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Generalized Fourier series with orthogonal polynomial bases have useful applications in several fields, including differential equations, pattern recognition, and image and signal processing. However, computing the generalized Fourier…
Generalised Morrey (function) spaces enjoyed some interest recently and found applications to PDE. Here we turn our attention to their discrete counterparts. We define generalised Morrey sequence spaces…
In this paper, we deal with a generalization of the geometry of parallelizable manifolds, or the absolute parallelism (AP-) geometry, in the context of generalized Lagrange spaces. All geometric objects defined in this geometry are not only…
Every maximal Hardy field has a proper elementary differential subfield that is Dedekind complete in the maximal Hardy field. This pair of Hardy fields is a transserial tame pair, shown to have a complete and model complete elementary…
The universal approximation property uniformly with respect to weakly compact families of measures is established for several classes of neural networks. To that end, we prove that these neural networks are dense in Orlicz spaces, thereby…
An abstract theory of Fourier series in locally convex topological vector spaces is developed. An analog of Fej\'{e}r's theorem is proved for these series. The theory is applied to distributional solutions of Cauchy-Riemann equations to…
We develop category-theoretic framework for universal homogeneous objects, with some applications in the theory of Banach spaces, linear orderings, and in topology of compact spaces.
In their paper Scholze and Weinstein show that a certain diagram of perfectoid spaces is Cartesian. In this paper, we generalize their result. This generalization will be used in a forthcoming paper of ours to compute certain non-trivial…
We investigate the Pad\'e approximation method for the analytic continuation of numerical data and its ability to access, from the Euclidean propagator, both the spectral function and part of the physical information hidden in the second…
Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order $n$ that are uniformly valid for unbounded complex values of the argument $z$, including the real interval $0 \leq z \leq 1$ in which the zeros in…
An attempt is made to describe random matrix ensembles with unitary invariance of measure (UE) in a unified way, using a combination of Tracy-Widom (TW) and Adler-Shiota-Van Moerbeke (ASvM) approaches to derivation of partial differential…
For a linear ODE with indeterminate coefficients, we explicitly exhibit a fundamental system of solutions, in terms of the coefficients. We show that the generalized Wronskians of the fundamental system are given by an action of the Schur…
Earlier, there were defined two generalized (``motivic'') versions of the Poincar\'e series of a collection of plane valuations on the algebra ${\mathcal O}_{{\mathbb C}^2,0}$ of germs of holomorphic functions in two variables. One of them…
We prove a formula for the Taylor series coefficients of a zero of the sum of a complex-exponent polynomial and a base function which is a general holomorphic function with a simple zero. Such a Taylor series is more general than a Puiseux…
We consider various collections of functions from the Baire space X into itself naturally arising in (effective) descriptive set theory and general topology, including computable (equivalently, recursive) functions, contraction mappings,…
We obtain local parametrizations of classical non-compact Lie groups where adjoint invariants under maximal compact subgroups are manifest. Extension to non compact subgroups is straightforward. As a by-product parametrizations of the same…
We consider generalized metric spaces taking distances in an arbitrary ordered commutative monoid, and investigate when a class $\mathcal{K}$ of finite generalized metric spaces satisfies the Hrushovski extension property: for any…
We give a geometrical characterization of $\lambda$-prolongations of vector fields, and hence of $\lambda$-symmetries of ODEs. This allows an extension to the case of PDEs and systems of PDEs; in this context the central object is a…
We introduce a model of the set of all Polish (=separable complete metric) spaces: the cone $\cal R$ of distance matrices, and consider geometric and probabilistic problems connected with this object. The notion of the universal distance…
We construct two families of orthogonal polynomials associated with the universal central extensions of the superelliptic Lie algebras. These polynomials satisfy certain fourth order linear differential equations, and one of the families is…