Several questions concerning the second order modulus of smoothness are addressed in this note. The central part is a refined analysis of a construction of certain smooth functions by Zhuk and its application to several problems in approximation theory, such as degree of approximation and the preservation of global smoothness. Lower bounds for some optimal constants introduced by Sendov are given as well. We also investigate an alternative approach using quadratic splines studied by Sendov.

Let $f$ be a power series with positive radius of convergence. In the present paper, we study the phenomenon of overconvergence of sequences of classical Pade' approximants pi{n,m_n} associated with f, where m(n)<=m(n+1)<=m(n) and m(n) = o(n/\log n), resp. m(n) = 0(n) as n is going to infiity. We extend classical results by J. Hadamard and A. A. Ostrowski related to overconvergent Taylor polynomials, as well as results by G. Lo'pez Lagomasino and A. Ferna'ndes Infante concerning overconvergent subsequences of a fixed row of the Pade' table.

In the present paper, results about the explicit asymptotics of sequences of rows and of closed to rows sequences of classical Pade approximants will be provided.

Nowadays, the research related to strong asymptotics of diagonal Pade approximants of different classes of functions is very topical. In the present paper, we pose the question about the strong asymptotics of row sequences pi(n,m), n -> infinity, m - fixed of classical Pade approximants. We show that under appropriate conditions on the power series f there is a sequence of row Fade approximants which behaves outside the disk of m-meromorphy of f like the Taylor series of the mth meromorphic continuation of f.

Given a regular compact set E in the complex plane C, a unit measure it supported by partial derivative E, a triangular point set ss :={{ss(n,k)}(k=1)(n)}(n=1)(infinity) , ss subset of partial derivative E and a function f, holomorphic on E, let pi(ss)(n),(f)(m) be the associated multipoint ss- Pade approximant of order (n, m). We show that if the sequence pi(ss)(n),(f)(m), n E Lambda,m- fixed, converges exact maximally to f relatively to the measure mu, then the points ss(n,k) are uniformly distributed on partial derivative E with respect to mu as n is an element of A. Furthermore, a result about the zeros behavior of the exact maximally convergent sequence Lambda is provided, under the condition that Lambda is "dense enough."

In the paper, we propose two new conjectures about the convergence of Hermite Approximants of multivalued analytic functions of Laguerre class ${\mathscr L}$. The conjectures are based in part on the numerical experiments, made recently by the authors in [26] and [27].

The paper investigates the distribution of interpolation points of m1-maximally convergent multipoint Padé approximants with numerator degree ≤n and denominator degree ≤mn for meromorphic functions f on a compact set E⊂C, where mn=o(n/logn) as n→∞. It is shown that the normalized counting measures (resp. their associated balayage measures onto the boundary of E) converge for a subsequence in the weak* sense to the equilibrium measure μE of E if the multipoint Padé approximants for one single function f converge exactly in m1-measure on the maximal Green domain of meromorphy Eρ(f).

In this paper are discussed the results of new numerical experiments on zero distribution of type I Hermite-Padé polynomials of order $n=200$ for three different collections of three functions $[1,f_1,f_2]$. These results are obtained by the authors numerically and do not match any of the theoretical results that were proven so far. We consider three simple cases of multivalued analytic functions $f_1$ and $f_2$, with separated pairs of branch points belonging to the real line. In the first case both functions have two logarithmic branch points, in the second case they both have branch points of second order, and finally, in the third case they both have branch points of third order. All three cases may be considered as representative of the asymptotic theory of Hermite-Padé polynomials. In the first two cases the numerical zero distribution of type I Hermite-Padé polynomials are similar to each other, despite the different kind of branching. But neither the logarithmic case, nor the square root case can be explained from the asymptotic point of view of the theory of type I Hermite-Padé polynomials. The numerical results of the current paper might be considered as a challenge for the community of all experts on Hermite-Padé polynomials theory.

Given a compact set E in \(\mathbb{C}\) and a function f holomorphic on E, we investigate the distribution of zeros of rational uniform approximants \(\{{r}_{n,{m}_{n}}\}\) with numerator degree≤n and denominator degree≤m n , where \({m}_{n} = o(n/\log n)\) as n→∞. We obtain a Jentzsch–Szegő type result, i.e., the zero distribution converges weakly to the equilibrium distribution of the maximal Green domain Eρ(f) of meromorphy of f if f has a singularity of multivalued character on the boundary ∂Eρ(f). Further, we show that any singular point of f on the boundary ∂Eρ(f), that is not a pole, is a limit point of zeros of the sequence \(\{{r}_{n,{m}_{n}}\}\).

We investigate the growth and the distribution of zeros of rational uniform approximations with numerator degree ≤n and denominator degree ≤m n for meromorphic functions f on a compact set E of ℂ where m n =o(n/log n) as n→∞. We obtain a Jentzsch–Szegő type result, i.e., the zero distribution converges weakly to the equilibrium distribution of the maximal Green domain E ρ(f) of meromorphy of f if f has a singularity of multivalued character on the boundary of E ρ(f). The paper extends results for polynomial approximation and rational approximation with fixed degree of the denominator. As applications, Padé approximation and real rational best approximants are considered.

We provide sufficient conditions for a sequence of rational functions, having no more than m poles in a domain D, to converge almost uniformly in m(1)-measure of Hausdorff inside D.