By Peter W. Hawkes (Ed.)
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Additional info for Advances in Electronics and Electron Physics, Vol. 75
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.