# Gilewicz J.

### Approximation of Smooth Functions by Weighted Means of *N*-Point Padé Approximants

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1410–1419

Let *f* be a function we wish to approximate on the interval [*x* _{1} *,x* _{ N }] knowing *p* _{1} *>* 1*,p* _{2} *, . . . ,p* _{ N } coefficients of expansion of *f* at the points *x* _{1} *,x* _{2} *, . . . ,x* _{ N } *.* We start by computing two neighboring *N* -point Padé approximants (NPAs) of *f,* namely *f* _{1} = [*m/n*] and *f* _{2} = [*m −* 1*/n*] of *f.* The second NPA is computed with the reduced amount of information by removing the last coefficient from the expansion of *f* at *x* _{1} *.* We assume that *f* is sufficiently smooth, (e.g. convex-like function), and (this is essential) that *f* _{1} and *f* _{2} bound *f* in each interval]*x* _{ i } *,x* _{ i+1}[ on the opposite sides (we call the existence of such two-sided approximants the two-sided estimates property of *f* ). Whether this is the case for a given function *f* is not necessarily known a priori, however, as illustrated by examples below it holds for many functions of practical interest. In this case, further steps become relatively simple. We select a known function *s* having the two-sided estimates property with values *s*(*x* _{ i }) as close as possible to the values *f*(*x* _{ i })*.* We than compute the approximants *s* _{1} = [*m/n*] and *s* _{2} = [*m −* 1*/n*] using the values at points *x* _{ i } and determine for all *x* the weight function *α* from the equation *s* = *αs* _{1} + (1 *− α*)*s* _{2} *.* Applying this weight to calculate the weighted mean *αf* _{1} + (1 *− α*)*f* _{2} we obtain significantly improved approximation of *f.*

### General algorithm of computation of $c$-table and detection of valleys

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 762–772

We present a review of all interesting results concerning the c-table obtained by the authors for the last two decades. These results are not widely known because they were presented in publications of limited circulation. We discuss different computational aspects of software producing the $c$-tables in the presence of blocs and their evolution following the evolution of the computer environment: effects of the use of 32-bit arithmetic .≈8 digits), 64-bit arithmetic (double precision, ≈16 digits), and Bailey’s Fortran multiprecision package .32 or 64 digits), competition between the ascending and descending algorithms, relationship between the complexity of computation and precision, overflow and underflow problems, competition between different formulas allowing one to overcome the blocs in the $c$-table, practical simple criterion of detecting numerical zeros in the c-table allowing to identify the blocs, and automatic detection of valleys.

### On the relation between measures defining the Stieltjes and the inverted Stieltjes functions

Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 327–331

A compact formula is found for the measure of the inverted Stieltjes function expressed by the measure of the original Stieltjes function.

### Negative result in pointwise 3-convex polynomial approximation

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 563-567

Let $Δ^3$ be the set of functions three times continuously differentiable on $[−1, 1]$ and such that $f'''(x) ≥ 0,\; x ∈ [−1, 1]$. We prove that, for any $n ∈ ℕ$ and $r ≥ 5$, there exists a function $f ∈ C^r [−1, 1] ⋂ Δ^3 [−1, 1]$ such that $∥f (r)∥_{C[−1, 1]} ≤ 1$ and, for an arbitrary algebraic polynomial $P ∈ Δ^3 [−1, 1]$, there exists $x$ such that $$|f(x)−P(x)| ≥ C \sqrt{n}ρ^r_n(x),$$ where $C > 0$ is a constant that depends only on $r, ρ_n(x) := \frac1{n^2} + \frac1n \sqrt{1−x^2}$.

### Coconvex Pointwise Approximation

Dzyubenko H. A., Gilewicz J., Shevchuk I. A.

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1200-1212

Assume that a function *f* ∈ *C*[−1, 1] changes its convexity at a finite collection *Y* := {*y* _{1}, ... *y* _{s}} of *s* points *y* _{i} ∈ (−1, 1). For each *n* > *N*(*Y*), we construct an algebraic polynomial *P* _{n} of degree ≤ *n* that is coconvex with *f*, i.e., it changes its convexity at the same points *y* _{i} as *f* and $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c{\omega }_{2} \left( {f,\frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in \left[ { - 1,1} \right],$$ where *c* is an absolute constant, ω_{2}(*f*, *t*) is the second modulus of smoothness of *f*, and if *s* = 1, then *N*(*Y*) = 1. We also give some counterexamples showing that this estimate cannot be extended to the case of higher smoothness.