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Additional info for Abel's theorem and the allied theory, including the theory of the theta functions
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. The sequences in Mn can therefore be considered the signals, as interpreted by both operators, and LnU n and U nLn can be considered to be extensions of the identity operator on the set Mn . The previous two theorems also imply that the sequences are all roots of the operators Mn , Mn2 and the (nonexistent) operator R = M ∞ .
Let x = xi = δi,j + δi,1+j be a sequence, that clearly has two consecutive impulses superimposed on the zero sequence. With L2 U2 x = U2 L2 x = 0, it is clear that 0 is not a best approximation in any norm with p < ∞ since y = yi = δi,j−1 + δi,j + δi,j+1 is better. 50 5. Smoothing and Approximation with Signals It is worth noting that a best approximation from M2 does not even lie between L2 x and U2 x, and does not remove the basic impulse for which U2 L2 , L2 U2 , M2 etc. were designed! ) The above example illustrates this since y is a best approximation as ||x − y||p = 1 cannot be improved.
The n-noise interval of a sequence x is [(I − LnU n)x, (I − U nLn)x]. 13. Let An be an operator such that An ∈ [U nLn, LnU n]. Then I − An ∈ [(I − LnU n)x, (I − U nLn)x]. Proof. If U nLnx ≤ Anx ≤ LnU nx, then −U nLnx ≥ −Anx ≥ −LnU nx and (I − U nLn)x ≥ (I − An)x ≥ (I − LnU n)x. Since this is true for each x ∈ X, the theorem is proved. Since I − An − (I − An)2 = An(I − An), the ambiguity as to the amount of signal still left in the residual (I − An)x can be considered to be in [U nLn(I − LnU n)x, LnU n(I − U nLn)], which is the union of two LU LU -intervals.