The Concepts of Discrete Convergence and Discrete Approximations
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The concepts of discrete convergence and discrete approximation make the idea of approximating elements or spaces by sequences of elements or spaces more precise in a general setting. These concepts are introduced in Section 5.1, and explained by means of introductory examples. In Section 5.2, we show how discrete approximations are constructed by restriction and imbedding operators. On the other hand, we demonstrate that restriction operators can always be defined from a given discrete approximation. In Section 5.3, we illustrate these concepts by studying bounded continuous functions defined on a domain G and associated discrete approximations by sequences of functions defined on perturbations of G. The properties of a discrete approximation are then characterized by conditions on the perturbed domains of definition. Since the supremum norm is the underlying norm, we can also term this convergence “discretely uniform convergence”, but we wish to point out that this convergence is nothing other than discrete convergence in the sense of Section 5.1 with respect to a particular norm. In Section 5.4, we explore the concept of discrete convergence further by considering the example of discrete approximations of Lp-spaces over perturbed regions of integration in ℝd, d ʬ ℕ. The concept of discrete approximation can be characterized in this setting by conditions on the regions of integration, or, equivalently, by the weak convergence of the Lebesgue measures associated with each (perturbed) region of integration.
KeywordsLebesgue Measure Weak Convergence Restriction Operator Quadrature Formula Discrete Approximation
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