Kenneth Harrison

There are many advantages in the use of Hadamard matrices in digital signal processing. However one possible disadvantage is the so-called overflow, as measured by the associated Lebesgue constants. We show that for certain classes of recursively generated Hadamard matrices, there are logarithmic upper bounds for these constants. On the other hand, for some Hadamard matrices the Lebesgue constants are of order √m. These results have natural analogues in classical Fourier analysis.

The theory underlying line transect and variable circular plot surveys-distance sampling-begins with an assumed detectability function, giving the probabilities of detecting animals at different distances from the observer's path. The nature of these probabilities is unspecified in the general development, leaving users to question whether the actual probability structure matters. In particular, may one use the methodology in surveys where animals at the same distance have different probabilities of detection? This paper presents three examples where probabilities come from different assumptions: from the random placement of transects; from the uniform distribution of animals over the study region; and from cues randomly detected by the observer. These exemplify situations where detectability may not be a function of distance alone. Horvitz-Thompson estimators are displayed which can be used in each example, but some estimators require measuring features other than distance. A result concerning optimally weighted Horvitz-Thompson estimators shows that all three can be brought under the same umbrella if detection areas are measured instead of detection distances and if animals are uniformly distributed.

In a matrix-completion problem the aim is to specify the missing entries of a matrix in order to produce a matrix with particular properties. In this paper we survey results concerning matrix-completion problems where we look for completions of various types for partial matrices supported on a given pattern. We see that the existence of completions of the required type often depends on the chordal properties of graphs associated with the pattern.

In time-domain identification of linear systems the aim is to estimate the impulse response or transfer function of a linear system to within a given tolerance using a finite number of noisy observations of the output. Whether this is possible depends on the model set, that is, a given set to which the system is assumed to belong a priori. We give necessary and sufficient conditions on the model set to ensure that such identification is possible in the continuous-time case.

There are two natural ways of defining the numerical range of a partial matrix. We show that for each partial matrix supported on a given pattern they give the same convex subset of the complex plane if and only if a graph associated with the pattern is chordal. This extends a previously known result (C.R. Johnson, M.E. Lundquist, Operator Theory: Adv. Appl. 50 (1991) 283–291) to patterns that are not necessarily reflexive and symmetric, and our proof overcomes an apparent gap in the proof given in the above-mentioned reference. We also define a stronger completion property that we show is equivalent to the pattern being an equivalence.

We study the complexity of worst-case time-domain identification of linear time-invariant systems using model sets consisting of degree-n rational models with poles in a fixed region of the complex plane. For specific noise level δ and tolerance levels τ, the number of required output samples and the total sampling time should be as small as possible. In discrete time, using known fractional covers for certain polynomial spaces (with the same norm), we show that the complexity is O(n 2) for the H ∞ norm, O(n) for the ℓ 2 norm, and exponential in n for the ℓ 1 norm, for each δ and τ. We also show that these bounds are tight. For the continuous-time case we prove analogous results, and show that the input signals may be compactly supported step functions with equally spaced nodes. We show, however, that the internodal spacing must approach 0 as n increases.