Abstract
This paper addresses the optimal recovery of functions from Hilbert spaces of functions on the unit disc. The estimation, or recovery, is performed from inaccurate information given by integration along radial paths. For a holomorphic function expressed as a series, three distinct situations are considered: where the information error in L2 norm is bound by δ＞0 or for a finite number of terms the error in l2N norm is bound by δ＞0 or lastly the error in the jth coefficient is bound by δj＞0. The results are applied to the Hardy-Sobolev and Bergman-Sobolev spaces.
Highlights
Ix y Y (1)Tx and m y in Z, i.e. minimize Tx m y
This paper addresses the optimal recovery of functions from Hilbert spaces of functions on the unit disc
The estimation, or recovery, is performed from inaccurate information given by integration along radial paths
Summary
Theorem 1 gives a constructive approach to finding an optimal method mfrom the information. It follows from results obtained in [1,2,3,4,5,6,7] (see [8] where this theorem was proven for one particular case.). In order to apply Theorem 1 the values of extremal problems (4) and the dual problem (3) must agree. When one encounters extremal problems, one approach is to construct the Lagrange function. If we wish to combine Theorems 1 and 2 to determine an optimal error and method we must show the posed problem is able to satisfy equating extremal problems (3) and (4).
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