Vol 210, No 6 (2019)

We give a description of the algebra of outer derivations of the group algebra of a finitely presented discrete group in terms of the Cayley complex of the groupoid of the adjoint action of the group. This problem is a smooth version of Johnson's problem on derivations of a group algebra. We show that the algebra of outer derivations is isomorphic to the one-dimensional compactly supported cohomology group of the Cayley complex over the field of complex numbers. Bibliography: 34 titles.

The Turan, Fejer and Bohman extremal problems for the multivariate Fourier transform in terms of the eigenfunctions of a Sturm-Liouville problem on the Cartesian product of half-lines are solved under natural conditions on a weight function defined as a product of one-dimensional weight functions. Extremal functions are constructed. A multivariate Markov quadrature formula is proved based on the zeros of eigenfunctions of the Sturm-Liouville problem. This quadrature formula is shown to be sharp on entire multivariate functions of exponential type. A Paley-Wiener type theorem is proved for the multivariate Fourier transform. A weighted $L^2$-analogue of the Kotel'nikov-Nyquist-Whittaker-Shannon sampling theorem is put forward. Bibliography: 42 titles.

A pro-nilpotent Lie algebra $\mathfrak g$ is said to be naturally graded if it is isomorphic to its associated graded Lie algebra $\operatorname{gr}\mathfrak g$ with respect to the filtration by the ideals in the lower central series. Finite-dimensional naturally graded Lie algebras are known in sub-Riemannian geometry and geometric control theory, where they are called Carnot algebras. We classify the finite-dimensional and infinite-dimensional naturally graded Lie algebras $\mathfrak g=\bigoplus_{i=1}^{+\infty}\mathfrak g_i$ with the property $$\dim\mathfrak g_i+\dim\mathfrak g_{i+1}\le3,\qquad i\ge1.$$An arbitrary Lie algebra $\mathfrak g=\bigoplus_{i=1}^{+\infty}\mathfrak g_i$ of this class is generated by the two-dimensional subspace $\mathfrak g_1$, and the corresponding growth function $F_\mathfrak g^\mathrm{gr}(n)$ satisfies the bound $F_\mathfrak g^\mathrm{gr}(n)\le3n/2+1$. Bibliography: 32 titles.