1. 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
2. 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
3. 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
4. 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
Keywords:Constrained approximation, polynomials, splines, degree of, approximation, $L_p$ space, Sobolev space
Categories:41A10, 41A15, 41A25, 41A29