Vol 51, No 3 (2017)

Given a G-invariant holomorphic 1-form with an isolated singular point on a germ of a complex-analytic G-variety with an isolated singular point (G is a finite group), its equivariant homological index and (reduced) equivariant radial index are defined as elements of the ring of complex representations of the group. We show that these indices coincide on a germ of a smooth complex analytic G-variety. This makes it possible to consider the difference between them as a version of the equivariant Milnor number of a germ of a G-variety with an isolated singular point.

It is well known how to construct a system of symmetric orthogonal polynomials in an arbitrary finite number of variables from an arbitrary system of orthogonal polynomials on the real line. In the special case of the big q-Jacobi polynomials, the number of variables can be made infinite. As a result, in the algebra of symmetric functions, there arises an inhomogeneous basis whose elements are orthogonal with respect to some probability measure. This measure is defined on a certain space of infinite point configurations and hence determines a random point process.

A convolution in a Hilbert space is defined, and its basic properties are studied.

Let O ⊂ R^d be a bounded domain of class C^1,1. Let 0 < ε - 1. In L₂(O;Cⁿ) we consider a positive definite strongly elliptic second-order operator B_D,ε with Dirichlet boundary condition. Its coefficients are periodic and depend on x/ε. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent (B_D,ε − ζQ₀(·/ε))⁻¹ as ε → 0. Here the matrix-valued function Q₀ is periodic, bounded, and positive definite; ζ is a complex-valued parameter. We find approximations of the generalized resolvent in the L₂(O;Cⁿ)-operator norm and in the norm of operators acting from L₂(O;Cⁿ) to the Sobolev space H¹(O;Cⁿ) with two-parameter error estimates (depending on ε and ζ). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation Q₀(x/ε)∂_tv_ε(x, t) = −(B_D,εv_ε)(x, t).

The notion of reduced synthesis in the context of harmonic analysis on general locally compact groups is introduced; in the classical situation of commutative groups, this notion means that a function f in the Fourier algebra is annihilated by any pseudofunction supported on f⁻¹(0). A relationship between reduced synthesis and compact synthesis (i.e., the possibility of approximating compact operators by pseudointegral ones without increasing the support) is determined, which makes it possible to obtain new results both in operator theory and in harmonic analysis. Applications to the theory of linear operator equations are also given.