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Definition. A sequence x is n-monotone if for each j, {xj , xj+1 , . . , xj+n+1 } is monotone. 22 3. 1. The output L1 x of a random sequence, showing no points larger than both neighbors. 2. For each n, Lnx = U nx if and only if x is n-monotone Proof. Let Lx = Lnx = U nx = U x and X = {xi−n , xi−n+1 , . . , xi , xi+1 }. Since Lx ≤ x ≤ U x, ∀x ∈ X, it follows that x = Lx = U x. Let xq be the first element in X that differs from xi−n . Assume that xq > xi−n . Then (Lx)q > (Lx)i−n and, by the previous theorem, it follows that (Lx)q+1 ≥ (Lx)q and therefore xq+1 ≥ xq .

The above theorem refines the order relation on the set of selectors considerably, and yields a proof that several other classes of selectors and compositions map into sets of locally monotone sequences, and do so by mapping a sequence x into a sequence between U mLmx and LmU mx. Examples are given by the following corollary. 3. LU LU -Smoothers, Signals and Ambiguity 27 Corollary. If m = max{n, k} then U mLm ≤ Mnj Mki ≤ LmU m for all i, j > 0, U nLn ≤ Mni ≤ LnU n and U kLk ≤ Mki since ≤ LkU k. Apart from the popular smoothers M n∞ and M n∗ , there are now a whole class of smoothers composed of various LnU n and M n that map consistently onto the class Mn of n-monotone sequences.

A) Ln ≤ Qn ≤ U n. (b) QnLn = U nLn and QnU n = LnU n. (c) (LnQn)2 = (LnQn)3 and (U nQn)2 = (U nQn)3 . Proof. (a) (U n + Ln − I)x ≤ U nx for each x, since Lnx − x ≤ 0. Similarly Qn ≥ Ln. (b) (U n + Ln − I)Ln = U nLn + LnLn − Ln = U nLn, since Ln is idempotent. Similarly QnU n = LnU n. (c) LnQn)2 = Ln(QnLn)Qn = LnU nLnQn = U nLnQn, (LnQn)3 = LnQn(U nLnQn) = Ln(QnU n)LnQn = Ln(LnU n)LnQn = LnU nLnQn = U nLnQn. The above properties have a remarkable similarity with those of the median smoother Mn .

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