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This happens if the approximation basis is orthonormal. The coeﬃcients are given by (4). While implementating, the basis functions are represented via their values in a ﬁnite range of discrete arguments. , en (r)], and r is a vector of M real entries from the range [a, b]. Adopting Matlab notation, e0 (r) means a vector of e0 function values calculated in the arguments speciﬁed by the vector r. Thus r contains the samples of a real axis (−∞, ∞). Obviously it must be truncated, so it also means that B(r)T B(r) = In+1 ,.

ICANN 2008, Part I, LNCS 5163, pp. 21–30, 2008. c Springer-Verlag Berlin Heidelberg 2008 22 Y. Ito, C. Srinivasan, and H. Izumi We observed that the diﬃculty in training of our networks arose from the optimization of the inner parameters. In the case of learning Bayesian discriminant functions, the teacher signals are dichotomous random variables. Learning with such teacher signals is diﬃcult because the approximation cannot be realized by simply bringing the output of the network close to the target function.

130–132]) which extends the concept of Lebesgue integrals to mappings into Banach spaces; the value of a Bochner integral is a function, not a number. The Bochner integral is suitable for dealing with arbitrary computational units. ) to Banach spaces of functions computable by units with such parameters; see Kainen [19]. A special case of application of the Bochner to estimation of rates of approximation by neural networks was sketched by Girosi and Anzellotti [9]. The paper is organized as follows.