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A Riemann-Hilbert problem with a vanishing coefficient and applications to Toeplitz operators

A. Perälä, J. A. Virtanen, L. Wolf (2013)

Concrete Operators

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.

Bounded projections in weighted function spaces in a generalized unit disc

A. H. Karapetyan (1995)

Annales Polonici Mathematici

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 M m , n : I ( m ) - Z Z * i s p o s i t i v e d e f i n i t e . Here I ( m ) M m , m is the unit matrix. If 1 ≤ p < ∞ and α > -1, then L α p ( R m , n ) is defined to be the space L p R m , n ; [ d e t ( I ( m ) - Z Z * ) ] α d μ m , n ( Z ) , where μ m , n is the Lebesgue measure in M m , n , and H α p ( R m , n ) L α p ( R m , n ) is the subspace of holomorphic functions. In [8,9] M. M. Djrbashian and A. H. Karapetyan proved that, if R e β > ( α + 1 ) / p - 1 (for 1 < p < ∞) and Re β ≥ α (for p = 1), then     f ( ) = T m , n β ( f ) ( ) , R m , n , where T m , n β is the integral operator defined by (0.13)-(0.14). In the present paper, given 1 ≤ p <...

Conical diffraction by multilayer gratings: A recursive integral equation approach

Gunther Schmidt (2013)

Applications of Mathematics

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 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...

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