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In the previous section we have shown that significantly improved resolution can be achieved when the size of the object support is not large with respect to the resolution distance. In a similar vein, the recovery and resolution of exponential object components may be improved by using a priori knowledge of the support of the object. e. the inversion of the Laplace transform of a function with bounded support within a given interval, say [a, b], 0 < a < b < oo. Because of the scaling properties of the Laplace transformation, it is not restrictive to assume that the support of f is [1, 3'], so that the finite Laplace transformation is defined as follows (Af)(p)= fl / e Ptf(t)dt, 0~

Zm), is devoid of zeros in 0 m. Define /3i = max(Iail, Idol). Then, by applying Theorem 1, it follows that Ai(zl, z , . . , zm) is stable if and only if z~ml fli < 1. COMMENT 3. Theorem 1 can be applied to a variety of polynomials generated from A ( z l , z2, . . , Zm). , Zm) = B ( z ~ ' , Z k2 Z , . . , Z m ) =k,~ I+ ~aiz~i, i=1 where the k~'s are positive integers, is devoid of zeros either inside or on 0 m if and only if Eima Jail < 1. COMMENT 4. It has been pointed out, [20], that the polynomial generated from Zm) = 1 + E~=l a~zi via the transformation, A(zx, z 2.....

1. 20) which is self-adjoint in L2(O, 2) if one looks for eigenfunctions which are bounded at the origin and square integrable at infinity; (iii) all the singular values o"k have multiplicity 1; (iv) the singular function u~ has exactly k zeros interior to [1, y]; the points 1 and 3/can never be zeros of u~. The previous results can be used to estimate the solution of the Laplace transform inversion as a truncated singular function expansion. However, the method has some unsatisfactory features.