James T. Lewis

Constrained Least-Squares Synthesis of Coefficients for Arrays of Sensors and FIR Digital Filters

A general method is presented for synthesizing weighting coefficients for arrays of sensors or for finite-impulse-response (FIR) digital filters. The beam pattern of the array or the frequency response of the digital filter is a weighted, least mean-square (LMS) approximation to a desired function, subject to constrained pattern values at specified points. The method is applied to the problem of producing beam patterns of a line array for the rejection of interfering point sources. For this case, we show that adaptive array processing can be based on modification of the roots of the array polynomial.

Best approximation using a peak norm

Abstract A family of norms ∥ g ∥ ( α ) , 0 α L 1 norms is defined. Best approximation of a continuous function from an n -dimensional subspace is characterized and (in case of a T -subspace) a uniqueness theorem is proven. The family, as well as the best approximation, is continuous in α. In particular, when α tends to zero or one, we get the uniform or the L 1 case, respectively.

The generalized Riemann, simple, dominated and improper integrals

A powerful generalization of the Riemann integral has been introduced by making an innocent-looking modification in the usual definition. This generalized Riemann integral was defined in 1957 by Kurzweil [6]. It was independently defined and extensively studied and generalized by Henstock [3-51 who called it the Riemann-complete integral. Although this integral has been popularized somewhat (cf. [7]), it still is not as well known as it deserves to be. Among the virtues of this powerful integral are the following:

Infinite Riemann Sums, the Simple Integral, and the Dominated Integral

Simple integrability of a function f (defined by Haber and Shisha in [2]) is shown to be equivalent to the convergence of the infinite Riemann sum

Avoiding-sequences with minimum sum

\sum\limits_{k = 1}^\infty {f\left( {{\xi _k}} \right)\left( {{x_k} - {x_{k - 1}}} \right)}

Circular avoiding sequences with prescribed sum

to the improper Riemann integral \( \int_0^\infty f \) f as the gauge of the partition \( \left( {{x_k}} \right)_{k = 0}^\infty \) of [0,∞)converges to O. An analogous result is obtained for dominant integrability (defined by Osgood and Shisha in [5]). Also certain results of Bromwich and Hardy [1] are recovered.

Lp convergence of monotone functions and their uniform convergence

A numerical method is presented for designing digital filters. The method allows one to minimize the mean-square error or noise power over some intervals of frequency, while simultaneously constraining the maximum error in other intervals of frequency. Thus, for example, one can minimize noise power from a stopband of frequencies while constraining signal fidelity in a passband of frequencies by limiting the maximum passband deviation.

Antenna Design as a Partial Basis Problem.

Abstract A circular sequence of positive integers of length n is a sequence of n terms in which the first and last are considered consecutive. For x≤n a positive integer, such a sequence is x-avoiding if no set of consecutive terms sums to x. We show that an x-avoiding circular sequence of length n satisfies a1+a2+⋯+an≥2n, and give a simple necessary and sufficient condition for equality. Minimizing sequences are exhibited when the minimum sum is known.