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Let us illustrate what we mean by the one-dimensional example. Let @(x,E), x E [0,1] be regular for x E [d, 11, Vd > 0. Then, by extension, a function Eim)@, x E (O,l] may be defined and, in any local variable t, the expansion EF)Eim)@ may be studied (if such an expansion exists). Suppose further, that in some local variable 5, we have E:'")@, ( E [O,A], V A > 0. Then, by extension a function Eim)@,5 E [O,co)can be defined. We can next study the regular expansion E y y ) @= E'"'T,E:"'@, In Chapter 3 we shall show that, for a class of functions, and with special provisions for the sequence d;), one has the relation Ep)E(m)E(m)@= E ( m ) E ( m ) @ .
4*, which defines regular and local expansions truncated to a prescribed order of accuracy. To facilitate further the study of the chapter we shall now briefly explain again the most frequently occurring operations, for the simple onedimensional case, and assuming non-trivial expansions. Consider functions @(x,&), x E [OJ], E E (O,cO], and the local variables where 6, are elements of some gauge set of order functions. A regular expansion is given by c 6fl(&)4fl(x), a(m) ELrn)@= n=O local expansions are given by c d:(&)$;)(l,).
In what follows we shall derive extension theorems under rather general conditions, for functions defined in D c R" of arbitrary dimension, and without specific choices for the norm of the function. This can easily be done, because, as we shall show, the extension theorems can be considered as consequences of two rather elementary lemmas on monotonic functions. We commence with the onedimensional situation, in the setting of uniform convergence. To define the ideas, and introduce the subject, we start with a classical result due to S .