Philip Feinsilver

Representations of Lie groups

With ξ = (ξ1,…,ξ d ), {ξ i } a basis for the Lie algebra ℊ, a basis for the universal enveloping algebra u(ℊ) is given by

Discrete analogues of the Heisenberg-Weyl algebra

\left[\kern-0.15em\left[ \text{n} \right]\kern-0.15em\right] = \xi ^n = \xi _1^{n1} \cdots \xi _d^{nd}

Appell systems on Lie groups

where the product is ordered, since the ξ i do not commute in general. As we saw for Appell systems, it is natural to look at generating functions to see how multiplication by the basis elements ξ i on u looks. We have

The spectrum of symmetric Krawtchouk matrices

\sum\limits_{n \geqslant 0} {\frac{{A^n }} {{n!}}} \,\xi ^n = \sum {\frac{{\left( {A_1 \xi _1 } \right)^{n_1 } }} {{n_1 !}} \ldots \frac{{\left( {A_d \xi _d } \right)^{n_d } }} {{n_d !}} = } \,e^{A_1 \xi _1 } \ldots e^{A_d \xi _d }

Hardware realization of Krawtchouk transform using VHDL modeling and FPGAs

(0.1) This is an element of the group, as it is a product of the one-parameter subgroups generated by the basis elements.

On the coherent states for the q-Hermite polynomials and related Fourier transformation

Abstract We establish a correspondence between the vanishing of a certain set of minors of a matrix A and the vanishing of a related set of minors of A ×1 . In particular, inverses of banded matrices are characterized. We then use our results to find patterns for Toeplitz matrices with banded inverses. Finally we give an interesting determinant formula for inverses of banded matrices, and show that in general a “banded partial” matrix may be completed in a unique way to give a banded inverse of the same bandwidth.