F. Alberto Grünbaum

Image Reconstruction of the Interior of Bodies That Diffuse Radiation

A method for reconstructing images from projections is described. The unique aspect of the procedure is that the reconstruction of the internal structure can be carried out for objects that diffuse the incident radiation. The method may be used with photons, phonons, neutrons, and many other kinds of radiation. The procedure has applications to medical imaging, industrial imaging, and geophysical imaging.

Orthogonal matrix polynomials satisfying second-order differential equations

We develop a general method that allows us to introduce families of orthogonal matrix polynomials of size N × N satisfying second-order differential equations. The presence of this extra property should make these orthogonal polynomials into useful tools in several areas of mathematics and its applications. Historically, this has certainly been the case for their scalar-valued versions. The subtlety of the noncommutative algebra of matrices can be exploited to yield many different such families, almost dwarfing the scalar situation by comparison. All these families form a nice and rich hierarchy starting from the classical Jacobi, Hermite, and Laguerre families, N=1, and increasing in number and variety as the size N increases. We illustrate the use of our method by giving large classes of generic examples of arbitrary size N.

Matrix valued Jacobi polynomials

Abstract For every value of the parameters α , β >−1 we find a matrix valued weight whose orthogonal polynomials satisfy an explicit differential equation of Jacobi type.

Variations on a theme of Heine and Stieltjes: an electrostatic interpretation of the zeros of certain polynomials

We show a way to adapt the ideas of Stieltjes to obtain an electrostatic interpretation of the zeroes of a large class of orthogonal polynomials.

A study of fourier space methods for “limited angle” image reconstruction *

The problem of recovering a function f(x 1,x 2) from a limited number of its one-dimensional projections is an ill-conditioned inverse problem arising in areas which include radio astronomy, electron microscopy, and X-ray tomography. The ill-conditioning of the problem is related to the availability of data only for angles 0 ⩽θ ⩽ α < π. In this paper we make a detailed study of small scale models of the practical implementation of some Fourier methods for the reconstruction of f(x 1 x 2). We concentrate on explaining the source of the ill-conditioning, as well as trying to give a qualitative connection between the amount of “angular data” a and the degree of well-posedness of the problem. Our study leads one naturally to the study of the detailed structure of the spectral properties of a certain selfadjoint positive definite operator, similar to the one encountered in the study of prolate spheroidal functions by Sepian, Pollak, and Landau. A careful look at these spectral properties as a function of the p...

A Theorem of Bochner, Revisited

Many hierarchies of the theory of solitons possess symmetries which do not belong to the hierarchy itself. These symmetries are known under the various names of additional, master or conformal symmetries. They were discovered by Fokas, Fuchssteiner and Oevel [9], [10], [25], Chen, Lee and Lin [4] and Orlov and Schulman [26]. They are intimately related to the bihamiltonian nature of the equations of the theory of solitons which was pioneered in the work of Magri [23] and Gel’fand and Dorfman [11].

Convolution Algorithms for Arbitrary Projection Angles

The point response function ¿ of a convolution algorithm for reconstructing a function from a finite set of its projections is the sum of the back-projections of the filters used. An effective method is given for choosing the filters so that ¿ is as close as possible to a specified point response ¿. The weighted mean square error in approximating ¿ by ¿ goes to 0 as the number of projection angles goes to infinity, independent of their placement. Compensation for additive noise in the projections is discussed and numerical results are presented.

A new property of reproducing kernels for classical orthogonal polynomials

Abstract We exhibit a second-order differential operator commuting with the reproducing kernel ∑ n − 0 T φ n (λ) φ n (μ) h n each time that { φ n ( λ )} is one of the classical orthogonal polynomials: Jacobi, Laguerre, Hermite and Bessel. This is the analog of a known property in the study of time and band-limited signals.

Toeplitz matrices commuting with tridiagonal matrices

Abstract we prove that if R is a nonscalar Toeplitz matrix Ri, j=r∦i−j∦ which commutes with a tridiagonal matrix with simple spectrum, then r k r 1 = u k-1 r 2 r 1 cos p u k-1 (cos p) , k=4, 5,…, with Uk the Chebychev polynomial of the second kind, where p is determined from cos p= 1 2 r 2 1 −r 1 r 3 r 2 2 −r 1 r 3 .

Orthogonal matrix polynomials satisfying first order differential equations: a collection of instructive examples

Abstract We describe a few families of orthogonal matrix polynomials of size N × N satisfying first order differential equations. This problem differs from the recent efforts reported for instance in [7] (Orthogonal matrix polynomials satisfying second order differential equations, Internat. Math. Research Notices, 2004 : 10 (2004), 461–484) and [15] (Matrix valued orthogonal polynomials of the Jacobi type, Indag. Math. 14 nrs. 3, 4 (2003), 353–366). While we restrict ourselves to considering only first order operators, we do not make any assumption as to their symmetry. For simplicity we restrict to the case N = 2. We draw a few lessons from these examples; many of them serve to illustrate the fundamental difference between the scalar and the matrix valued case.