Mock period functions in higher level cases

Generalizing the results of Bringmann, Diamantis and Raum [3] we investigate a generating series of non-critical values which can be interpreted as a mock period function and prove that non-critical values can be encoded into a sesquiharmonic Maass form in higher level cases. Moreover we prove Eichler–Shimura-type isomorphism for the space of period polynomials and for that of mock period functions in level p   with p∈{2,3}$p\in \{2,3\}$.     

d-Symmetric d-orthogonal polynomials of Brenke type

In this paper, we characterize the d-symmetric d-orthogonal polynomials of Brenke type. We obtain two new families of polynomials and the moments of the corresponding d-dimensional vectors of linear functionals. Weights, providing integral representations for these moments, were also given.     

Interpolation by neural network operators activated by ramp functions

In this paper, a family of interpolation neural network operators are introduced. Here, ramp functions as well as sigmoidal functions generated by central B-splines are considered as activation functions. The interpolation properties of these operators are proved, together with a uniform approximation theorem with order, for continuous functions defined on bounded intervals. The relations with the theory of neural networks and with the theory of the generalized sampling operators are discussed.     

Commutativity of the Valdivia–Vogt table of representations of function spaces

In this article, we show that the Valdivia–Vogt structure table—containing the sequence space representations of the most used spaces of smooth functions appearing in the theory of distributions—can be interpreted as a commutative diagram, i.e., there is an isomorphism between the space of infinitely differentiable functions and the space , where s is the space of rapidly decreasing sequences, such that its restriction to the other function spaces in the structure table yields an isomorphism between these spaces of smooth functions and their sequence space representation. This result answers the corresponding question of Prof. Dietmar Vogt formulated on the conference “Functional Analysis: Applications to Complex Analysis and Partial Differential Equations” held in B?dlewo in May 2012.     

Quadrangulations of sphere and ball quotients

We give a classification of sphere quadrangulations satisfying a condition of non‐negative curvature, following Thurston's classification of sphere triangulations under the same condition. The generic family of quadrangulations is parametrized by the points of positive square‐norm of an integral Gaußian lattice in the six‐dimensional complex Lorentz space. There is a subgroup of automorphisms of acting on this lattice whose orbits parametrize sphere quadrangulations in a one‐to‐one manner. This group acts discretely on the corresponding five‐dimensional complex hyperbolic space; is of finite co‐volume; its ball quotient is the moduli space of unordered 8 points on the Riemann sphere, and also appears in Picard‐Terada‐Deligne‐Mostow list. Both Thurston's lattice and our lattice may be thought of as parametrizations of certain families of subgroups of the modular group; equivalently, of certain families of dessins. These families also parametrize points of a moduli space.     

Isometries on and monotone functions

We study necessary and sufficient conditions on a bounded operator T defined on the Hilbert space to be an isometry and show that, under suitable hypotheses, it suffices to restrict T to a smaller class of functions (e.g., if , to the cone of positive and decreasing functions). We also consider the problem of characterizing the sets for which the orthogonal projection of the operator T on is also an isometry. Finally, we illustrate our results with several examples involving classical operators on different settings.     

Hyponormality of bounded‐type Toeplitz operators

In this paper we deal with the hyponormality of Toeplitz operators with matrix‐valued symbols. The aim of this paper is to provide a tractable criterion for the hyponormality of bounded‐type Toeplitz operators (i.e., the symbol is a matrix‐valued function such that Φ and are of bounded type). In particular, we get a much simpler criterion for the hyponormality of when the co‐analytic part of the symbol Φ is a left divisor of the analytic part.