We transpose the parametric geometry of numbers, recently created by Schmidt and Summerer, to fields of rational functions in one variable and analyze, in that context, the problem of simultaneous approximation to exponential functions.

In this work type II Hermite-Padé approximants for a vector of Cauchy transforms of smooth Jacobi-type densities are considered. It is assumed that densities are supported on mutually disjoint intervals (an Angelesco system with complex weights). The formulae of strong asymptotics are derived for any ray sequence of multi-indices.

Despite the efficiency of trajectory piecewise-linear (TPWL) model order reduction (MOR) for nonlinear circuits, it needs large amount of expansion points for large-scale nonlinear circuits. This will inevitably increase the model size as well as the simulation time of the resulting reduced macromodels. In this paper, subspace TPWL-MOR approach is developed for the model order reduction of nonlinear circuits. By breaking the high-dimensional state space into several subspaces with much lower dimensions, the subspace TPWL-MOR has very promising advantages of reducing the number of expansion points as well as increasing the effective region of the reduced-order model in the state space. As a result, the model size and the accuracy of the TWPL model can be greatly improved. The numerical results have shown dramatic reduction in the model size as well as the improvement in accuracy by using the subspace TPWL-MOR compared with the conventional TPWL-MOR approach.

We construct bivariate polynomial approximations of the Lerch function that for certain specialisations of the variables and parameters turn out to be Hermite–Padé 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 Prévost. 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.

We prove convergence of diagonal multipoint Padé approximants of Stieltjes-type functions when a certain moment problem is determinate. This is used for the study of the convergence of Fourier–Padé and nonlinear Fourier–Padé approximants for such type of functions.

It is well known that many polynomials which solve extremal problems on a single interval as the Chebyshev or the Bernstein-Szegö 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-Szegö 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 etc. are derived.

We show that if $m,n\ge 0,\lambda >1$, and $R$ is a rational function with numerator, denominator of degree $\le m,n$, respectively, then there exists a set $S\subset [0,1]$ of linear measure $\ge \,\frac{1}{4}\,\exp \left( -\frac{13}{\log \,\text{ }\!\!\lambda\!\!\text{ }} \right)$ such that for $r\in S$,

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$$_{|z|=r\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,|z|=r}^{\max |R(z)|/min|R(z)|\le {{\lambda }^{m+n}}.}$$

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 Padé approximants for functions meromorphic in the unit ball.

Let m, n be nonnegative integers and B(m+n) be a set of m + n + 1 real interpolation points (not necessarily distinct). Let Rm,n = Pm,n/Qm.n be the unique rational function with deg Pm,n ≤ m, deg Qm,n ≤ n, that interpolates ex in the points of B(m+n). If m = mv, n = nv with mv + nv → ∞, and mv / nv → λ as v → ∞, and the sets B(m+n) are uniformly bounded, we show that locally uniformly in the complex plane C, where the normalization Qm,n(0) = 1 has been imposed. Moreover, for any compact set K ⊂ C we obtain sharp estimates for the error |ez — Rm,n(z)| when z ∈ K. These results generalize properties of the classical Padé approximants. Our convergence theorems also apply to best (real) Lp rational approximants to ex on a finite real interval.