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.

this paper, we suggest a method for numerical differentiation that we believe avoids some of the limitations mentioned above. Given a real valued, smooth function u defined on the closed interval [a; b], we construct an associated functional,

Several questions of approximation theory are discussed: 1) can one approximate stably in L ∞ norm f ′ given approximation fδ, � fδ −f �L∞< δ, of an unknown smooth function f(x), such that � f ′ (x) �L∞ ≤ m1? 2) can one approximate an arbitrary f ∈ L2 (D), D ⊂ Rn, n ≥ 3, is a bounded domain, by linear combinations of the products u1u2, where um ∈ N(Lm), m = 1, 2, Lm is a formal linear partial differential operator and N(Lm) is the null-space of Lm in D, N(Lm): = {w: Lmw = 0 in D}? 3) can one approximate an arbitrary L2 (D) function by an entire function of exponential type whose Fourier transform has support in an arbitrary small open set? Is there an analytic formula for such an approximation?

Abstract. Theories of monochromatic high-frequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude — in other words, are complex-valued. The following mathematical principle is ultimately behind the scenes: any geometric optical eikonal, which conventional rays engender in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an alternative eikonal, endowed with a non-zero imaginary part, comes on stage. In the present paper we explore such a principle in dimension 2. We investigate a partial differential system that governs the real and the imaginary parts of complex-valued two-dimensional eikonals, and an initial value problem germane to it. In physical terms, the problem in hand amounts to detecting waves that rise beside, but on the dark side of, a given caustic. In mathematical terms, such a problem shows two main peculiarities: on the one hand, degeneracy near the initial curve; on the other hand, ill-posedness in the sense of Hadamard. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, and a suitable discretization. Approximate differentiation and a parody of the quasi-reversibility method are also involved. We offer an algorithm that restrains instability and produces effective approximate solutions. 1.