Keith F. Taylor

Wavelets from square-integrable representations

The continuous wavelet decompositions that arise from square-integrable representations of certain Lie groups on

Convergence of lattice sums and Madelung’s constant

L^2 (\mathbb{R}^n )

A direct integral decomposition of the wavelet representation

are investigated. The groups are formed as the semidirect product of

Compact open sets in duals and projections in L1-algebras of certain semi-direct product groups

\mathbb{R}^n

A Finer Aspect of Eigenvalue Distribution of Selfadjoint Band Toeplitz Matrices

with an n-dimensional subgroup H of

Practical version of the asymptotic linearity theorem with applications to the additivity problems of thermodynamic quantities

GL_n (\mathbb{R})

Aspects of form and general topology: Alpha Space Asymptotic Linearity Theorem and the spectral symmetry of alternants

. There is a natural “translation and dilation” representation of such groups on

Explicit Cross-Sections of Singly Generated Group Actions

L^2 (\mathbb{R}^n )

Projections inC*-algebras of nilpotent groups

. The basic formulas of Duflo and Moore, which lead to the resolution of the identity via a square-integrable representation, are given an elementary proof for this special case. Several two-dimensional examples are described. A method for discrete decompositions via frames is given using the representations under study.

C*-Algebras of Crystal Groups

The lattice sums involved in the definition of Madelung’s constant of an NaCl‐type crystal lattice in two or three dimensions are investigated. The fundamental mathematical questions of convergence and uniqueness of the sum of these, not absolutely convergent, series are considered. It is shown that some of the simplest direct sum methods converge and some do not converge. In particular, the very common method of expressing Madelung’s constant by a series obtained from expanding spheres does not converge. The concept of analytic continuation of a complex function to provide a basis for an unambiguous mathematical definition of Madelung’s constant is introduced. By these means, the simple intuitive direct sum methods and the powerful integral transformation methods, which are based on theta function identities and the Mellin transform, are brought together. A brief analysis of a hexagonal lattice is also given.