Limit of a function

Summary

Although the function \tfrac{\sin x}{x} is not defined at zero, as x becomes closer and closer to zero, \tfrac{\sin x}{x} becomes arbitrarily close to 1. In other words, the limit of \tfrac{\sin x}{x}, as x approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very clo

Official source

https://en.wikipedia.org/wiki/Limit_of_a_function

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Pickands constants play a crucial role in the asymptotic theory of Gaussian processes. They are commonly defined as the limits of a sequence of expectations involving fractional Brownian motions and, as such, their exact value is often unknown. Recently, Dieker and Yakir (Bernoulli, 20(3), 1600-1619, 2014) derived a novel representation of Pickands constant as a simple expected value that does not involve a limit operation. In this paper we show that the notion of Pickands constants and their corresponding Dieker-Yakir representations can be extended to a large class of stochastic processes, including general Gaussian and Levy processes. We furthermore develop a link to extreme value theory and show that Pickands-type constants coincide with certain constants arising in the study of max-stable processes with mixed moving maxima representations.

Springer Verlag2017

Generalized Sampling: Stability and Performance Analysis

Michaël Unser

Generalized sampling provides a general mechanism for recovering an unknown input function

f(x) \in H

from the samples of the responses of m linear shift-invariant systems sampled at 1 ⁄ mth the reconstruction rate. The system can be designed to perform a projection of f(x) onto the reconstruction subspace

V(\varphi) = span \{\varphi(x - k)\} _{ k \in Z }

; for example, the family of bandlimited signals with

\varphi(x) = sinc(x)

. This implies that the reconstruction will be perfect when the input signal is included in V(φ): the traditional framework of Papoulis' generalized sampling theory. Otherwise, one recovers a signal approximation

f ^{ ~ } (x) \in V(\varphi)

that is consistent with f(x) in the sense that it produces the same measurements. To characterize the stability of the algorithm, we prove that the dual synthesis functions that appear in the generalized sampling reconstruction formula constitute a Riesz basis of V(φ), and we use the corresponding Riesz bounds to define the condition number of the system. We then use these results to analyze the stability of various instances of interlaced and derivative sampling. Next, we consider the issue of performance, which becomes pertinent once we have extended the applicability of the method to arbitrary input functions, that is, when H is considerably larger than V(φ), and the reconstruction is no longer exact. By deriving general error bounds for projectors, we are able to show that the generalized sampling solution is essentially equivalent to the optimal minimum error approximation (orthogonal projection), which is generally not accessible. We then perform a detailed analysis for the case in which the analysis filters are in

L _{ 2 }

and determine all relevant bound constants explicitly. Finally, we use an interlaced sampling example to illustrate these various calculations.

IEEE1997