## Journal of Integral Equations and Applications

### Solvability and Spectral Properties of Integral Equations on the Real Line: II. $L^p$-Spaces and Applications

#### Article information

Source
J. Integral Equations Applications Volume 15, Number 1 (2003), 1-35.

Dates
First available in Project Euclid: 5 June 2007

http://projecteuclid.org/euclid.jiea/1181074943

Digital Object Identifier
doi:10.1216/jiea/1181074943

Mathematical Reviews number (MathSciNet)
MR2004792

Zentralblatt MATH identifier
1043.45001

#### Citation

Arens, Tilo; Chandler-Wilde, Simon N.; Haseloh, Kai O. Solvability and Spectral Properties of Integral Equations on the Real Line: II. $L^p$-Spaces and Applications. J. Integral Equations Applications 15 (2003), no. 1, 1--35. doi:10.1216/jiea/1181074943. http://projecteuclid.org/euclid.jiea/1181074943.

#### References

• P.M. Anselone and I.H. Sloan, Integral equations on the half-line, J. Integral Equations Appl. 9 (1985), 3-23.
• --------, Spectral approximations for Wiener-Hopf operators, J. Integral Equations Appl. 11 (1990), 237-261.
• T. Arens, A new integral equation formulation for the scattering of plane elastic waves by diffraction gratings, J. Integral Equations Appl. 11 (1999), 275-297.
• --------, The scattering of elastic waves by rough surfaces, Ph.D. Thesis, Brunel University, 2000.
• --------, Existence of solution in elastic wave scattering by unbounded rough surfaces, Math. Methods Appl. Sci. 25 (2002), 507-528.
• T. Arens, S.N. Chandler-Wilde and K. O. Haseloh, Solvability and spectral properties of integral equations on the real line: I. Weighted spaces of continuous functions, J. Math. Anal. Appl. 272 (2002), 276-302.
• A. Böttcher, Y.I. Karlovich and I.M. Spitkovsky, Convolution operators and factorization of almost periodic matrix functions, Birkhäuser Verlag, Basel, 2002.
• A. Böttcher and B. Silbermann, Analysis of Toeplitz operators, Springer-Verlag, New York, 1990.
• S.N. Chandler-Wilde, On the behavior at infinity of solutions of integral equations on the real line, J. Integral Equations Appl. 4 (1992), 153-177.
• --------, Some uniform stability and convergence results for integral equations on the real line and projection methods for their solution, IMA J. Numer. Anal. 13 (1993), 509-535.
• --------, The impedance boundary value problem for the Helmholtz equation in a half-plane, Math. Methods Appl. Sci. 20 (1997), 813-840.
• S.N. Chandler-Wilde and D.C. Hothersall, Efficient calculation of the Green's function for acoustic propagation above a homogeneous impedance plane, J. Sound Vibration 180 (1995), 705-724.
• S.N. Chandler-Wilde and C.R. Ross, Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane, Math. Methods Appl. Sci. 19 (1996), 959-976.
• S.N. Chandler-Wilde, C.R. Ross and B. Zhang, Scattering by infinite one-dimensional rough surfaces, Proc. Roy. Soc. London 455 (1999), 3767-3787.
• S.N. Chandler-Wilde and B. Zhang, On the solvability of a class of second kind integral equations on unbounded domains, J. Math. Anal. Appl. 214 (1997), 482-502.
• --------, Electromagnetic scattering by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate, Proc. Roy. Soc. London 454 (1998), 519-542.
• --------, A uniqueness result for scattering by infinite rough surfaces, SIAM J. Appl. Math. 58 (1998), 1774-1790.
• S.N. Chandler-Wilde, B. Zhang and C.R. Ross, On the solvability of second kind integral equations on the real line, J. Math. Anal. Appl. 245 (2000), 28-51.
• D. Colton and R. Kress, Integral equation methods in scattering theory, Wiley, New York, 1983.
• D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, 2nd. ed., Springer, Berlin, 1983.
• I. Gohberg and I.A. Fel'dman, Convolution equations and projection methods for their solution, Amer. Math. Soc., Providence, RI, 1974.
• K.O. Haseloh, On the solvability of second kind integral equations on unbounded domains with an application to an acoustic waveguide problem, MSc Dissertation, Brunel University, 1998.
• --------, Lösbarkeit Fredholmscher Integralgleichungen zweiter Art in gewich-teten Räumen, Diplomarbeit, Universität Hannover, 2000.
• K. Jörgens, Linear integral operators, Pitman, London, 1982.
• N.K. Karapetiants and S.G. Samko, The index of certain classes of integral operators, Soviet Math. Dokl. 11 (1970), 1229-1233.
• --------, Equations with involutive operators, Birkhäuser Boston, Inc., Boston, MA, 2001.
• S. Prössdorf and B. Silbermann, Numerical analysis for integral and related operator equations, Akademie-Verlag, Berlin, 1991.
• W. Rudin, Functional analysis, 2nd ed., McGraw-Hill, New York, 1991.
• E.M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, 1971.