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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 56 (26 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 14 (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 11 (5 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.
Singularities of the Radon transform
 Bull. Amer. Math. Soc
, 1993
"... Abstract. Singularities of the Radon transform of a piecewise smooth function f(x), x ∈ R n, n ≥ 2, are calculated. If the singularities of the Radon transform are known, then the equations of the surfaces of discontinuity of f(x) are calculated by applying the Legendre transform to the functions, w ..."
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Cited by 5 (4 self)
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Abstract. Singularities of the Radon transform of a piecewise smooth function f(x), x ∈ R n, n ≥ 2, are calculated. If the singularities of the Radon transform are known, then the equations of the surfaces of discontinuity of f(x) are calculated by applying the Legendre transform to the functions, which appear in the equations of the discontinuity surfaces of the Radon transform of f(x); examples are given. Numerical aspects of the problem of finding discontinuities of f(x), given the discontinuities of its Radon transform, are discussed. I.