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עמוד:*25
. order n Then in the presence of a sampling noise with the maximal magnitude ε 0 the accuracy � � z i − f ( i ) � � 0 � i L � ε 0 γ ≤ � L � ( nd − i + 1 ) nd + 1 ) ( ) 6 ( is obtained in finite - time for some γ i ≥ 1 only depending on the coefficients →− ˜ λ n . Whereas γ i depend on the parameters ˜ λ i of ( 4 ) , the asymptotics structure ( 6 ) ( i . e . the powers ) is fixed and cannot be improved by any exact differentiation algorithm on func - n d ( L ) . tions f 0 ∈ Lip It is proved that ∀ ε such that for any ε 0 , ∗ > 0 ∃ t 0 > 0 ( also ∀ t 0 > 0 ∃ ε ) 0 > ∗ ε ≤ 0 ε < 0 and any f 0 , f 1 ∈ Lip , ∗ t ≥ 0 | f 1 ( t ) − f 0 ( t ) | ≤ ε 0 implies n d Lthe inequality sup sup t ≥ t 0 | f ( i ) t ) | ≤ Ki,n d ( 2 L ) i 1 ( t ) − f ( i ) ( 0 nd + 1 ε nd + 1 − i nd + 1 ) 7 ( 0 for i = 0 , 1 , . . . , n d . Here Ki,n d are the Kolmogorov constants, and ( 7 ) turn into equalities on certain functions . It is known that Ki,n d ∈ [ 1 , π / 2 ] , in particular, K1 , 1 = √ Taking η = f 1 − f 0 . 2 obtain an unremovable restriction on the best possible accuracy of noisy differentiation . Adifferentiator is called asymptotically optimal, if for some μ i > 0 , any f 0 ∈ n d ( L ) , bounded noise η , ess sup | η ( t ) | ≤ ε 0 , it in FTprovides the estimation accu - Lip racy z i ( t ) − f ( i ) | t ) | ≤ μ i L i ( 0 n d + 1 ε n d + 1 − i n d + 1 i = 0 , 1 , . . . , n d . ( 8 ) , 0 Obviously, μ i ≥ 2 i n d + 1 holds . n d + 1 Ki,n d ≥ 2 i * 25
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אדמוני, אריאל
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