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On generalized convolution rengs of arithmetic functions

1980; University of Tsukuba; Volume: 4; Issue: 2 Linguagem: Inglês

10.21099/tkbjm/1496159171

2423-821X

Miguel Ferrero,

Digital Filter Design and Implementation

The set of all functions defined in the semi-group of the natural numbers whose values are in a commutative ring $R$ , becomes an associative ring when addition and multiplication are defined by the functional addition and the $\gamma$ -convolution $(f*g)(n)=\sum_{rs=n}f(r)g(s)\gamma(r, s)$ .In this paper, we will give some characterization of these rings.Finally, we will study on the extension of the base ring.Let $N$ be the multiplicative semi-group of the natural numbers and let $K$ be a field.In [4] the ring of arithmetic functions is defined as the set of all func- tions $f:N\rightarrow K$ whose addition and multiplication are defined by the functional addition and the convolution $(f*g)(n)=\sum_{rs=n}f(r)g(s)$ for every $n\in N$ .In [3], the generalized convolution ring is studied.The concept of convolution is generalized by a weighting kernel $\gamma:N\times N\rightarrow K$ and the multiplication is defined by $(f*g)(n)$ $=\sum_{rs=n}f(r)g(s)\gamma(r, s)$ .All weighting kernels $\gamma$ are characterized by the requirement that the set of all arithmetic functions still remains as an associative integral ring.We consider here a commutative unitary ring $R$ and we define the generalized convolution ring of arithmetic functions over $R$ , in an analogous way.If $\gamma(r, s)$ $=1$ for every $r,$ $s$ in $N$ , the ring of arithmetic functions, which is denoted by $\mathcal{F}(N, R)$ , is naturally isomorphic to the formal power series ring with infinitely many indeterminates $R[[X_{1}, \cdots, X_{n}, \cdots]]$ via the the application $\Gamma$ which is defined as follow.If $\{p_{1}, \cdots, p_{n}\cdots\}$ is the sequence of all prime numbers, we put $\Gamma(X_{i})=f_{pi}$ where $f_{pj}(n)=\delta_{pi^{n}},$ ( $\delta$ is the Kronecker's delta).In section 1, we characterize these rings as solution of an universal problem.Section 2 is devoted to characterize them in an intrinsic way.In section 3 we concern with the extension of the base ring.This paper is originated in the unpublished paper "Sur les anneaux de fonc-