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I've got an assignment to consider boundary problems in space of finite measures W, where the boundary conditions are considered in sense of weak approximation.

For example Hilbert's problem with boundary condition in sense of weak approx. will have following definition. $$ \lim_{r \to 1-0}\int\limits_T \ ({\Phi^+(rt)}-a(t){\Phi^-(r^{-1}t)})g(t)dt=\int\limits_Tg(t)d\mu $$ Where $$ T = \{ z: |z|=1 \},\ \forall g \in C,\ d\mu \in W \\\Phi^+(z)\ - \text{analytic function inside unit circle}\\ \Phi^-(z)\ -\text{analytical function outside the unit circle}\\ a(t) - \text{is a piecewise continuous function from Hölder's space.} $$

Questions :

  • I would like to understand what is the idea of formulating the problem in measure space ?
  • What generalization it does ?
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I guess the idea is that we look for a measure that is the weak limit of something related to the limit of the integrand on the left as $r\to 1$. But much remains unclear here. For example, how is $g$ related to $a$, if at all? Generally, what are the quantifiers here: the equation must hold for some ..., for all ... such that ..., etc. – user53153 Mar 5 '13 at 22:39
just updated the question, $$ a(t) - is\ a\ piecewise\ continues\ function\ from\ Hölder's\ space $$ and $$ g - is\ any\ function\ from\ C $$ – deimus Mar 6 '13 at 7:55
This is better but I still want a clarification about $r$. There is no $r$ on the right. Should there be $\lim\limits_{r\to 1}$ on the left? – user53153 Mar 6 '13 at 18:15
Correct, sorry for non-full info. – deimus Mar 7 '13 at 8:42

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