Related papers: Adelic versions of the Weierstrass approximation t…

The pointwise estimates of the deviations $\widetilde{T}_{n,A,B}^{\text{}%}f\left(\cdot \right) -\widetilde{f}(\cdot)$ and $\widetilde{T}_{n,A,B}^{% \text{}}f\left(\cdot \right) -\widetilde{f}(\cdot,\varepsilon)$ in terms of moduli of…

The universal C*-algebras of discrete product systems generalize the Toeplitz- Cuntz algebras and the Toeplitz algebras of discrete semigroups. We consider a semigroup P which is quasi-lattice ordered in the sense of Nica, and, for a…

We introduce a class of real algebraic varieties characterised by a simple rationality condition, which exhibit strong properties regarding approximation of continuous and smooth mappings by regular ones. They form a natural counterpart to…

Given a porous compact $K \subset \mathbb{R}^d$ and a continuity modulus $\omega$, we prove a quantitative Jackson-Bernstein type theorem on harmonic approximation. That is, a function $f$ belongs to the class $\mathrm{Lip}_{\omega}(K)$ if…

A theorem of Hoischen states that given a positive continuous function $\varepsilon:\mathbb{R}\to\mathbb{R}$, an integer $n\geq 0$, and a closed discrete set $E\subseteq\mathbb{R}$, any $C^n$ function $f:\mathbb{R}\to\mathbb{R}$ can be…

In terms of the best approximations of functions and generalized moduli of smoothness, direct and inverse approximation theorems are proved for Besicovitch almost periodic functions whose Fourier exponent sequences have a single limit point…

We study rings over which an analogue of the Weierstrass preparation theorem holds for power series. We show that a commutative ring $R$ admits a factorization of every power series in $R[[x]]$ as the product of a polynomial and a unit if…

Let $E/\mathbb{Q}_p$ be an elliptic curve whose mod $p$ Galois image is contained in the normaliser of a non-split Cartan. We classify the possible $p$-adic images of $E$ using tools from $p$-adic Hodge theory via a careful analysis of the…

We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some methods of this approximation for functions in a class induced by the convolution with a given function,…

Let $p\in\mathbb Z$ be a prime, $\overline{\mathbb Q_p}$ a fixed algebraic closure of the field of $p$-adic numbers and $\overline{\mathbb Z_p}$ the absolute integral closure of the ring of $p$-adic integers. Given a residually algebraic…

Given an elliptic curve E/Q and a prime p at which E has good reduction, let e_p be the exponent of the group E_p(F_p) of F_p-rational points on the reduction of E modulo p. Under the Generalized Riemann Hypothesis (GRH) for the Dedekind…

For primes $p>3$ we produce a new derivation of the universal $p$-adic sigma function and $p$-adic Weierstrass zeta functions of Mazur and Tate for ordinary elliptic curves by a method that highlights congruences among coefficients in…

We define a Fr\'echet topology on the space $C^\infty(X)[[\hbar]]$ of formal smooth functions on a symplectic manifold $X$, by constructing a sequence of semi-norms on it. For any star product $\star$ on $C^\infty(X)[[\hbar]]$ making it a…

In this paper, we establish that the space $ \mathbb{P}_p $ of all periodic function of fundamental period $ p $ can be expressed as a direct sum of the space $ \mathbb{P}_{p/2} $ of all periodic functions with fundamental period $ p/2 $…

We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstras P-function using two…

Several new inequalities for moduli of smoothness and errors of the best approximation of a function and its derivatives in the spaces $L_p$, $0<p<1$, are obtained. For example, it is shown that for any $0<p<1$ and $k,\,r\in \mathbb{N}$ one…

We establish asymptotic estimates for the least upper bounds of approximations in the uniform metric by Fourier sums of order $n-1$ of classes of $2\pi$-periodic Weyl--Nagy differentiable functions, $W^r_{\beta,p}, 1\le p\le \infty,…

Let $S$ be a dense subring of the real numbers. In this paper we prove a polynomial version of Van der Waerden's theorem near zero. In fact, we prove that if $p_1,\ldots,p_m \in \mathbb{Z}[x]$ are polynomials such that $p_i(0) = 0$ and…

Let $G$ be a profinite group. A strongly admissible smooth representation $\rho$ of $G$ over $\mathbb{C}$ decomposes as a direct sum $\rho \cong \bigoplus_{\pi \in \mathrm{Irr}(G)} m_\pi(\rho) \, \pi$ of irreducible representations with…

Let $E/\mathbb Q$ be an elliptic curve and for each prime $p$, let $N_p$ denote the number of points of $E$ modulo $p$. The original version of the Birch and Swinnerton-Dyer conjecture asserts that $\prod \limits _{p \leq x} \frac{N_p}{p}…