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R $^{d}$; it is also possible to consider a nonparametric class for α with restricted support, ¯ W . Of course without any restrictions ¯ W = R $^{d}$. Recall the differ- entiation operator, ∂ $^{m}$, for m = ( m$_{1}$, ...m$_{d}$ ˙ ) and denote by supp ( β, ∂ ) the set ∪ ∞ Σ m i $_{=0}$supp ( ∂ $^{m}$β ); where supp ( ∂ $^{m}$β ) is an open set where a continuous derivative ∂ $^{m}$β exists. Any point where β is zero belongs to this set if some finite-order partial continuous derivative of β is not zero at that point (and in some open neighborhood); for β itself supp ( β ) ≡ supp ( β, 0) . 

Define the functions 

<img src='content_image/321754.jpg'>

with any ˜ α such that α$_{1}$α$_{2}$ ∈ F t ( A ) . 

Consider the case when α, β and thus γ are continuous. For any point x$_{0}$ if β ( x$_{0}$ ) ̸ = 0 , there is a neighborhood N ( x$_{0}$ ) where β ̸ = 0 , and division by β is possible. If β ( x$_{0}$ ) has a zero, it could only be of finite order and in some neighborhood, N ( x$_{0}$ ) ∈ supp ( ∂ $^{m}$β ) a representation 

<img src='content_image/321753.jpg'>

holds for some continuous function η in S $^{∗}$, such that η > c$_{η}$ > 0 on supp ( η ) . Then η − $^{1}$γ in N ( x$_{0}$ ) is a non-zero continuous function; division of such a function by Π d i $_{=1}$( x$_{i}$ − x$_{0}$$_{i}$ ) m i in S ∗ is defined (Schwartz, 1967, pp. 125-126), thus division by β is defined in this neighborhood N ( x$_{0}$ ). For the 

