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University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > Wiener-Hopf factorisation through an intermediate space and applications to diffraction theory

## Wiener-Hopf factorisation through an intermediate space and applications to diffraction theoryAdd to your list(s) Download to your calendar using vCal - Frank Speck (Universidade de Lisboa)
- Monday 12 August 2019, 10:00-11:00
- Seminar Room 1, Newton Institute.
If you have a question about this talk, please contact info@newton.ac.uk. WHTW01 - Factorisation of matrix functions: New techniques and applications An operator factorisation conception is investigated for $ where $A : X ightarrow Z$ and $A_-: Z ightarrow Y$ have certain invariance properties and the cross factor $C : Z ightarrow Z$ splits the “intermediate space” $Z$ into complemented subspaces closely related to the kernel and cokernel of $W$, such that $W$ is equivalent to a “simpler” operator, $W sim P C | $.+ cup mathbb{R}The main result shows equivalence between the generalised invertibility of the Wiener-Hopf operator and this kind of factorisation (provided $P_1 sim P_2$) which implies a formula for a generalised inverse of $W$. It embraces I.B. Simonenko's generalised factorisation of matrix measurable functions in $L ^{p$ spaces and various other factorisationapproaches.As applications we consider interface problems in weakformulation for the n-dimensional Helmholtz equation in $Omega =mathbb{R}}n^{n_-$ (due to $x_n > 0$ or $x_n < 0$,respectively), where the interface $Gamma = partial Omega$ is identifiedwith $mathbb{R}}{n-1}$ and divided into two parts, $Sigma$ and $Sigma'$,with different transmission conditions of first and second kind. These two parts are half-spaces of $mathbb{R} ^{$ (half-planes for $n = 3$). Weconstruct explicitly resolvent operators acting from the interface data intothe energy space $H}1(Omega)$. The approach is based upon the presentfactorisation conception and avoids an interpretation of the factors as unbounded operators. In a natural way, we meet anisotropic Sobolev spaces which reflect the edge asymptotic of diffracted waves. This talk is part of the Isaac Newton Institute Seminar Series series. ## This talk is included in these lists:- All CMS events
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## Other listsMathematics Theory of Condensed Matter DAMTP Jubilee Celebration## Other talksThe Oxford Expedition 2016 (South Africa) On the Wiener-Hopf technique and its applications in science and engineering: Lecture 2 Towards a multivariable Wiener-Hopf method: Lecture 4 Orthogonal polynomials, singular integrals, and solving Riemann–Hilbert problems: Lecture 2 JCTS PRESENTATIONS Understanding dynamic crack growth in structured systems with the Wiener-Hopf technique: Lecture 1 |