By Peter W. Hawkes (Ed.)

ISBN-10: 0120146754

ISBN-13: 9780120146758

**Read or Download Advances in Electronics and Electron Physics, Vol. 75 PDF**

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**Sample text**

As a consequence the operator (87) is not self-adjoint, though it is still possible to show that it is a compact operator from L2(- 1,l) into L2(- 00, + a). Moreover, its singular system can be given again in terms of PSWF. More precisely, the singular values a, and the singular functions uk are given again by Eq. (86), while the singular functions uk are now given by uk(x)= $k(c,x). , 1980a). G . Laplace Transform Inversion In several domains of experimental science, the experimenter is concerned with the problem of recovering and resolving exponential relaxation rates.

1979). Since we consider a bounded object, situated, for instance, inside the sphere of radius a, f(r) is zero for r > a. Then the total field u (incident plus scattered) is a solution of the wave equation Au + k2u = k’f(r)u, (53) where k = 2n/A is the wave number (in free space) of the incident radiation. r)- (k2/4n) s G(r - yr)eik(so,r‘lf(r’)dr’. s. of Eq. (54) (Born approximation of scattered wave). s. ,t,,,(kz - t: - 5,2)”’). It follows from Eq. ( 5 5 ) that knowledge of ijLB)({, z) is equivalent to knowledge off on the surface of the Ewald sphere with centre ks, and radius k and, therefore, from the mathematical point of view, this problem is analogous to the inverse source problem discussed in Section II,B.

On the other hand, the singular functions u k are the eigenfunctions of the fourth-order differential operator + which is also self-adjoint in L2(0, 00) if one looks for eigenfunctions that are bounded at the origin and square-integrable at infinity. Using these results it is possible to prove (Bertero and Griinbaum, 1985) that: a) all the singular values have multiplicity 1; b) the singular function uk has exactly k zeros interior to [l, y]; the points 1 and y can never be zeros of uk. 34 M. , 1986a).

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