Related papers: Sieve functions in arithmetic bands
The arithmetic function of two variables is defined. Some properties of the function are given along with the formula that is an analog of the so-called Mobius' inversion formula. A heuristic statement is suggested.
We investigate the boundedness of the $H^\infty$-calculus by estimating the bound $b(\varepsilon)$ of the mapping $H^{\infty}\rightarrow \mathcal{B}(X)$: $f\mapsto f(A)T(\varepsilon)$ for $\varepsilon$ near zero. Here, $-A$ generates the…
We develop a wavelet like representation of functions in $L^p(\mathbb{R})$ based on their Fourier--Hermite coefficients; i.e., we describe an expansion of such functions where the local behavior of the terms characterize completely the…
Suppose that $(P,Q) \in \mathbb{N}_2^{\mathbb{N}} \times \mathbb{N}_2^{\mathbb{N}}$ and $x=E_0.E_1E_2\cdots$ is the $P$-Cantor series expansion of $x \in \mathbb{R}$. We define $\psi_{P,Q}(x):=\sum_{n=1}^\infty \frac {\min(E_n,q_n-1)} {q_1…
Jensen's inequality is ubiquitous in measure and probability theory, statistics, machine learning, information theory and many other areas of mathematics and data science. It states that, for any convex function $f\colon K \to \mathbb{R}$…
A classical problem in number theory is showing that the mean value of an arithmetic function is asymptotic to its mean value over a short interval or over an arithmetic progression, with the interval as short as possible or the modulus as…
In this paper we collect some results about arithmetic progressions of higher order, also called polynomial sequences. Those results are applied to $(m,q)$-isometric maps.
Let $f$ be an $\mathbb{F}_q$-linear function over $\mathbb{F}_{q^n}$. If the $\mathbb{F}_q$-subspace $U= \{ (x^{q^t}, f(x)) : x\in \mathbb{F}_{q^n} \}$ defines a maximum scattered linear set, then we call $f$ a scattered polynomial of index…
In this paper we study a new family of sinc--like functions, defined on an interval of finite width. These functions, which we call ``little sinc'', are orthogonal and share many of the properties of the sinc functions. We show that the…
We extend the Matom\"{a}ki-Radziwi\l\l{} theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a…
In this paper, we consider a quaternionic short-time Fourier transform (QSTFT) with normalized Hermite functions as windows. It turns out that such a transform is based on the recent theory of slice polyanalytic functions on quaternions.…
Cyclic sieving is a well-known phenomenon where certain interesting polynomials, especially $q$-analogues, have useful interpretations related to actions and representations of the cyclic group. We propose a definition of sieving for an…
This paper studies the approximation of generalized ridge functions, namely of functions which are constant along some submanifolds of $\mathbb{R}^N$. We introduce the notion of linear-sleeve functions, whose function values only depend on…
We classify the $Q$-homogeneous skew Schur $Q$-functions, i.e., those of the form $Q_{\lambda/\mu} = k \cdot Q_{\nu}$. On the way we develop new tools that are useful also in the context of other classification problems for skew Schur…
We introduce a type of surgery on metric spaces. This surgery, in some sense, seeks to replace a subspace $S$ of a metric space $X$ with another metric space $T$ via a function $f : S \to T$. When $T$ is a discrete space, this amounts to…
In this paper, we develop some basic techniques towards the Riemann hypothesis for higher rank non-abelian zeta functions of an integral regular projective curve of genus $g$ over a finite field $\mathbb F_q$. As an application of the…
The set B of geodesic rays avoiding a suitable obstacle in a complete negatively curved Riemannian manifold determines a spectrum S. While various properties of this spectrum are known, we define and study dimension functions on S in terms…
A subequation on an open subset $X\subset \mathbb R^n$ is a subset $F$ of the space of $2$-jets on $X$ with certain properties. A smooth function is said to be $F$-subharmonic if all of its $2$-jets lie in $F$, and using the viscosity…
We analyse the role of vertex operator algebra and 2d amplitudes from the point of view of the representation theory of infinite dimensional Lie algebras, MacMahon and Ruelle functions. A p-dimensional MacMahon function is the generating…
It is known that backward iterations of independent copies of a contractive random Lipschitz function converge almost surely under mild assumptions. By a sieving (or thinning) procedure based on adding to the functions time and space…