The main goal of this paper is to extend the approximation theorem of contiuous functions by Haar polynomials (see Theorem A) to infinite matrices (see Theorem C). The extension to the matricial framework will be based on the one hand on the remark that periodic functions which belong to$L$^{∞}($T$) may be one-to-one identified with Toeplitz matrices from$B$($l$_{2}) (see Theorem 0) and on the other hand on some notions given in the paper. We mention for instance:$ms$—a unital commutative subalgebra of$l$^{∞},$C$($l$_{2}) the matricial analogue of the space of all continuous periodic functions$C$($T$), the matricial Haar polynomials, etc.

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For an integral domain$R$and a non-zero non-unit$a$ε$R$we consider the number of distinct factorizations of$a$^{$n$}into irreducible elements of$R$for large$n$. Precise results are obtained for Krull domains and certain noetherian domains. In fact, we prove results valid for certain classes of monoids which then apply to the above-mentioned classes of domains.

Let$f$and$g$be nonlinear entire functions. The relations between the dynamics of$f⊗g$and$g⊗f$are discussed. Denote byℐ (·) and$F$(·) the Julia and Fatou sets. It is proved that if$z$∈$C$, then$z$∈ℐ8464 ($f⊗g$) if and only if$g(z)$∈ℐ8464 ($g⊗f$); if$U$is a component of$F$($f$○$g$) and$V$is the component of$F$($g$○$g$) that contains$g(U)$, then$U$is wandering if and only if$V$is wandering; if$U$is periodic, then so is$V$and moreover,$V$is of the same type according to the classification of periodic components as$U$. These results are used to show that certain new classes of entire functions do not have wandering domains.

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