### A new convergence rate for the quadrature method for solving singular integral equations

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We study the homogeneous Riemann-Hilbert problem with a vanishing scalar-valued continuous coefficient. We characterize non-existence of nontrivial solutions in the case where the coefficient has its values along several rays starting from the origin. As a consequence, some results on injectivity and existence of eigenvalues of Toeplitz operators in Hardy spaces are obtained.

Let ${M}_{m,n}$ be the space of all complex m × n matrices. The generalized unit disc in ${M}_{m,n}$ is >br> ${R}_{m,n}=Z\in {M}_{m,n}:{I}^{\left(m\right)}-ZZ*ispositivedefinite$. Here ${I}^{\left(m\right)}\in {M}_{m,m}$ is the unit matrix. If 1 ≤ p < ∞ and α > -1, then ${L}_{\alpha}^{p}\left({R}_{m,n}\right)$ is defined to be the space ${L}^{p}{R}_{m,n};{\left[det({I}^{\left(m\right)}-ZZ*)\right]}^{\alpha}d{\mu}_{m,n}\left(Z\right)$, where ${\mu}_{m,n}$ is the Lebesgue measure in ${M}_{m,n}$, and ${H}_{\alpha}^{p}\left({R}_{m,n}\right)\subset {L}_{\alpha}^{p}\left({R}_{m,n}\right)$ is the subspace of holomorphic functions. In [8,9] M. M. Djrbashian and A. H. Karapetyan proved that, if $Re\beta >(\alpha +1)/p-1$ (for 1 < p < ∞) and Re β ≥ α (for p = 1), then $f\left(\right)={T}_{m,n}^{\beta}\left(f\right)\left(\right),\in {R}_{m,n},$where ${T}_{m,n}^{\beta}$ is the integral operator defined by (0.13)-(0.14). In the present paper, given 1 ≤ p <...

The paper is devoted to an integral equation algorithm for studying the scattering of plane waves by multilayer diffraction gratings under oblique incidence. The scattering problem is described by a system of Helmholtz equations with piecewise constant coefficients in ${\mathbb{R}}^{2}$ coupled by special transmission conditions at the interfaces between different layers. Boundary integral methods lead to a system of singular integral equations, containing at least two equations for each interface. To deal with...