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For each sequence x ∈ X, U nLnx and LnU nx are n-monotone. 3. LU LU -Smoothers, Signals and Ambiguity 24 Proof. By the above corollary it follows that U n(U nLnx) = (U nU n)(Lnx) = U n(Lnx) = U nLnx and Ln(U nLnx) = (LnU n)(Lnx) = (LnU nLn)x = U nLnx. 2. U nLnx is n-monotone. A similar proof follows for LnU nx. It is illuminating to take stock of what has been achieved. The operators LnU n and U nLn share the range Mn , given by the following deﬁnition. Deﬁnition. Mn is the set of all sequences in x that are n-monotone.
A) If λ ≥ 0, then P 2 e = P (λe) = λP e = λ2 e = P e = λe. But (λ2 − λ)e = 0 implies λ2 − λ = 0 or λ = 1, 0. (b) If λ < 0, then (I −P )e = e−λe = (1−λ)e. Since P (I −P )e = 0 = P (1−λ)e = (1 − λ)P e = (1 − λ)λe, it follows that λ(1 − λ) = 0, which is impossible for λ < 0. Corollary. t. 1, unless the range contains only the zero sequence. Similarly, the range of I − P is the set of eigensequences of 0. t. t. the eigenvalue 0 (“noise”). It is important to note that, if x = y + z, with y in the range of P , and therefore a “signal”, it does not follow that y = P z and z = (I − P )x.
The operators are all syntone and LU ≥ U L so that LnU n(LkU k) ≥ LnU nU kLk ≥ U nLnU kLk. Suppose that m = n. 9. Similarly LnU nLkU k ≤ LnU nIU k, since Lk ≤ I, ≤ LnU nU n, since U k ≤ U n, = LnU n, since U n is idempotent. A similar argument when m = k completes the proof of the theorem. 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.