Skip to main content
SpringerLink
Log in

Commentary

  • Chapter
An Introduction to Echo Analysis

Part of the book series: Springer Monographs in Mathematics ((SMM))

  • 873 Accesses

Abstract

From the outset it has been emphasised that this book is an introductory text intended for the use of those wishing to begin studying the scattering of waves by time-dependent perturbations. For this reason we offer in this chapter some additional remarks on the material that has been presented in previous chapters. The main intentions are, on the one hand, to give some indications of the work that either has been or is being done to cater for more general situations than those considered here and, on the other hand, to suggest further reading directions. Whilst it is recognised that it is impossible to give credit to all sources, nevertheless, those cited in the extended Bibliography provided here will, in turn, give additional references.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References and Further Reading

  1. R. Adams: Sobolev Spaces, Academic Press, New York, 1975.

    MATH  Google Scholar 

  2. Z.S. Agranovich and V.A. Marchenko: The Inverse Problem of Scattering Theory, Gordon and Breach, New York, 1963.

    MATH  Google Scholar 

  3. A.C. Aitken: Determinants and Matrices, Oliver and Boyd, Edinburgh, 1951.

    Google Scholar 

  4. N.I. Akheizer and L.M. Glazman: Theory of Linear Operators in Hilbert Space, Pitman-Longman, London, 1981.

    Google Scholar 

  5. H. Amann: Ordinary Differential Equations, An Introduction to Nonlinear Analysis, W. de Gruyter, Berlin, 1990.

    MATH  Google Scholar 

  6. W.O. Amrein, J.M. Jauch and K.B. Sinha: Scattering Theory in Quantum Mechanics, Lecture Notes and Supplements in Physics, Benjamin, Reading, 1977.

    Google Scholar 

  7. J. Arsac: Fourier Transforms and Theory of Distributions, Prentice Hall, New York, 1966.

    Google Scholar 

  8. C. Athanasiadis, G.F. Roach and I.G. Stratis: A time domain analysis of wave motions in chiral materials, Math. Nachr. 250, 2003, 3–16.

    Article  MATH  MathSciNet  Google Scholar 

  9. G.N. Balanis: The plasma inverse problem, Jour. Math. Physics 13(7), 1972, 1001–1005.

    Article  MathSciNet  Google Scholar 

  10. G.N. Balanis, Inverse scattering: Determination of inhomogeneities in sound speed, Jour. Math. Physics 23(12), 1982, 2562–2568.

    Article  MATH  MathSciNet  Google Scholar 

  11. G.R. Baldock and T. Bridgeman: Mathematical Theory of Wave Motion, Ellis Horwood, Chichester, 1981.

    MATH  Google Scholar 

  12. H. Baumgärtel and M. Wollenberg: Mathematical Scattering Theory, Operator Theory: Advances and Applications, Birkhaüser-Verlag, Stuttgart, 1983.

    Google Scholar 

  13. A. Belleni-Morante: Applied Semigroups and Evolution Equations, Clarendon Press, Oxford, 1979.

    MATH  Google Scholar 

  14. A. Belleni-Morante and A.C. McBride: Applied Nonlinear Semigroups, Mathematical Methods in Practice, 3, Wiley, Chichester, 1998.

    Google Scholar 

  15. A. Belleni-Morante and G.F. Roach: A mathematical model for gamma ray transport in the cardiac region, Jour. Math. Anal Appl., 244, 2000, 498–514.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Belleni-Morante and G.F. Roach: Gamma ray transport in the cardiac region: an inverse problem, Jour. Math. Anal Appl. 269, 2002, 200–2115.

    Article  MATH  MathSciNet  Google Scholar 

  17. A.M. Berthier: Spectral Theory and Operators for the Schrödinger Equation, Pitman Research Notes in Mathematics No. 71, Pitman, London, 1982.

    MATH  Google Scholar 

  18. R. Burridge: The Gel’fand-Levitan, the Marchenko and the Gopinath-Sondhi integral equations of the inverse scattering theory regarded in the context of inverse impulse-response problems, Wave Motion 2, 1980, 305–323.

    Article  MATH  MathSciNet  Google Scholar 

  19. K. Chadan and P. Sabatier: Inverse Problems in Quantum Scattering Theory, Springer-Verlag, New York, 1977.

    MATH  Google Scholar 

  20. D. Colton and R. Kress: Inverse Acoustic and Electromagnetic Scacttering Theory, Applied Mathematical Sciences No. 93, Springer-Verlag, Berlin, 1991.

    Google Scholar 

  21. R. Courant and D. Hilbert: Methods of Mathematical Physics Vol. II, Wiley Interscience, New York, 1962.

    MATH  Google Scholar 

  22. H.L. Cyon, R.G. Froese, W. Kirsh and B. Simon: Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1981.

    Google Scholar 

  23. J.W. Dettman: Mathematical Methods in Physics and Engineering, McGraw-Hill, New York, 1962.

    MATH  Google Scholar 

  24. N. Dunford and J.T. Schwartz: Linear Operators, Vol. 1-3, Wiley Interscience, New York, 1958.

    MATH  Google Scholar 

  25. M.S.P. Eastham: Theory of Ordinary Differential Equations, Van Nostrand, London, 1970.

    MATH  Google Scholar 

  26. M.S.P. Eastham: The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh, 1973.

    MATH  Google Scholar 

  27. D.E. Edmunds and W.D. Evans: Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.

    MATH  Google Scholar 

  28. D.M. Eidus: The principle of limiting absorption, Math. Sb., 57(99), 1962 and AMS Transl., 47(2), 1965, 157-191.

    Google Scholar 

  29. D.M. Eidus: The principle of limiting amplitude, Uspekhi Mat. Nauk. 24(3), 1969, 91–156 and Russ. Math. Surv. 24(3), 1969, 97-167.

    MathSciNet  Google Scholar 

  30. L.D. Faddeev: On the relation between S-matrix and potential for the onedimensional Schrödinger operator, Dokl. Akad. Nauk SSSR Ser. Mat. Fiz. 121(1), 1958, 63–66 (in Russian).

    MathSciNet  Google Scholar 

  31. L.D. Faddeev: The inverse problem in the quantum theory of scattering, Uspekhi Mat. Nauk 14, 1959, 57–119 (in Russian); English translation: Jour. Math. Physics 4, 1963, 72-104.

    MATH  MathSciNet  Google Scholar 

  32. J. Fawcett: On the stability of inverse scattering problems, Wave Motion 6, 1984, 489–499.

    Article  MATH  MathSciNet  Google Scholar 

  33. K.O. Friedrichs: Spectral Theory of Operators in Hilbert Space, Springer-Verlag, Berlin, 1973.

    MATH  Google Scholar 

  34. I.M. Gel’fand and B.M. Levitan: On the determination of a differential equation from its spectral function, Izv. Akad. Nauk SSSR Ser. Mat. 15, 1951, 309–360 (in Russian); English translation: Amer. Math. Soc. Transl. 1, 1955, 253-304.

    MATH  MathSciNet  Google Scholar 

  35. M. Gelfand and G.E. Shilov: Generalised Functions, Academic Press, New York, 1964.

    Google Scholar 

  36. J.A. Goldstein: Semigroups and second order differential equations, J. Functional Anal. 4, 1969, 50–70.

    Article  MATH  Google Scholar 

  37. J.A. Goldstein: Abstract evolution equations, Trans. American Math. Soc. 141, 1969, 159–185.

    Article  MATH  Google Scholar 

  38. J.A. Goldstein: Semigroups of Linear Operators and Applications, Oxford University Press, Oxford, 1986.

    Google Scholar 

  39. E. Goursat: Cours d’analyse Mathématique, Paris, 1927.

    Google Scholar 

  40. S.H. Gray: Inverse scattering for the refl ectivity function, Jour. Math. Physics 24(5), 1983, 1148–1151.

    Article  MATH  Google Scholar 

  41. K.E. Gustafson: An Introduction to Partial Differential Equations and Hilbert Space Methods, John Wiley and Sons, New York, 1980.

    Google Scholar 

  42. G. Helmberg: Introduction to Spectral Theory in Hilbert Space, Elsevier, New York, 1969.

    MATH  Google Scholar 

  43. T. Ikebe: Eigenfunction expansions associated with the Schrodinger Operators and their application to scattering theory. Arch. Rat. Mech. Anal. 5, 1960, 2–33.

    Article  MathSciNet  Google Scholar 

  44. D.S. Jones: Acoustic and Electromagnetic Waves, Clarendon Press, Oxford, 1986.

    Google Scholar 

  45. T.F. Jordan: Linear Operators for Quantum Mechanics, John Wiley & Sons, New York, 1969.

    MATH  Google Scholar 

  46. L.V. Kantorovich and G.P. Akilov: Functional Analysis in Normed Spaces, Pergamon, Oxford, 1964.

    MATH  Google Scholar 

  47. T. Kato: On linear differential equations in Banach spaces, Comm. Pure Appl. Math. 9, 1956, 479–486.

    Article  MATH  MathSciNet  Google Scholar 

  48. T. Kato: Perturbation Theory for Linear Operators, Springer, New York, 1966.

    MATH  Google Scholar 

  49. I. Kay: The inverse scattering problem when the reflection coefficient is a rational function, Comm. Pure Appl. Math. 13, 1960, 371–393.

    Article  MATH  MathSciNet  Google Scholar 

  50. R.E. Kleinman and R.B. Mack: Scattering by linearly vibrating objects, IEEE Trans. Antennas & Propagation AP-27(3), 1979, 344–352.

    Article  MathSciNet  Google Scholar 

  51. R.E. Kleinman and G.F. Roach: Boundary integral equations for the three dimensional Helmholtz equation, SIAM Reviews, 16(2), 1974, 214–236.

    Article  MATH  MathSciNet  Google Scholar 

  52. E. Kreyszig: Introductory Functional Analysis with Applications, Wiley, Chichester, 1978.

    MATH  Google Scholar 

  53. R.J. Krueger: An inverse problem for a dissipative hyperbolic equation with discontinuous coefficients, Quart. Appl. Math. 34(2), 1976, 129–147.

    MATH  MathSciNet  Google Scholar 

  54. A. Kufner and J. Kadelec: Fourier Series, Illiffe, London, 1971.

    MATH  Google Scholar 

  55. S.T. Kuroda: On the existence and the unitary property of the scattering operator, Nuovo Cimento, 12, 1959, 431–454.

    Article  MATH  Google Scholar 

  56. J.A. LaVita, J.R. Schulenberg and C.H. Wilcox: The scattering theory of Lax and Phillips and propagation problems of classical physics, Applic. Anal. 3, 1973, 57–77.

    Article  MATH  Google Scholar 

  57. P.D. Lax and R.S. Phillips: Scattering Theory, Academic Press, New York, 1967.

    MATH  Google Scholar 

  58. H. Levin: Unidirectional Wave Motions, North Holland, Amsterdam, 1978.

    Google Scholar 

  59. B.M. Levitan: Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, 1987.

    MATH  Google Scholar 

  60. R. Leis: Initial Boundary Value Problems of Mathematical Physics, John Wiley & Sons, Chichester, 1986.

    Google Scholar 

  61. R. Leis and G.F. Roach: A transmission problem for the plate equation, Proc. Roy. Soc. Edinburgh 99A, 1985, 285–312.

    MathSciNet  Google Scholar 

  62. A.E. Lifschitz: Magnetohydrodynamics and Spectral Theory, Kluwer, Dordrecht, 1988.

    Google Scholar 

  63. J. Lighthill: Introduction to Fourier Analysis and Generalised Functions, Cambridge University Press, Cambridge, 1958.

    MATH  Google Scholar 

  64. B.V. Limaye: Functional Analysis, Wiley Eastern, New Dehli, 1981.

    MATH  Google Scholar 

  65. W. Littman: Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69, 1963, 766–770.

    Article  MATH  MathSciNet  Google Scholar 

  66. E.R. Lorch: Spectral Theory, Oxford University Press, Oxford, 1962.

    MATH  Google Scholar 

  67. M. Matsumura: Uniform estimates of elementary solutions of first order systems of partial differential equations, Publ. RIMS, Kyoto Univ. 6, 1970, 293–305.

    Article  MathSciNet  Google Scholar 

  68. M. Mabrouk and Z. Helali: The scattering theory of C. Wilcox in elasticity, Math. Meth. in Appl. Sci. 25, 2002, 997–1044.

    Article  MATH  MathSciNet  Google Scholar 

  69. M. Mabrouk and Z. Helali: The elastic echo problem, Math. Meth. in Appl. Sci. 26, 2003, 119–150.

    Article  MATH  MathSciNet  Google Scholar 

  70. V.A. Marchenko: Some problems in the theory of one-dimensional linear differential operators of second order. I. Trudy Mosk. Ob. 1, 1952, 327–420 (in Russian).

    Google Scholar 

  71. V.A. Marchenko: The construction of the potential energy from the phases of the scattered wave, Dokl. Akad. Nauk SSSR 104, 1955, 695–698 (in Russian).

    MATH  MathSciNet  Google Scholar 

  72. A.C. McBride: Semigroups of Linear Operators; an Introduction, Pitman Research Notes in Mathematics No156, Pitman, London, 1987.

    MATH  Google Scholar 

  73. I.V. Melnikova and A. Filinkov: Abstract Cauchy Problems: Three Approaches, Monographs and Surveys in Pure and Applied Mathematics, Vol. 120, Chapman and Hall, London, 2001.

    Google Scholar 

  74. S.G. Mikhlin: Integral Equations and their Applications to Certain Problems in Mechanics, Physics and Technology, Pergamon Press, Oxford, 1957.

    Google Scholar 

  75. K. Morgenröther and P. Werner: Resonances and standing waves, Math. Meth. Appl. Sci. 9, 1987, 105–126.

    Article  MATH  Google Scholar 

  76. P.M. Morse and H. Feshbach: Methods of Theoretical Physics Vol 1,2 McGraw-Hill, New York, 1953.

    MATH  Google Scholar 

  77. N.F. Mott and M.S.W. Massey: The Theory of Atomic Collisions, Oxford University Press, Oxford, 1949.

    MATH  Google Scholar 

  78. A.W. Naylor and G.R. Sell: Linear Operator Theory in Engineering and Science, Holt Rinehart and Winston, New York, 1971.

    Google Scholar 

  79. R.G. Newton: Scattering Theory of Waves and Particles, McGraw-Hill, New York, 1966.

    Google Scholar 

  80. R.G. Newton: Inverse scattering, J. Math. Phys., 23, 1982, 594–604.

    Article  MathSciNet  Google Scholar 

  81. L.P. Niznik: Inverse problem of nonstationary scattering, SSSDokl. Akad. Nauk R.196(5), 1971, 1016–1019 (in Russian).

    MathSciNet  Google Scholar 

  82. A. Olek: Inverse Scattering Problems for Moving, Penetrable Bodies, PhD Thesis, University of Strathclyde, Glasgow, 1997.

    Google Scholar 

  83. A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.

    MATH  Google Scholar 

  84. D.B. Pearson: Quantum Scattering and Spectral Theory, Academic Press, London, 1988.

    MATH  Google Scholar 

  85. C.L. Perkeris: Theory of propagation of explosive sound in shallow water, Geo. Soc. Am. Memoir 27, 1948.

    Google Scholar 

  86. E. Prugovecki: Quantum Mechanics in Hilbert Space, Academic Press, New York, 1981.

    MATH  Google Scholar 

  87. M. Reed and B. Simon: Methods of Mathematical Physics, Vols 1-4, Academic Press, New York, 1972-1979.

    MATH  Google Scholar 

  88. F. Riesz and B. Sz-Nagy: Functional Analysis, Ungar, New York, 1981.

    Google Scholar 

  89. M.A. Rincon and I. Shih Lin: Existence and uniqueness of solutions of elastic string with moving ends, Math. Meth. Appl Sci., 27, 2004.

    Google Scholar 

  90. G.F. Roach: Greens Functions (2nd Edn), Cambridge Univ. Press, London, 1970/1982.

    Google Scholar 

  91. G.F. Roach: An Introduction to Linear and Nonlinear Scattering Theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 78, Longman, Essex, 1995.

    Google Scholar 

  92. G.F. Roach and B. Zhang: A transmission problem for the reduced wave equation in inhomogeneous media with an infi nite interface, Proc. Roy. Soc. Lond. A436, 1992, 121–140.

    MathSciNet  Google Scholar 

  93. G.F. Roach and B. Zhang: The limiting amplitude principle for wave propagation with two unbounded media, Proc. Camb. Phil. Soc. 112, 1993, 207–223.

    Article  MathSciNet  Google Scholar 

  94. G.F. Roach and B. Zhang: Spectral representations and scattering theory for the wave equation with two unbounded media, Proc. Camb. Phil. Soc. 113, 1993, 423–447.

    Article  MATH  MathSciNet  Google Scholar 

  95. G.F. Roach and B. Zhang: On Sommerfeld radiation conditions for wave propagation with two unbounded media, Proc. Roy. Soc. Edinburgh, 1992, 149–161.

    Google Scholar 

  96. J. Rose, M. Cheney and B. De Facio: Three dimensional inverse plasma and variable velocity wave equations, J. Math. Phys., 26, 1985, 2803–2813.

    Article  MATH  MathSciNet  Google Scholar 

  97. J. Rose, M. Cheney and B. De Facio: Determination of the wave field from scattering data, Phys. Rev. Lett. 57, 1986, 783–786.

    Article  MathSciNet  Google Scholar 

  98. W. Rudin: Principles of Mathematical Analysis, (3rd Ed), McGraw-Hill, New York, 1976.

    MATH  Google Scholar 

  99. D.E. Rutherford: Vector Methods, Oliver and Boyd, Edinburgh, 1951.

    MATH  Google Scholar 

  100. Y. Saito: An inverse problem in potential theory and the inverse scattering problem. J. Math. Kyoto Univ. 22, 1982, 307–321.

    MATH  MathSciNet  Google Scholar 

  101. Y. Saito: Some properties of the scattering amplitude and the inverse scattering problem, Osaka J. of Math. 19(8), 1982, 527–547.

    MATH  Google Scholar 

  102. Y. Saito: An asymptotic behaviour of the S-matrix and the inverse scattering problem, J. Math. Phys. 25(10), 1984, 3105–3111.

    Article  MATH  MathSciNet  Google Scholar 

  103. Y. Saito: Inverse scattering for the plasma wave equation starting with large-t data, J. Phys. A: Math. Gen. 21, 1988, 1623–1631.

    Article  MATH  Google Scholar 

  104. E. Sanchez-Palencia: Non-homogeneous Media and Vibration Theory, Lecture Notes in Physics Vol. 127, Springer-Verlag, Berlin, 1980.

    MATH  Google Scholar 

  105. J. Sanchez Hubert and E. Sanchez-Palencia: Vibration and Coupling of Continuous Systems: Asymptotic Methods, Springer-Verlag, Berlin, 1989.

    MATH  Google Scholar 

  106. M. Schechter: Spectra of Partial Differential Operators, North Holland, Amsterdam, 1971.

    MATH  Google Scholar 

  107. E.J.P. Schmidt: On scattering by time-dependent potentials, Indiana Univ. Math. Jour. 24(10) 1975, 925–934.

    Article  MATH  Google Scholar 

  108. J.R. Schulenberger and C.H. Wilcox: Eigenfunction expansions and scattering theory for wave propagation problems of classical physics, Arch. Rat. Mech. Anal. 46, 1972, 280–320.

    Article  MATH  MathSciNet  Google Scholar 

  109. L. Schwartz: Mathematics for the Physical Sciences, Hermann, Paris, 1966.

    MATH  Google Scholar 

  110. N.A. Shenk: Eigenfunction expansions and scattering theory for the wave equation in an exterior domain, Arch. Rational Mechanics and Anal., 21, 1966, 120–1506.

    MATH  MathSciNet  Google Scholar 

  111. E. Skudrzyk: The Foundations of Acoustics, Springer-Verlag, New York, 1977.

    Google Scholar 

  112. V.I. Smirnov: Course of Higher Mathematics, Pergamon, New York, 1965.

    Google Scholar 

  113. I.N. Sneddon: Fourier Transforms, McGraw-Hill, New York, 1951.

    Google Scholar 

  114. P.E. Sobolevski: Equations of parabolic type in a Banach space, Amer. Math. Soc. Transl. 49, 1996, 1–62.

    Google Scholar 

  115. B. Spain: Tensor Calculus, Oliver and Boyd, Edinburgh, 1956.

    Google Scholar 

  116. M.H. Stone: Linear Transformations in Hilbert Space and their Appliations to Analysis, Amer. Math. Soc. Coll. Publ. 15, Providence, RI, 1932.

    Google Scholar 

  117. W.A. Strauss: Partial Differential Equations: An Introduction, Wiley, New York, 1992.

    MATH  Google Scholar 

  118. W.W. Symes: Inverse boundary value problems and a theorem of Gel’fand and Levitan, Jour. Math. Anal. Applic., 71, 1979, 379–402.

    Article  MATH  MathSciNet  Google Scholar 

  119. H. Tanabe: On the equation of evolution in a Banach space, Osaka Math. J. 12, 1960, 363–376.

    MATH  MathSciNet  Google Scholar 

  120. H. Tanabe: Evolution Equations, Pitman Monographs and Studies in Mathematics, Vol. 6, Pitman, London, 1979.

    Google Scholar 

  121. J.R. Taylor: Scattering Theory: The Quantum Theory of Non-Relativistic Collisions, University of Boulder, Colorado.

    Google Scholar 

  122. E.C. Titchmarsh: Introduction to the Theory of Fourier Integrals, Oxford Univ. Press, 1937.

    Google Scholar 

  123. F. Tricomi: Integral Equations, Interscience, New York, 1957.

    MATH  Google Scholar 

  124. B. Vainberg: Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York, 1989.

    Google Scholar 

  125. V.S. Vladymirov: Equations of Mathematical Physics, Marcel Dekker, New York, 1971.

    Google Scholar 

  126. H.F. Weinberger: A First Course in Partial Differential Equations, Blaisdell, Waltham, MA, 1965.

    MATH  Google Scholar 

  127. P. Werner: Regularity properties of the Laplace operator with respect to electric and magnetic boundary conditions, J. Math. Anal. Applic. 87, 1982, 560–602.

    Article  MATH  Google Scholar 

  128. P. Werner: Resonances in periodic media, Math. Methods in Appl. Sciences, 14, 1991, 227–263.

    Article  MATH  Google Scholar 

  129. V.H. Weston: On the inverse problem for a hyperbolic dispersive partial differential equation, Jour. Math. Physics 13(12), 1972, 1952–1956.

    Article  MATH  MathSciNet  Google Scholar 

  130. C.H. Wilcox: Scattering states and wave operators in the abstract theory of scattering, Jour. Functional Anal. 12, 1973, 257–274.

    Article  MATH  MathSciNet  Google Scholar 

  131. C.H. Wilcox: Scattering Theory for the d’Alembert Equation in Exterior Domains, Lecture Notes in Mathematics, No.442, Springer, Berlin, 1975.

    Google Scholar 

  132. C.H. Wilcox: Transient Electromagnetic Wave Propagation in a Dielectric Waveguide, Proc. Conf. on the Mathematical Theory of Electromagnetism, Instituto Nazionale di Alto Mathematica, Rome, 1974.

    Google Scholar 

  133. C.H. Wilcox: Spectral analysis of the Perkeris operator in the theory of acoustic wave propagation in shallow water, Arch. Rat. Mech. Anal. 60, 259–300, 1976.

    Article  MathSciNet  Google Scholar 

  134. J. Wloka: Partial Differential Equations, Cambridge University Press, Cambridge, 1987.

    MATH  Google Scholar 

  135. K. Yosida: Functional Analysis, Springer-Verlag, Berlin, 1971.

    MATH  Google Scholar 

Download references

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag London Limited

About this chapter

Cite this chapter

(2008). Commentary. In: An Introduction to Echo Analysis. Springer Monographs in Mathematics. Springer, London. https://doi.org/10.1007/978-1-84628-852-4_12

Download citation

Publish with us

Policies and ethics

Search

Navigation