Shayne Waldron

Generalized Welch bound equality sequences are tight frames

This article shows what are called the Welch (1974) bound equality (WBE) sequences by the signal processing community are precisely the isometric/equal norm-normalized/uniform tight frames which are currently being investigated for a number of applications, and in the real case are the spherical 2-designs of combinatorics. Recent applications include wavelet expansions, Grassmannian frames, frames robust to erasures, and quantum measurements. This is done by giving an elementary proof of a generalization of Welchs inequality to vectors which need not have equal energy, and then showing that equality occurs in this exactly when the vectors form a tight frame.

A generalised beta integral and the limit of the Bernstein-Durrmeyer operator with Jacobi weights

We give a generalisation of the multivariate beta integral. This is used to show that the (multivariate) Bernstein-Durrmeyer operator for a Jacobi weight has a limit as the weight becomes singular. The limit is an operator previously studied by Goodman and Sharma. From the elementary proof given, it follows that this operator inherits many properties of the Bernstein-Durrmeyer operator in a natural way. In particular, we determine its eigenstructure and give a differentiation formula for it which is new.

On the spacing of Fekete points for a sphere, ball or simplex

Suppose that K ⊂ ℝd is either the unit ball, the unit sphere or the standard simplex. We show that there are constants c1, c2 > 0 such that for a set of Fekete points (maximizing the Vandermonde determinant) of degree n, Fn ⊂ K, Download full-size image for all a ∈ Fn. Here dist(a, b) is a natural distance on K that will be described in the text.

Tight frames generated by finite nonabelian groups

Let

ISOMETRIC TIGHT FRAMES

\cal H

Signed frames and Hadamard products of Gram matrices

be a Hilbert space of finite dimension d, such as the finite signals ℓ2(d) or a space of multivariate orthogonal polynomials, and n ≥ d. There is a finite number of tight frames of n vectors for

The diagonalisation of the multivariate Bernstein operator

\cal H

Multipoint Taylor formulæ

which can be obtained as the orbit of a single vector under the unitary action of an abelian group G (of symmetries of the frame). Each of these so called harmonic frames or geometrically uniform frames can be obtained from the character table of G in a simple way. These frames are used in signal processing and information theory. For a nonabelian group G there are in general uncountably many inequivalent tight frames of n vectors for

On the Bernstein-Bézier form of Jacobi polynomials on a simplex

\cal H

Pseudometrics, distances and multivariate polynomial inequalities

which can be obtained as such a G-orbit. However, by adding an additional natural symmetry condition (which automatically holds if G is abelian), we obtain a finite class of such frames which can be constructed from the character table of G in a similar fashion to the harmonic frames. This is done by identifying each G-orbit with an element of the group algebra ℂG (via its Gramian), imposing the condition in the group algebra, and then describing the corresponding class of tight frames.