[en] By using the concept of Choquet nonlinear integral with respect to a submodular set function, we introduce the nonlinear Picard–Choquet operators, Gauss–Weierstrass–Choquet operators and Poisson–Cauchy–Choquet operators with respect to a family of submodular set functions. Quantitative approximation results of the order ${\mathrm{\omega <!--\; ??\; -->}}_1(f;t{)}_{\mathbb{R}}$, $t>0$, are obtained with respect to the Choquet measure $\mathrm{\mu <!--\; ??\; -->}(A)=\sqrt{M(A)}$ where M represents the Lebesgue measure, and with respect to some families of possibility measures. Also, due to the possibilities of choice for the submodular set functions, for some subclasses of functions we prove that these Choquet type operators have essentially better approximation properties than their classical correspondents.

[en] We will present an estimation of the generalized strong mean $(H,\mathrm{\Phi <!--\; ??\; -->})$ as an approximation version of the Totik type generalization of the results of J. Marcinkiewicz and A. Zygmund on strong summability of Fourier series of integrable functions. As a measure of such approximation we will use the function constructed by function $\mathrm{\Psi <!--\; ??\; -->}$ complementary to $\mathrm{\Phi <!--\; ??\; -->}$ on the base of definition of the Gabisoniya points. Some corollary and remark will also be given.

[en] Suppose that Q is a family of seminorms on a locally convex space E which determines the topology of E. We study the existence of Q-nonexpansive retractions for families of Q-nonexpansive mappings and prove that a separated and sequentially complete locally convex space E has the weak fixed point property for commuting separable semitopological semigroups of Q-nonexpansive mappings. This proves the Bruck’s problem (Pacific J Math 53:59–71, 1974) for locally convex spaces. Moreover, we prove the existence of Q-nonexpansive retractions for the right amenable Q-nonexpansive semigroups.

[en] We study the regularity of solutions to the p-Poisson equation, $1<p<\mathrm{\infty <!--\; ???\; -->}$, in the vicinity of a vertex of a polygonal domain. In particular, we are interested in smoothness results in the adaptivity scale of Besov spaces $B_{\mathrm{\tau <!--\; ??\; -->}}^{{\mathrm{\sigma <!--\; ??\; -->}}_{\mathrm{\tau <!--\; ??\; -->}}}(L_{\mathrm{\tau <!--\; ??\; -->}}(\mathrm{\Omega <!--\; ??\; -->}))$, $1/\mathrm{\tau <!--\; ??\; -->}={\mathrm{\sigma <!--\; ??\; -->}}_{\mathrm{\tau <!--\; ??\; -->}}/2+1/p$, since the regularity in this scale is known to determine the maximal approximation rate that can be achieved by adaptive and other nonlinear approximation methods. We prove that under quite mild assumptions on the right-hand side f, solutions to the p-Poisson equation possess Besov regularity ${\mathrm{\sigma <!--\; ??\; -->}}_{\mathrm{\tau <!--\; ??\; -->}}$ for all $0<{\mathrm{\sigma <!--\; ??\; -->}}_{\mathrm{\tau <!--\; ??\; -->}}<2$. In case f vanishes in a small neighborhood of the corner, the solutions even admit arbitrary high Besov smoothness. The proofs are based on singular expansion results and continuous embeddings of intersections of Babuska–Kondratiev spaces ${\mathcal{K}}_{p,a}^{\mathrm{\ell <!--\; ???\; -->}}(\mathrm{\Omega <!--\; ??\; -->})$ and Besov spaces $B_p^s(L_p(\mathrm{\Omega <!--\; ??\; -->}))$ into the specific scale of Besov spaces we are interested in. In regard of these embeddings, we extend the existing results to the limit case $\mathrm{\ell <!--\; ???\; -->}\to <!--\; ???\; -->\mathrm{\infty <!--\; ???\; -->}$ by showing that the Fréchet spaces ${\cap <!--\; ???\; -->}_{\mathrm{\ell <!--\; ???\; -->}=1}^{\mathrm{\infty <!--\; ???\; -->}}{\mathcal{K}}_{p,a}^{\mathrm{\ell <!--\; ???\; -->}}(\mathrm{\Omega <!--\; ??\; -->})\cap <!--\; ???\; -->B_p^s(L_p(\mathrm{\Omega <!--\; ??\; -->}))$ are continuously embedded into the metrizable complete topological vector space ${\cap <!--\; ???\; -->}_{{\mathrm{\sigma <!--\; ??\; -->}}_{\mathrm{\tau <!--\; ??\; -->}}>0}{B}_{\mathrm{\tau <!--\; ??\; -->}}^{{\mathrm{\sigma <!--\; ??\; -->}}_{\mathrm{\tau <!--\; ??\; -->}}}(L_{\mathrm{\tau <!--\; ??\; -->}}(\mathrm{\Omega <!--\; ??\; -->}))$.

[en] The symbol ${\mathrm{\delta <!--\; ??\; -->}}^k(f(x))$ for an infinitely differentiable function f having a simple root is meaningless in the theory of Schwartz distributions. In this work we first of all give meaning to the symbol ${\mathrm{\delta <!--\; ??\; -->}}^k(f(x))$ via neutrix calculus due to van der Corput (J Anal Math 7:291–398, 1959). Then we consider the case that f is the s-th power of a function g(x) having a simple single root and the particular case ${\mathrm{\delta <!--\; ??\; -->}}^k(x_{+}^{\mathrm{\lambda <!--\; ??\; -->}})$ for $\mathrm{\lambda <!--\; ??\; -->}>0,k\in <!--\; ???\; -->\mathbb{N}.$ Finally we also give meaning to the symbols $H({\mathrm{\delta <!--\; ??\; -->}}^k),H^{(r)}({\mathrm{\delta <!--\; ??\; -->}}^k),{\mathrm{\delta <!--\; ??\; -->}}^{(r)}({\mathrm{\delta <!--\; ??\; -->}}^k)$ and $G({\mathrm{\delta <!--\; ??\; -->}}^k),$ where H denotes the Heaviside function and G is a bounded continuous summable function on $\mathbb{R}$.

[en] For a non-conformal harmonic surface in $R^3$, we give transforms to get holomorphic maps to the 2-sphere, which is a generalization of the classical fact that the Gauss map of a minimal surface in $R^3$ is holomorphic.

[en] We discuss to what extent the local techniques of resolution of singularities over fields of characteristic zero can be applied to improve singularities in general. For certain interesting classes of singularities, this leads to an embedded resolution via blowing ups in regular centers. We illustrate this for generic determinantal varieties. The article is partially expository and is addressed to non-experts who aim to construct resolutions for other special classes of singularities in positive or mixed characteristic.

[en] The characteristic interval plays a vital role on the existence of iterative roots of PM functions with height less than or equal to one. In this paper, we define the characteristic interval for continuous functions and prove theorems on extension and nonexistence of iterative roots for a class of continuous non-PM functions on a closed and bounded interval I. Also, we prove that a class of continuous non-PM functions, which do not possess any iterative roots, is dense in C(I, I).

[en] We study the existence and the properties of the solution to a stochastic partial differential equation with multiplicative time-space fractional noise. The equation we consider involves a pseudo-differential operator that generates a stable-like process and it extends the standard heat equation. Our techniques are based on stochastic analysis, Malliavin calculus and Wiener-Itô chaos expansion.

[en] In the present paper, we introduce two kinds of complex Bernstein–Stancu polynomials and complex Kantorovich–Stancu polynomials in movable compact disks. Their approximation properties for analytic functions in the movable compact disks are considered.