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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.

Let$X$be a complex manifold and let$f$:$X$→ℂ^{$p$}be a holomorphic mapping defining a complete intersection. We prove that the iterated Mellin transform of the residue integral associated with$f$has an analytic continuation to a neighborhood of the origin in ℂ^{$p$}.

In this paper, we compare some deterministic and probabilistic techniques in the study of upper bounds in problems related to certain mean square discrepancie with respect to balls in the$d$-dimensional unit torus, and show that the quality of these techniques depends in an intricate way on the dimension$d$under consideration.

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