From the spectral approximation point of view, the problem becomes using M spectral samples to approximate complex sinusoid e j 2 π M mτ up to half a cycle.
Many would defend utility theory (as well as other aspects of their standard model of behaviour) as a useful approximation, pointing out that exceptions to approximations can always be found.
Moreover, we establish a sequence { x n } which converges strongly to the unique best approximation point.
In the first best approximation theorem, we establish a sequence { x n } which converges strongly to the unique best approximation point; while in the second best approximation theorem, we obtain the existence of a minimum best approximation point and a maximum best approximation point in order intervals.
The point x in the theorem above is called a best approximation point of T in A. Note that if (xin A) is a best approximation point, then (|x-Tx|) need not be the optimum.
Then, by Theorem 3.2, every increasing f : [ u 0, v 0 ] → ℓ 2 has a minimum best approximation point and a maximum best approximation point with respect to W ( x, y ) in [ u 0, v 0 ].
Step 2 : (Iteration): Given the current approximation point (u_{n} in K), (n inmathbb{N} cup{0}), compute u_{n+1}=P_{K} u_{n} - lambda Su_{n}), satisfying Theorem 4.1(a).
which is a contradiction so that z = T2z is a best approximation point in A of T A ∪ B → A ∪B.
Then, f has a minimum best approximation point x ∗ and a maximum best approximation point x ∗ with respect to W ( x, y ) in [ u 0, v 0 ], such that u 0 ⪯ u 1 ⪯ ⋯ ⪯ u n ⪯ ⋯ ⪯ x ∗ ⪯ x ∗ ⪯ ⋯ ⪯ v n ⪯ ⋯ ⪯ v 1 ⪯ v 0, (3.18).
Given u 0, v 0 ∈ L 2 such that u 0 ≺ v 0 ; then, by Theorem 3.2, every increasing f : [ u 0, v 0 ] → L 2 has a minimum best approximation point and a maximum best approximation point with respect to W ( x, y ) in [ u 0, v 0 ].
