Approximation theory and classical inequalities, harmonic analysis and function theory, extremal problems in one and several variables, analytical methods of metric geometry.
Main publications:
D. Gorbachev, V. Ivanov, S. Tikhonov, “Uncertainty principles for eventually constant sign bandlimited functions”, DOI: 10.1137/20M131566X, SIAM J. Math. Anal., 52:5 (2020), 4751–4782
F. Dai, D. Gorbachev, S. Tikhonov, “Nikolskii constants for polynomials on the unit sphere”, DOI: 10.1007/s11854-020-0084-9, J. dAnal. Math., 140:1 (2020), 161–185
L. De Carli, D. Gorbachev, S. Tikhonov, “Pitt inequalities and restriction theorems for the Fourier transform”, DOI: 10.4171/RMI/955, Revista Matematica Iberoamericana, 33:3 (2017), 789–808
D. V. Gorbachev, “Extremum problem for periodic functions supported in a ball”, DOI: 10.4213/mzm508, Math. Notes, 69:3 (2001), 313–319
D. V. Gorbachev, “Extremal problem for entire functions of exponential spherical type, connected with the Levenshtein bound on the sphere packing density in $\mathbb{R}^{n}$”, Izvestiya of the Tula State University Ser. Mathematics, 6:1 (2000), 71–78