Abstract. A new approach to the construction of finite-difference methods is presented. It is shown how the multi-point 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.

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.

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 ap-proximate 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.