The aim of this paper is to obtain Grüss and GrüssVoronovskaya inequalities with exact quantitativeestimates (with respect to the degree) for the complex convolution polynomial operators of de la Vallée Poussin, ofZygmundRiesz and of Jackson, acting on analytic functions.Keywords: Complex convolution polynomials, de la ValléePoussin kernel, RieszZygmund kernel, Jackson kernel,Grüsstype estimate, GrüssVoronovskayatype estimate, analytic functions.

. We provide a version of Korovkintype theorems for monotone sublinear operators in vector latticesand discuss the possibilities of further extensions and generalizations.

In this paper, by using generalized Cesàro means based on qintegers, we study on approximatingcontinuous and periodic functions by their Fourier series. We also discuss its connection with the concept of statisticalconvergence. At the end of the paper, some applications and graphical illustrations are also provided.Keywords: Fourier analysis, Cesàro summability, Fejér’s kernel, qintegers, statistical convergence.

We generalize the classical Lebesgue’s theorem to multidimensional functions. We prove that theCesàro means of the Fourier series of the multidimensional function f ∈ L1(log L)d−1(Td) ⊃ Lp(Td) (1 < p ≤ ∞)converge to f at each strong Lebesgue point.

In this paper, some recent applications of the socalled Generalized Bernstein polynomials are collected.This polynomial sequence is constructed by means of the samples of a continuous function f on equispaced points of[0, 1] and depends on an additional parameter which can be suitable chosen in order to improve the rate of convergenceto the function f, as the smoothness of f increases, overcoming the wellknown low degree of approximation achievedby the classical Bernstein polynomials or by the piecewise polynomial approximation. The applications consideredhere deal with the numerical integration and the simultaneous approximation. Quadrature rules on equidistant nodesof [0, 1] are studied for the numerical computation of ordinary integrals in one or two dimensions, and usefully employed in Nyström methods for solving Fredholm integral equations. Moreover, the simultaneous approximationof the Hilbert transform and its derivative (the Hadamard transform) is illustrated. For all the applications, somenumerical details are given in addition to the error estimates, and the proposed approximation methods have beenimplemented providing numerical tests which confirm the theoretical estimates. Some open problems are also introduced.

In this paper, we establish a quantitative estimate for multivariate sampling Kantorovich operators bymeans of the modulus of smoothness in the general setting of Orlicz spaces. As a consequence, the qualitative order ofconvergence can be obtained, in case of functions belonging to suitable Lipschitz classes. In the particular instance ofLpspaces, using a direct approach, we obtain a sharper estimate than that one that can be deduced from the generalcase.

The authors provide a complete asymptotic expansion for a class of functions in terms of the completeBell polynomials. In particular, they obtain known asymptotic expansions of some Keller type sequences.Keywords: Asymptotic expansions, Bell polynomials.

T. We laconically describe the great contributions of Professor Francesco Altomare to mathematical research and Ph.D education, and his unique status in the mathematical community. In particular, we present and giveexamples of his innovative and great achievements related to the following areas of mathematics: Functional Analysis,Operator Theory, Potential Theory, Approximation Theory, Probability Theory, Function Spaces, Choquet’s Theory,Dirichlet’s Problem and Semigroup Theory. Moreover, we report on and give concrete examples of his unique way towork together with Ph.D students, both before and sometimes also after their dissertation. Finally, we shortly describehis remarkable “class travel” from “simple” conditions with no academic traditions in his family in the small townGiovinazzo to finally become the broad, ingenious, and powerful mathematician he is regarded to be today.Keywords: Francesco Altomare, History of Mathematics, Functional Analysis, Operator Theory, Potential Theory, Approximation Theory, Probability Theory, Function Spaces, Choquet’s Theory, Dirichlet’s Problem, Semigroup Theoryand Evolution Equations, Ph.D education.

We present uniform and Lp mixed CaputoBochner abstract generalized fractional Landau inequalitiesover R of fractional orders 2 < α ≤ 3. These estimate the size of first and second derivatives of a composition with aBanach space valued function over R. We give applications when α = 2.5.

T. In this paper, we give a survey about recent versions of Korovkintype theorems for modular functionspaces, a class which includes Lp, Orlicz, MusielakOrlicz spaces and many others. We consider various kinds ofmodular convergence, using certain summability processes, like triangular matrix statistical convergence, and filterconvergence (which are generalizations of the statistical convergence). Finally, we consider an abstract axiomaticconvergence which includes the previous ones and even almost convergence, which is not generated by any filter, aswe show by an example.
