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By Bellman

This can be quantity 40-11 inMATHEMATICS IN technological know-how AND ENGINEERINGA sequence of monographs and textbooksEdited by way of RICHARD BELLMAN, college of Southern California

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8) where A(0) = {S(h) I h E E, 0 E N(h)}. 6), and linearity of S. 7). 8) is also valid. 8), we first show that the set A(0) is balanced. Indeed, let h E E, 0 E N(h). 6) we have h = (fl - f_1)/2, where f_1i fl E E and N(f_1) fl N(f1) # 0. 5) we get 0 E N((f_1 - fl)/2) = N(-h). Hence S(h) E A(0) implies -S(h) = S(-h) E A(0). 1. 1 yields the following theorem which is the main result of this section. 3 Suppose that the solution operator S is linear, information N is linear with uniformly bounded noise, N(f) = I IIy - N(f)IIY < 61, 2 Worst case setting 18 and the set E of problem elements is convex.

6 Special splines 47 Note that for a = 0, the operator A,, = 8-2N*N may not be oneto-one. In this case, if ker N 0 ker S then we formally set max{X I A E Sp(SAO1S*)} = +oo. If kerN c kerS then we treat AO = S-2N*N as an operator acting in the space V = (ker N)1. Since S(kerN) = {0}, we have S*(V) C V and SAO 1 S* : V - V is a well defined self-adjoint nonnegative definite operator. 31). 31) can be rewritten as sup{ IISAa1/2(A /2h)II I = sup f IISAa112hII I IIA1112hIIr < 1 } IIhIIF < 1 } = max{VI AESp(SAa1S*)} (this also holds for a = 0), which completes the proof.

Y,,,] E R', where yi = f (i/n) + xi, 1 < i < n, and the noise IIxI12 = ( x? 3 yields radwor(N) = sup { f f E E, E f 2(i/n) < 62 }. ) Hence w = n-1/2(1,1, ... ,1) and d = n-1/2. The unique optimal linear algorithm is the well known arithmetic mean Wlin(y) = 1 n n yii=1 Note that in this case the optimal linear algorithm is independent of the noise level 6. However, its error does depend on 6. 1 The problem of the existence of optimal linear or affine algorithms for approximating linear functionals has a long history.

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