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Band and Time Limiting, Recursion Relations and Some Nonlinear Evolution Equations

  • F. Alberto Grünbaum
Part of the Mathematics and Its Applications book series (MAIA, volume 18)

Abstract

Let Af the Finite Fourier Transform of a function f(x) ∈ L2 (R), given by
$$\left( {Af} \right)\left(\lambda \right) = \int_{ - T}^T {{e^{i\lambda x}}} f\left( x \right)dx,\lambda \in \left[{ - \Omega ,\Omega } \right]$$
One can consider A as the result of first chopping f to the interval [-T,T], then taking its Fourier transform and then chopping again, this time to the interval [-Ω, Ω]. This explains the title of this section.

Keywords

Recursion Relation Normalization Constant Tridiagonal Matrix Hermite Function Schroedinger Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 6.
    F.A. Grünbaum. A study of Fourier space methods for “limited angle” image reconstruction. Numerical Functional Analysis and Optimization 2(1), 31–42.Google Scholar
  2. 9.
    F.A. Grünbaum. ‘Toeplitz matrices commuting with a tridiagonal matrix’, Linear Algebra and its Applications40, (1981) 25–36.Google Scholar
  3. 10.
    F.A. Grünbaum. ‘A remark on Hilbert’s matrix’. Linear Algebra and its Applications, 43, (1982), 119–124.Google Scholar
  4. 11.
    F.A. Grünbaum. ‘The eigenvectors of the discrete Fourier transform: a version of the Hermite functions’, J. Math Anal. Applic. 88, (1982), 355–363.Google Scholar
  5. 12.
    E.T. Whittaker. 1915, Proc. London Math. Soc. (2) 14, 260–268.zbMATHGoogle Scholar
  6. 13.
    E.L. Ince. 1922, On the connection between linear diff. systems and integral equations. Proc. of the Royal Society of Edinburg (42), pp. 43–53.Google Scholar
  7. 14.
    M.L. Mehta. Random matrices, Academic press, N.Y., 19. 67.Google Scholar
  8. 15.
    F.A. Grünbaum. The limited angle problem in tomography and some related mathematical problems. Internat. Colloq. Luminy ( France ), May 1982, North Holland.Google Scholar
  9. 16.
    F.A. Grünbaum. To appear, ‘A new property of reproducing kernels for classical orthogonal polynomials’, J. Math. Anal. Applic. 95, (1983).Google Scholar
  10. 17.
    J. Duistermaat and F.A. Grünbaum. In preparation, Differential equations in the eigenvalue parameter.Google Scholar
  11. 18.
    V. Bargmann. On the connection between phase shifts and scattering potential. Rev. Mod. Phys. 21, 1949, 489–493.Google Scholar
  12. 19.
    H. Airault, H. McKean, and J. Moser. 1977, Rational and elliptic solutions of the Korteweg-deVries equation and a related many body problem. Communications in Pure and Applied Math. (30), 95–148.Google Scholar
  13. 20.
    D. Chudnovski and G. Chudnovski. 1977, Pole expansions for nonlinear partial differental equations. Nuovo Cimento, 40B, 339–353.CrossRefGoogle Scholar
  14. 21.
    M. Adler and J. Moser. 1978. On a class of polynomials connected with the Korteweg-deVries equation. Communications in Mathematical physics (61), 1–30.Google Scholar
  15. 22.
    M. Ablowitz and H. Airault. 1981. Perturbations finies et forme particulière de certaines solutions de l’equation de Korteweg-deVries. C.R. Acad. Sci. Paris, t. 292, 279–281.MathSciNetzbMATHGoogle Scholar
  16. 23.
    P. Lax. 1968. Integrals of nonlinear equations of evolutions and solitary waves. Communication on Pure and Aoolies Mathematics 21,; 467–490.Google Scholar
  17. 24.
    I. Gelfand and Dikii. 1976. Fractional powers of operators and Hamiltonian systems. Funkts. Anal. Prilozhen, 10, 4, 13–39.MathSciNetGoogle Scholar
  18. 25.
    F.A. Grünbaum. Recursion relations and a class of isospectral manifolds for Schroedinger’s equation, to appear in Nonlinear Waves, E. Debnath, Editor.Google Scholar

Copyright information

© D. Reidel Publishing Company 1984

Authors and Affiliations

  • F. Alberto Grünbaum

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