By Alan Jeffrey
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Therefore, the solution to the inverse source problem in electrodynamics is not unique and may not exist, in general. 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]).
Let Then, and Au = p. , u is not the minimal-norm solution to the equation Au = p. 20 is proved. 22. 19 hold and and if where then where and is the minimal-norm solution to the equation A(u) = f. In particular, if 0 < a < l, then Indeed, where is the unique solution to the equation It is well known that provided that and, clearly, one multiplies the identity by and uses the monotonicity of A and the inequality The result similar to the one in the above remark can be found in [ARy]. 9 Regularizers for ill-posed problems must depend on the noise level In this Section we prove the following simple claim: Claim 2.
27) imply that In this argument we assume that the function f (x) satisfies the inequality 37 This inequality is satisfied if, for example, the derivative of f (x) exists except at a finite number of points and is uniformly bounded. 27). 8) is the Dirichlet kernel. 9) holds in and at any point at which f (x) is differentiable. In practice it is advisable to choose the system in such a way that tends rapidly to zero. Note that depends only on the system and therefore we can control this quantity to some extent by choosing the system Let us note that one can estimate f (x) at a given point optimally using the same procedure.