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On stable numerical differentiation
 Mathem. of Computation
, 1968
"... Abstract. A new approach to the construction of finitedifference methods is presented. It is shown how the multipoint differentiators can generate regularizing algorithms with a stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iterative ..."
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Cited by 52 (24 self)
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Abstract. A new approach to the construction of finitedifference methods is presented. It is shown how the multipoint differentiators can generate regularizing algorithms with a stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iteratively regularized scheme for solving the numerical differentiation problem in the form of Volterra integral equation is developed. 1.
Reconstructing singularities of a function from its Radon transform
 MATH. COMPUT. MODELLING
, 1993
"... We study the relation between the singularities of a function f and its Radon transform R(f). We prove that their singular loci are related via Legendre transform. Geometric properties of the singular locus of R(f) are studied. The problem of computing the Legendre transform from approximately know ..."
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Cited by 12 (5 self)
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We study the relation between the singularities of a function f and its Radon transform R(f). We prove that their singular loci are related via Legendre transform. Geometric properties of the singular locus of R(f) are studied. The problem of computing the Legendre transform from approximately known data is discussed.
Inequalities for the derivatives
 MATHEM. INEQUALITIES AND APPLICATIONS, 3, N1, (2000), PP.129132
, 2000
"... The following question is studied and answered: Is it possible to stably approximate f ′ if one knows: 1) fδ ∈ L ∞ (R) such that �f − fδ � < δ, and 2) f ∈ C ∞ (R), �f � + �f ′ � ≤ c? Here �f �: = sup x∈R f(x)  and c> 0 is a given constant. By a stable approximation one means �Lδfδ − f ′ � ..."
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Cited by 10 (4 self)
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The following question is studied and answered: Is it possible to stably approximate f ′ if one knows: 1) fδ ∈ L ∞ (R) such that �f − fδ � < δ, and 2) f ∈ C ∞ (R), �f � + �f ′ � ≤ c? Here �f �: = sup x∈R f(x)  and c> 0 is a given constant. By a stable approximation one means �Lδfδ − f ′ � ≤ η(δ) → 0 as δ → 0. By Lδfδ one denotes an estimate of f ′. The basic result of this paper is the inequality for �Lδfδ − f ′ �, a proof of the impossibility to approximate stably f ′ given the above data 1) and 2), and a derivation of the inequality η(δ) ≤ cδ a 1+a if 2) is replaced by �f�1+a ≤ m1+a, 0 < a ≤ 1. An explicit formula for the estimate Lδfδ is given.