By Alan Jeffrey
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31). 4. 18). 18). 18) is unique when it exists. 18), be not empty. 18). 25). 18). 18), is where and are the Lagrange multipliers. 40) into 39 yields an equation for The roots of Eq. 37). Finally choose the for which (Bh, h) = min. 18). 5. 18) in the following way. 46) in with one constraint. 46) can be solved by the Lagrange multipliers method. One has where is the Lagrange multiplier. Thus, 40 2. 49) gives the corresponding H. 46). 3 ITERATIVE METHODS There is a vast literature on iterative methods [VV], [BG], [R65].
The antenna synthesis problems are inverse source-type problems of electrodynamics. For example, if j is the current along a linear antenna (which is a wire along is the length of the antenna, then so is determined uniquely by the data A. Finding which produces the desired diagram is the problem of linear antenna synthesis. There is a large body of literature on this subject. Let for The inverse source problem is: given find f (x, t), The questions mentioned in this subsection were discussed in many papers and books ([AVST], [MJ], [ZK], [R11], [R21], [R26], [R27], [R28], [R73], [Is2]).
More generally, consider a system where is a constant matrix, u (0) = v. If there are points such that the system is a linearly independent system of vectors, then u (t) determines the matrix uniquely. The reader can easily prove this. One can find more details in [Den]. 3 ILL-POSED PROBLEMS Why are ill-posed problems important in applications? How are they related to inverse problems? Let where X and Y are Banach spaces, or metric spaces, and A is a nonlinear operator, in general. 1) is called well-posed if A is a homeomorphism of X onto Y .