Convolutions with unbounded unity

The Cauchy-type product of two arithmetic functions f and g on nonnegative integers is defined by

Acta Mathematica Sinica, English Series, 2019

In this paper, we consider a multiplicative convolution operator M f acting on a Hilbert spaces 2 (N, ω). In particular, we focus on the operators M 1 and M μ , where μ is the Möbius function. We investigate conditions on the weight ω under which the operators M 1 and M μ are bounded. We show that for a positive and completely multiplicative function f , M 1 is bounded on 2 (N, f 2) if and only if f 1 < ∞, in which case M 1 2,ω = f 1 , where ω n = f 2 (n). Analogously, we show that M μ is bounded on 2 (N, 1/n 2α) with M μ 2,ω = ζ(α) ζ(2α) , where ω n = 1/n 2α , α > 1. As an application, we obtain some results on the spectrum of M * 1 M 1 and M * μ M μ. Moreover, von Neumann algebra generated by a certain family of bounded operators is also considered.

2018

A b s t r ac t. A convolution is a mapping C of the set Z + of positive integers into the set P(Z +) of all subsets of Z + such that, for any n ∈ Z + , each member of C(n) is a divisor of n. If D(n) is the set of all divisors of n, for any n, then D is called the Dirichlet&#39;s convolution [2]. If U (n) is the set of all Unitary(square free) divisors of n, for any n, then U is called unitary(square free) convolution. Corresponding to any general convolution C, we can define a binary relation C on Z + by &#39;m C n if and only if m ∈ C(n)&#39;. In this paper, we present a characterization of regular convolution.

2014

The Cauchy-type product of two arithmetic functions f and g on nonnegative integers is defined by (f • g)(k) := ∑k m=0 ( k m ) f(m)g(k −m). We explore some algebraic properties of the aforementioned convolution, which is a fundamental characteristic of the identities involving the Bernoulli numbers, the Bernoulli polynomials, the power sums, the sums of products, and so forth.

Journal of Mathematical Analysis and Applications, 1978

We find the automorphisms and the spectra of several different topological convolution algebras of Cm-functions on the real line. Starting with the convolution algebra of compactly supported Cm-functions, equipped with the usual LF-topology, we define a corresponding convolution algebra of Cm-functions of arbitrarily fast exponential decay at CO; and convolution algebras of a given finite degree T of exponential decay at 00. These algebras may be described topologically as "hyper Schwartz spaces." With a natural Frechet topology, which we define, they get a structure as locally m-convex algebras. The continuous automorphisms and spectra of these algebras are described completely. We show that the algebra of Cm-functions of infmitly fast exponential decay at cc, .%Y, on the one hand, and the algebra of Cm-functions of only a finite degree e-'Jr1 decay at co, Yr", on the other hand, have quite different automorphisms, although 3E"Y = n, Y,". As an application, we show that the conformal group is canonically represented as the full group of automorphisms of .YYr', and that this representation does not extend to a representation on the Banach algebra L'(R).

Proceedings of the American Mathematical Society, 2007

We investigate the solvability of polynomial equations on the Calgebra of arithmetic functions g : N → C.

2018

A convolution is a mapping \(\mathcal{C}\) of the set \(Z^{+}\) of positive integers into the set \({\mathscr{P}}(Z^{+})\) of all subsets of \(Z^{+}\) such that, for any \(n\in Z^{+}\), each member of \(\mathcal{C}(n)\) is a divisor of \(n\). If \(\mathcal{D}(n)\) is the set of all divisors of \(n\), for any \(n\), then \(\mathcal{D}\) is called the Dirichlet&#39;s convolution [2]. If \(\mathcal{U}(n)\) is the set of all Unitary(square free) divisors of \(n\), for any \(n\), then \(\mathcal{U}\) is called unitary(square free) convolution. Corresponding to any general convolution \(\mathcal{C}\), we can define a binary relation \(\leq_{\mathcal{C}}\) on \(Z^{+}\) by `\(m\leq_{\mathcal{C}}n\) if and only if \( m\in \mathcal{C}(n)\)&#39;. In this paper, we present a characterization of regular convolution.

Mathematica

Let $E$ be the open unit disk $\{z\in \mathbb{C}: |z|&lt;1\}$. Let $A$ be the class of analytic functions in $E$, which have the form $f(z)=z+a_2z^2+...$. We define operators $L_n^\sigma\colon A\to A$ using the convolution *. Using these operators, we define and study new classes of functions in the unit disk. Moreover, we obtain some basic properties of the new classes, namely inclusion, growth, covering, distortion, closure under certain integral transformation and coefficient inequalities. Comment: 10 pages. Published

International Journal of Mathematics and Mathematical Sciences, 1990

Continuing a line of research initiated by Larsen, Liu and Wang [12], Martin and Yap [13], Gürkanli [15], and influenced by Reiter&#39;s presentation of Beurling and Segal algebras in Reiter [2,10] this paper presents the study of a family of Banach ideals of Beurling algebrasLw1(G),Ga locally compact Abelian group. These spaces are defined by weightedLp-conditions of their Fourier transforms. In the first section invariance properties and asymptotic estimates for the translation and modulation operators are given. Using these it is possible to characterize inclusions in section 3 and to show that two spaces of this type coincide if and only if their parameters are equal. In section 4 the existence of approximate identities in these algebras is established, from which, among other consequences, the bijection between the closed ideals of these algebras and those of the corresponding Beurling algebra is derived.

2008

Let n = p p νp(n) denote the canonical factorization of n ∈ N. The binomial convolution of arithmetical functions f and g is defined as (f • g)(n) = d|n p νp(n) νp(d) f (d)g(n/d), where a b is the binomial coefficient. We provide properties of the binomial convolution. We study the C-algebra (A, +, •, C), characterizations of completely multiplicative functions, Selberg multiplicative functions, exponential Dirichlet series, exponential generating functions and a generalized binomial convolution leading to various Möbius-type inversion formulas. Throughout the paper we compare our results with those of the Dirichlet convolution *. Our main result is that (A, +, •, C) is isomorphic to (A, +, * , C). We also obtain a "multiplicative" version of the multinomial theorem.