A Course in Mathematics for Students of Physics, vol. 1 by Bamberg P., Sternberg S.

By Bamberg P., Sternberg S.

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This scheme was implemented by Wang [4] in a software package for finding solutions to Volterra integral equations of the first kind. We view this approach particularly suited to the approximate solution of ill-posed problems. For ill-posed problems the objections raised in Section 1 are always significant, so that it is not surprising that the classical error analysis has proved to be a weak tool for the treatment of ill-posed problems. While there are many effective ways of producing Reliability of Information from Approximately-Solved Problems 35 "good-looking" s o l u t i o n s , very l i t t l e is normally said about t h e i r error.

1 ,Ψ 1 > . . ·*>) 2 + λ||/ , ι*ιι 2 exists uniquely for every λ>0 and n M p 8λ= Σ ^ ι η , ) + Σ^νΦν+ ΣΘ,Ψ,·. 4) j=\ If ξ(· is the représenter of N-t in Hx then P ^ = P fa so it is not necessary to know ( or define ) η,- . ,<:„)', d = {d^dMÏ, and Θ = (filt . . , Θ / . G. Wahba 40 A simple one dimensional example is H =Hìespan{Oì, . . ^m} is the Sobolev space Η^ΙΌ,Ι] of functions with absolutely continuous m-1st derivatives and square integrable m th derivative with seminorm Jm (f ) given by 1 Jm(f) = l(f(m\x))2dx.

Over the years, a certain formalism has been developed which helps us in studying these questions in a very general setting. Restricted to linear operator equations, we consider the solution of Lx = y (1) f o r the unknown x, given L and y . The usual assumption is t h a t L is a l i n e a r operator with domain in a normed l i n e a r space X and range in another normed l i n e a r space Y. it When X and Y are i n f i n i t e - d i m e n s i o n a l is not possible t o solve (1) e x a c t l y , we resort t o d i s c r e t i z a t i o n and in which we replace (1) by Here L„ : X„ -► Y„ , n n n ' where now X„ and Y„ are f i n i t e - d i m e n s i o n a l n n of X and Y, r e s p e c t i v e l y .

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