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By A.I. Mal'cev

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Then, there exist positive integers ill, l = 1, ... , q, such that for every probability measure 7r and for every VI ~ ill, l = 1, ... ,q, there exist weight vectors 10, l = 1, ... Ql) ) and positive scalars {3 and c;, l = 1, ... ,q, such that where C; = (2rct)2 and r is the radius of the smallest closed sphere (centered in the origin) containing I (see {34} for the expression of (3). 4, it is worth noting that, in a sense, Proposition 2 specifies quantitatively the content of Proposition 1. More specifically, with reference to the l-th component 1'1 (I) of the optimal control function 1° (I) , if assumptions A6, A7, and A8 are verified and the constants Cl, l = 1, ...

The sets of nodes that are upstream and downstream neighbors of node i are denoted by P(i) and S(i) , respectively. We assume that at each node i EN there is a queue in which messages are stored for the destination node D. Therefore, the node D must be reachable from each node of N. We assume that the rate of messages that are sent from one node to another is synchronously updated by control variables at discrete periodic instants 0,1, ... , T - 1. Up to this point, the communication network may be considered as the discretetime version of the continuous-time model proposed by Segall in [35] and used in subsequent works.

Q) . In order to characterize the ability of the functions the functions lQ ~ col(1lli, I 1ivd (I, 1lli) = 1, ... , q) to approximate ,nn, we introduce the integrated square error hI,? -1i V ') 12 1r( d n, evaluated on the domain I (1r is a probability measure). We assume such a domain to contain the origin. 4, we need to assume that the optimal control functions It to be approximated belong to classes of the type (10). However, the functions It have been considered on the domain I, which is a subset of the space IRs, where s ~ dim (I).

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