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Theory of approximation
01 Jan 1956-
About: The article was published on 1956-01-01 and is currently open access. It has received 883 citations till now. The article focuses on the topics: Spouge's approximation & Born–Huang approximation.
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TL;DR: In this article, the authors discuss certain basic considerations such as the nonoptimality of the classical symmetric (balanced) designs for hypothesis testing, the optimality of designs invariant under an appropriate group of transformations, etc.
Abstract: After some introductory remarks, we discuss certain basic considerations such as the nonoptimality of the classical symmetric (balanced) designs for hypothesis testing, the optimality of designs invariant under an appropriate group of transformations, etc. In section 3 we discuss complete classes of designs, while in section 4 we consider methods for verifying that designs satisfy certain specific optimality criteria, or for computing designs which satisfy such criteria. Some of the results are new, while part of the paper reviews pertinent results of the author and others.
565 citations
IBM1
TL;DR: A new method of farthest point strategy for progressive image acquisition-an acquisition process that enables an approximation of the whole image at each sampling stage-is presented, retaining its uniformity with the increased density, providing efficient means for sparse image sampling and display.
Abstract: A new method of farthest point strategy (FPS) for progressive image acquisition-an acquisition process that enables an approximation of the whole image at each sampling stage-is presented. Its main advantage is in retaining its uniformity with the increased density, providing efficient means for sparse image sampling and display. In contrast to previously presented stochastic approaches, the FPS guarantees the uniformity in a deterministic min-max sense. Within this uniformity criterion, the sampling points are irregularly spaced, exhibiting anti-aliasing properties comparable to those characteristic of the best available method (Poisson disk). A straightforward modification of the FPS yields an image-dependent adaptive sampling scheme. An efficient O(N log N) algorithm for both versions is introduced, and several applications of the FPS are discussed.
407 citations
334 citations
TL;DR: In this article, it was shown that the required number of elements is closely related to the desired sidelobe level and is almost independent of the aperture dimension, and the resolution or the beamwidth depends mainly on the aperture dimensions.
Abstract: Various probabilistic properties of a large antenna array with randomly spaced elements have been studied. It is found that for almost all cases of practical interest the required number of elements is closely related to the desired sidelobe level and is almost independent of the aperture dimension, the resolution (or the beamwidth) depends mainly on the aperture dimension, and the directive gain is proportional to the number of elements used if the average spacing is large. As a consequence the number of elements required is considerably less than that with uniform spacings. Starting with a given number of elements and a given aperture size, it is possible to improve the resolution by a factor of ten, a hundred, or more by spreading these elements over a larger aperture with little risk in obtaining a much higher sidelobe level and a lower directive gain. In fact, this method offers a solution which is optimum in a certain statistical sense, i.e., all sidelobes are of equal level with equal probability. In addition, this analysis also gives a simple estimate of the sidelobe level of most nonuniformly spaced antenna arrays. In a number of such arrays studied by various investigators with high speed computers, the agreement found is remarkable.
330 citations
TL;DR: In this paper, the authors discuss the application of Case's sum rules for Jacobi matrices and present a linear combination of two of their sum rules which has strictly positive terms.
Abstract: We discuss the proof of and systematic application of Case's sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms. Among our results are a complete classification of the spectral measures of all Jacobi matrices J for which J - J(0) is Hilbert-Schmidt, and a proof of Nevai's conjecture that the Szego condition holds if J - J(0) is trace class.
316 citations