords:Constrained approximation, polynomials, splines, degree of, approximation, $L_p$ space, Sobolev space Categories:41A10, 41A15, 41A25, 41A29 |