Approximate identity

An approximate identity refers to a set of functions, typically a one-parameter family, that exhibit desirable properties. These functions are used to approximate other functions, particularly in the context of continuous functions. The goal of an approximate identity is to produce an output that is as close as possible to the input. In other words, an approximate identity function is designed to learn how to mimic the input function as closely as possible.From: Linear and Complex Analysis for Applications [2017], NDE 4.0 compatible ultrasound inspection of butt-fused joints of medium-density polyethylene gas pipes, using chord-type transducers supported by customized deep learning models [2020]

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Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017

John P. D’Angelo

Remark. The proof of (23) relies on a profound idea in mathematics and physics. The Gaussian factor is being used as an approximate identity. An approximate identity is a one-parameter family of well-behaved functions Gϵ such that (for continuous f) limϵ(Gϵ * f)(x) = f(x). We do not specify where ∈ is tending, because in various situations the parameter tends to different values. Often it tends to infinity. For the Poisson kernel, mentioned in Section 6 of Chapter 7, the parameter tends to 1. In all these cases, evaluation of a function at x becomes a limit of convolutions. Readers who know about the Dirac delta “function” should note that approximate identities are one way to make this notion precise. See Exercise 4.10 for the simplest example of an approximate identity used in electrical engineering. We define and discuss approximate identities in more detail in subsection 4.1. See also subsection 4.3.

General system of cubic–quartic functional equations in quasi-β-normed spaces

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Published in International Journal of General Systems, 2022

Abasalt Bodaghi

A left (right) bounded approximate identity for a normed algebra is a bounded net in such that ( ) for each . A bounded approximate identity for is a bounded net which is both a left and a right bounded approximate identity. It is well-known that every -algebra has a bounded approximate identity.

Approximate identity

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General system of cubic–quartic functional equations in quasi-β-normed spaces