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Problems A, Aa and Ap) cannot be described by one mathematical formulation, because they have different restrictions on the required functions. On the other hand, physical problems similar to those considered in the previous chapter do not only arise in electromagnetic wave theory. They may differ only in the operator describing the devices or processes to be optimized. Here we give a general mathematical formulation of the problems. The problems considered in this chapter are divided onto three groups located in three sections.
They provide (exactly or approximately) the ﬁeld distribution u 1 on the antenna boundary determined in the ﬁrst step. The procedures which can be applied in this step are different for Problems Vs and Vw and are beyond the scope of this book. We refer readers to the books  and  for more details of them. 1 Waveguide mode convertors There is a problem in waveguide theory which consists of the conversion of one mode (or combination) into another mode (or combination). , [34, 35]). The ﬁrst work concerning the application of quasi-optical elements for this purpose was as expected .
50) It turns out that no stabilizing procedure is required to determine the greatest lower bound of the functional σ in this case. This fact is established by the theorem below. 2 Sequence fu n g is minimizing for the functional σ (u) if σ (u n ) ! µ as n ! 1. 1 If A is an isometric operator ˚acting « from H 1 into H2 , then any algorithm generating a minimizing sequence u δn for the functional σ δ (u) possesses the property σ δ u δn ! µ as δ ! 0, n ! 1. Proof: Let us consider the elementary inequality ˇ ˇ ˇ ˇ Á Á ˇ ˇ ˇ ˇ µ ˇ Ä ˇσ δ u δn µ δ ˇ C jµ δ µj .