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We give a representation of the classical theory of multiplicative arithmetic functions (MF)in the ring of symmetric polynomials. The basis of the ring of symmetric polynomials that we use is the isobaric basis, a basis especially sensitive…
In this article, we study spectral Barron spaces whose elements are made up of some vector-valued functions on a compact group whose Fourier transforms admit a certain summability property. We investigate their functional properties and…
Some concepts, such as non-compactness measure and condensing operators, defined on metric spaces are extended to uniform spaces. Such extensions allow us to locate, in the context of uniform spaces, some classical results existing in…
In this paper we introduce and study the integral means spectrum (IMS) functionals on Teichm\"uller spaces. We show that the IMS functionals on the closure of the universal Teichm\"uller space and the universal asymptotic Teichm\"uller…
Hecke symmetries generalize the usual tensor symmetry of vector spaces $v\otimes w\arrow w\otimes v$ as well as the symmetry of vector superspaces. To a Hecke symmetry $R$ there associates a quadratic algebra which can be interpreted as the…
We study the representations of the quantum Galilei group by a suitable generalization of the Kirillov method on spaces of non commutative functions. On these spaces we determine a quasi-invariant measure with respect to the action of the…
We give a review of modern approaches to constructing formal solutions to integrable hierarchies of mathematical physics, whose coefficients are answers to various enumerative problems. The relationship between these approaches and…
We study the properties based on the space generated by the translates of square integrable function using C-bracket on Vilenkin group. We give the necessary and sufficient condition for frame sequences by the family of translates of the…
Given a dilation matrix M, a so-called space of M-positive vectors in the Euclidean space is introduced and studied. An algebraic structure of this space is similar to the positive half-line equipped with the termwise addition modulo 2,…
The two-dimensional quantum harmonic oscillator is modified with reflection terms associated with the action of the Coxeter group $B_2$, which is the symmetry group of the square. The angular momentum operator is also modified with…
We study the connection between block Krylov subspaces and matrix orthogonal functions. Under a no-deflation assumption, we show that polynomial block Krylov subspaces are isometrically isomorphic to spaces of matrix polynomials of bounded…
In this study, we define a brief description of the hyperbolic and elliptic rotational surfaces using a curve and matrices in 4-dimensional semi Euclidean space. That is, we provide different types of rotational matrices, which are the…
We construct interpolation operators for functions taking values in a symmetric space -- a smooth manifold with an inversion symmetry about every point. Key to our construction is the observation that every symmetric space can be realized…
We construct a set of quaternionic metamonogenic functions (that is, in $\mbox{Ker}(D+\lambda)$ for diverse $\lambda$) in the unit disk, such that every metamonogenic function is approximable in the quaternionic Hilbert module $L^2$ of the…
We survey recent results on the representation theory of symplectic reflection algebras, focusing particularly on connections with symplectic quotient singularities and their resolutions, spaces of representations of quivers, and on…
We study the operator-valued positive definite functions on a group using positive block matrices. We give an alternative proof to Brehmer positivity for doubly commuting contractions. We classify all commuting unitary representations over…
In this note the smooth (i.e. with open stabilizers) linear and {\sl semilinear} representations of certain permutation groups (such as infinite symmetric group or automorphism group of an infinite-dimensional vector space over a finite…
An integral representation of the intertwining operator for the Dunkl operators associated with symmetric groups is derived for the class of functions of a single component. The expression provides a closed form formula for the reproducing…
This work rests upon the certainty that only fields of real and complex numbers, quaternions and octonions have algebras of all four arithmetical operations. Also quaternions are good to represent 3-dimensional Euclid space and…
We apply matrix methods to arithmetic functions by associating matrices to the functions in a manner drawn from the theory of symmetric functions. Then we study the characteristic polynomials of the associated matrices.