By Hebe de Azevedo Biagioni

ISBN-10: 0387524088

ISBN-13: 9780387524085

ISBN-10: 3540524088

ISBN-13: 9783540524083

This ebook offers an easy advent to a nonlinear conception of generalized features brought by way of J.F. Colombeau, which provides a intending to any multiplication of distributions. This conception extends from natural arithmetic (it offers a devoted generalization of the classical thought of C? capabilities and gives a synthesis of such a lot current multiplications of distributions) to physics (it allows the solution of ambiguities that seem in items of distributions), passing in the course of the conception of partial differential equations either from the theoretical point of view (it furnishes an idea of susceptible answer of pde's resulting in existence-uniqueness leads to many circumstances the place no distributional answer exists) and the numerical perspective (it introduces new and effective equipment constructed lately in elastoplasticity, hydrodynamics and acoustics). this article provides easy ideas and effects which earlier have been purely released in article shape. it truly is in- tended for mathematicians yet, because the thought and functions are usually not dissociated it could even be worthwhile for physicists and engineers. The wanted must haves for its examining are primarily lowered to the classical notions of differential calculus and the idea of integration over n-dimensional euclidean spaces.

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Additional info for A Nonlinear Theory of Generalized Functions

Example text

9 - As inclusion xE~1, of where HEAVISIDE we have C(O) (or of G] GENERALIZED noticed in Cf(O)) into ~ is a to subspaces. is a n o p e n s u b s e t of ~l=II1 (~f]jl ( F ) ) / g and E ~s ( ~ ) = R(~i/n,(x,0 R E @ M , s[~] functions .... is given by 0)) representative of G (see FUNCTIONS. 3, ~s(~). there Since is no c a n o n i c a l a generalized function in ~(~) may be c o n s i d e r e d as a " l i m i t " of a f a m i l y { R ~ } e > o of C ® f u n c t i o n s o n ~, w e p r e s e n t b e l o w s o m e e x a m p l e s of functions R~ which, at the limit ~0, give back the classical Heaviside function Y E C t ( ~ ) d e f i n e d b y Y ( x ) = 0 if x < 0 a n d Y ( x ) = I if x>0 : 50 2 / I/ If we consider the c l a s s in ~ ( ~ ) of a p o w e r n > l { R e } o ( e < I d r a w n , for i n s t a n c e , at the left hand then, representing both families { R e} a n d { R ~ } below (R~ b e i n g r e p r e s e n t e d by the d o t t e d l i n e ) , / / /', i of t h e side, on the family above, figure R~ Rt / #, > we see that {Re} and {R~} represent different generalized functions (lock, for instance, at the difference (R~-R~)(O) which does not necessarily t e n d to z e r o w h e n ~*0) in ~ s ( ~ ) , each of t h e m b e i n g c a l l e d Heaviside generalized function.

Generalized function ~(x2-a2). ~ , ( ~ ) be a D i r a c g e n e r a l i z e d function Let R~ be a r e p r e s e n t a t i v e of $ . S e t R~ (£,x)-- and R s (~,A) dA b for some b<0. The class of Rl~ in ~9, (~) is a Heaviside generalized function H, and H' z$. 4, H" ( x ~ - a ~) is the class in ~ , ( R ) of the m a p (~,x) ~ R~(~,x~-a'). Let ~ 6 ~(~) be given. Then (*) same remark as in the f o o t n o t e of p . 5 0 behaviours of R~ ( g , x ) ( s u c h as a number N(~) between -A(~) and +A(~), with N(~)~+~ if eliminated in a p h y s i c a l context.

F g(x+8"r/)¢p(rt)d'O ] ~(x) dx when ~*0. 7 [] - Proposition. If f E @ , ( R 2p) a n d g , , ' ' ' J g p E C(fl), w h e r e fl C En is open, then f(gl,'--,g~) E ~(fl) is a s s o c i a t e d with f(gx,---,gp) w h e r e gi E ~(fl) is the f u n c t i o n gi v i e w e d as an e l e m e n t of ~(fl), 1~i~p. 37 Proof Let, for e ~ c h i = l , . . p , Ri be the r e p r e s e n t a t i v e of given by (6)or (6'). If R denotes the representative f(gi,-'',gp) given by (ii) and R the r e p r e s e n t a t i v e of continuous f u n c t i o n x ~-* f ( g i ( x ) , .

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