Advances in Electronics and Electron Physics, Vol. 75 by Peter W. Hawkes (Ed.)

By Peter W. Hawkes (Ed.)

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Eq. (6). Then, if the responses of the detectors are linear, their outputs g, will be proportional to the values of g at some points x , . By neglecting the constant related to the efficiency of the instrument, we may write Yn = g(xn) = s K(xn,Y)S(Y)d y - (1 12) Such an equation does not take into account the fact that any detector integrates over some region in the domain of the physical variable x . When this effect cannot be neglected, Eq. (1 12) must be replaced by the equation Here P,(x) is an averaging function that describes the integration effect of the n-th detector.

131) This inequality is precise in the sense that equality can hold true. In spite of this fact, however, the condition number may be a rather pessimistic estimate of error propagation. Equality holds in a very special case which, in general, is not satisfied in practice. For this reason, the condition number can be called the “worst magnification” of relative errors (Twomey, 1974). A more realistic estimate is given by the “average magnification” of relative errors which we will denote (cond(L)).

Gn = jo+m n = 1,2,. . e-("- li2)'u(t)dt; (107) using the change of variables x = e-' and introducing the function f(x) = -In x), we transform this problem into the problem (106). Moreover, if the function u ( t ) belongs to L2(0,+a), then the function f ( x ) belongs to L2(0,1). , the problem of determining a function f ( x ) defined on (0,+a)from knowledge of its moments x-%( gn = jo+mx"-'f(x)dx; n = 1,2,3, ... ( 108) or the Hamburger moment problem, which is the problem of determining a function defined on ( - 03, + m) always from knowledge of its moments.

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