## 2 Theoretical Background
### Hollow Waveguide of Arbitrary Cross Section
To this end, introduce the following integral operator \(\mathcal{L}:X\to Y\), with a logarithmic singularity of the kernel and integration over a smooth closed contour \(L_{0}\),  
\[U(q)\rfloor_{q\in L_{0}}=0.\]2  
where \(X\) denotes the Banach space \(X=X(L_{0})=H_{\beta}(L_{0})\) of Holder-continuous functions of index \(\beta\in(0,1)\) and \(Y=Y(L_{0})=H_{\beta}^{1}(L_{0})\) the space of continuously differentiable functions with Holder-continuous derivatives of index \(\beta\) on \(L_{0}\). Using the smooth parametrization of closed smooth curve \(L_{0}\) introduced above, one can separate the logarithmic singularity of the integral operator \(\mathcal{L}(L_{0})\) in the form  
\[U(q)\rfloor_{q\in L_{0}}=0.\]3  
obtained with the help of polar coordinates, or alternatively as  
\[U(q)\rfloor_{q\in L_{0}}=0.\]4  
Here, \(S_{1}(\theta_{0},\tau_{0})\) is a continuously differentiable function of its arguments and integral operators \(\mathcal{L}_{0}\) and \(S\) are independent of \(L_{0}\) and the spectral parameter \(k\). The following theorem is the main theoretical result, which provides a secure basis for the algorithm developed by the MAR. **Theorem 1**. : _The operator \(S:X\to Y\) is continuously invertible with continuous inverse \(S^{-1}:X\to Y\). The norm \(S^{-1}\) is a constant independent of \(L_{0}\)and the spectral parameter \(k\); in particular, for the map \(S^{-1}:\ W_{2}^{1}\to L_{2}\) (where \(L_{2}\) and \(W_{2}^{1}\) are standard notations for the spaces of square-integrable functions and functions with the square-integrable derivative), the norm \(S^{-1}\) equals 2. If contour \(L_{0}\) is such that the homogeneous equation \(\mathcal{L}\phi=0\), \(\phi\in X\), has only the trivial solution then \(\mathcal{L}:X\to Y\)is continuously invertible and the continuous inverse \(\mathcal{L}^{-1}:Y\to X\)is constructed explicitly in [30]._  
The key idea of the MAR is the splitting of the kernel of integral Equation (7) into a singular part and a regular (\"smooth\") part \(H_{00}(k;\theta_{0},\tau_{0})\) via  
\[U(q)\rfloor_{q\in L_{0}}=0.\]5  
It will be supposed that contour \(L_{0}\) is such that \(H_{00}(k;\theta_{0},\tau_{0})\) is smooth and continuously differentiable with respect to \(\theta_{0}\) and \(\tau_{0}\). Then, the OVF \(G(k)\) can be represented in the form  
\[U(q)\rfloor_{q\in L_{0}}=0.\]6  
and classified therefore as a Fredholm integral OVF with a logarithmically singular kernel. The homogeneous Equation (7) can be reduced to the equivalent form  
\[U(q)\rfloor_{q\in L_{0}}=0.\]7  
involving a canonical Fredholm operator of the second kind. Here, \(\hat{H}_{00}(k):Y\to Y\) is a completely-continuous integral OVF [30], which admits analytical continuation to the Riemann surface \(\mathcal{R}\) of the logarithmic function \(\ln(k)\) with a branch point in the origin. One can show, following [30], that the resolvent set of \(\hat{G}(k)\) is not empty, which implies that \(\hat{G}(k)\) is a Fredholm OVF and the (spectral) set \(\sum_{\hat{G}}\) of its characteristic numbers (CNs) is a countable set of isolated points in \(\mathcal{R}\). The properties of the kernel of the IE (7) allow the expansion of \(H_{00}(k;\,\theta_{0},\tau_{0})\) in a double Fourier series,  
\[U(q)\rfloor_{q\in L_{0}}=0.\]8  
where \(h^{00}(k)=\left\{h_{np}^{00}(k)\right\}_{n,p=-\infty}^{\infty}\) may be considered as an infinite matrix with the entries depending on the spectral parameter \(k\) (considered on \(\mathcal{R}\)) and satisfying the condition  
\[U(q)\rfloor_{q\in L_{0}}=0.\]9  
(The factor \(-\frac{i}{\pi}\) has been chosen to simplify further transforms.) Finally, we represent the singular part of the kernel (8) and unknown density function \(z_{0}(\tau_{0})\) in the Fourier series  
\[U(q)=\int_{L_{0}}G_{2}(k|p_{0}-q|)Z_{0}(p_{0})dl_{p_{0}} \tag{1}\]0  
and  
\[U(q)=\int_{L_{0}}G_{2}(k|p_{0}-q|)Z_{0}(p_{0})dl_{p_{0}} \tag{1}\]1  
where the prime over the summation sign in (11) excludes the term with \(n=0\).