The main research directions: approximation in functional spaces, numerical methods, wavelets. Method of investigation is the synthesis of numerical analysis, theory of functions and geometrical number theory. 1) The approximation properties of subspaces generated by translations on a lattice of fixed function: a) definition of projection-net widths describing the approximation properties of such subspaces; b) the relationship between these widths and geometrical properties of Lebesgue sets packings; c) necessary and sufficient conditions for the optimality and the construction of optimal subspaces; d) the definition of average dimension and the comparison of projection-net and Kolmogorov widths of the same average dimension; e) direct and inverse projection-net theorems. 2) Wavelets: a) construction of wavelet bases which are universally optimal for the whole scale of Sobolev spaces; b) the constructive description of wavelets generated by some n-dimensional lattice tilings; c) the investigation of relationship between B–splines and wavelets; d) construction of the family of biorthonormal wavelet bases with compact supports; e) the calculation of exact constants for spline inequalities. 3) The investigation of the family of spline-trigonometrical bases: a) construction of biorthonormal pair of functional systems such that functions of the first system are the productions of trigonometrical and algebraical polinomials and functions of the second system are the productions of trigonometrical polinomials and derivatives of Schoenberg B–splines; b) interpolate representations of the entire functions of exponential type based on the using of spline-trigonometrical bases (these representations are similar to the well-known Shannon–Kotel"nikov theorem); c) construction and investigation of projection-difference analogs of differential operators by means of representations of the entire functions of exponential type; d) consideration of asymptotical behaviour of spline-trigonometrical bases generating some new integral transforms; e) the investigation of Lebesgue functions and constants for these bases. 4) Difference and finite-element approximation: a) the necessary and sufficient conditions of coincidence of difference and finite element operators; b) the complete description of the family of coordinate functions with asymptotically optimal approximative properties which generate the projection-difference approximations of n–dimensional Laplace operator coinciding with its simplest difference (2n+1)–point "cross-type" analog; c) the projection-net and difference approximations of Laplacian using wide-spaced nets.
Main publications:
Strelkov N. A. Proektsionno-setochnye poperechniki i reshetchatye ukladki // Matem. sbornik, 1991, 182(10), 1513–1533.
Strelkov N. A. Ermitovy poperechniki, srednyaya razmernost i kratnye ukladki // Matem. sbornik, 1996, 187(1), 121–142.
Strelkov N. A. Proektsionno-setochnye analogi operatora Laplasa tipa "krest" // Zhurn. vychislit. matem. i matem. fiz., 1996, 36(10), 46–55.
Strelkov N. A. Universalno optimalnye vspleski // Matem. sbornik, 1997, 188(1), 147–160.
Strelkov N. A. Splain-trigonometricheskie bazisy i ikh svoistva // Matem. sbornik, 2001, 192(7), 125–160.
