Necessary and sufficient conditions are obtained on the exponents $\alpha$ and $\beta$ under which the Fourier series in the Sobolev system of polynomials $\mathcal{P}_r^{\alpha,\beta}$ associated with the Jacobi polynomials converge uniformly on $[-1,1]$ to functions in the Sobolev space $W^r_{L^1_{\rho(\alpha,\beta)}}$, where $\rho(\alpha,\beta)$ is the Jacobi weight. Bibliography: 11 titles.
A construction of small and large Nijenhuis cohomologies is presented. For small cohomologies an isomorphism theorem is proved, relating de Rham cohomologies to small Nijenhuis cohomologies for a nondegenerate operator. Poncaré's lemma is deduced as a consequence. For operators with singularities a theorem on connections between the logarithmic and Nijenhuis cohomology is established. The cohomologies of balls containing differentially nondegenerate singularities are calculated. Relations between the cohomologies of different operators in Nijenhuis pencils are established. Bibliography: 16 titles.
We examine the set of irreducible germs fP: UP -> Vp of generic morphisms f : S -> P2 to the projective plane whose branch curve germs BP subset of Vp have singularities equisingular deformation equivalent to the singularities given by equations xk1-yk2 = 0 with coprime k1, k2 is an element of N. Bibliography: 16 titles.
We find a sharp constant in the inequality parallel to psi(& centerdot;)x(& centerdot;)parallel to Lq (T,& micro;) <= C max 1 <= j <= l parallel to phi j(& centerdot;)x(& centerdot;)parallel to gamma Lp(T,& micro;)max l+1 <= j <= n parallel to phi j(& centerdot;)x(& centerdot;)parallel to 1-gamma Lr (T,& micro;), where T is a cone in Rd, and the weights psi( & centerdot; ) and phi j ( & centerdot; ), j = 1,... , n, are homogeneous measurable functions satisfying certain additional symmetry conditions. The sharp inequality is a consequence of a more general problem of optimal recovery in weighted Lq (T, & micro;)-spaces from incompletely given functions, whose solution is also presented. The results obtained are used for the optimal recovery of powers of the generalized Laplace operator and for the derivation of relevant sharp inequalities. Bibliography: 11 titles.
