Topic

# Orthogonal complement

About: Orthogonal complement is a research topic. Over the lifetime, 549 publications have been published within this topic receiving 7932 citations.

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TL;DR: In this article, the authors give new bounds in certain inequalities concerning mutually reciprocal lattices in R". To formulate the problem, they have introduced some notation and terminology, and treat IR" as an n-dimensional euclidean space with the norm II II II and metric d.

Abstract: The aim of this paper is to give new bounds in certain inequalities concerning mutually reciprocal lattices in R". To formulate the problem, we have to introduce some notation and terminology. We shall treat IR" as an n-dimensional euclidean space with the norm II II and metric d. The inner product of vectors u, v will be denoted by uv; we shall usually write u 2 instead of uu. The closed and open unit balls in F," will be denoted by B, and B'., respectively. If A c IR ", then span A and A z denote the linear subspace spanned over A and the orthogonal complement of A in P~". A lattice in IR" is an additive subgroup of IR" generated by n linearly independent vectors. The family of all lattices in IR" will be denoted by A.. Given a lattice L e A., we define the polar (dual, reciprocal) lattice L* in the usual way: L* = {u ~ ~ " : u v ~ Z for each v ~ L } .

514 citations

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430 citations

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TL;DR: A new method for source localization is described that is based on a modification of the well-known MUSIC algorithm, and a general form of the RAP-MUSIC algorithm is described for the case of diversely polarized sources.

Abstract: A new method for source localization is described that is based on a modification of the well-known MUSIC algorithm. In classical MUSIC, the array manifold vector is projected onto an estimate of the signal subspace. Errors in the estimate of the signal subspace can make localization of multiple sources difficult. Recursively applied and projected (RAP) MUSIC uses each successively located source to form an intermediate array gain matrix and projects both the array manifold and the signal subspace estimate into its orthogonal complement. The MUSIC projection to find the next source is then performed in this reduced subspace. Special assumptions about the array manifold structure, such as Vandermonde or shift invariance, are not required. Using the metric of principal angles, we describe a general form of the RAP-MUSIC algorithm for the case of diversely polarized sources. Through a uniform linear array simulation with two highly correlated sources, we demonstrate the improved Monte Carlo error performance of RAP-MUSIC relative to MUSIC and two other sequential subspace methods: S and IES-MUSIC. We then demonstrate the more general utility of this algorithm for multidimensional array manifolds in a magnetoencephalography (MEG) source localization simulation.

365 citations

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244 citations

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TL;DR: In this article, necessary and sufficient conditions for the translates of functions x, r, y/x, y, r y/r to form a Riesz basis for V are derived.

Abstract: A multiresolution approximation (Km)m€Z of L2(R) is of multi- plicity r > 0 if there are r functions x, ... , r whose translates form a Riesz basis for V$ . In the general theory we derive necessary and sufficient conditions for the translates of x, ... , r, y/x, ... , y/r to form a Riesz basis for V\ . The resulting reconstruction and decomposition sequences lead to the construction of dual bases for V0 and its orthogonal complement W0 in Vx . The general theory is applied in the construction of spline wavelets with mul- tiple knots. Algorithms for the construction of these wavelets for some special cases are given.

216 citations