A Good kernel (bounded approximate identity) in $L^1(\mathbb{T})$ is a sequence of functions $\{g_n\}$ satisfying: (i) $\frac{1}{2\pi}\int_0^{2\pi}g_n(t)dt=1~~$ (ii) $||g_n||_1=O(1)~~$ (iii) $~~\lim_{n\to \infty}\int_{\delta}^{2\pi-\delta}g_n=0$.
The Fejer's kernel and Poussin kernels are two examples of Good kernels.
Question. Do there exist ant other items? Regardless of the theoretical importance, what is a practical necessity for the consideration?
Remark. Let me add something more. Let $G$ be a locally compact and Hausdorff group. Let $E_n$ be sequence of nbhds of $e$ (the identity of $G$) with $\cap E_n=\{e\}$ and $|E_n|=O(1)$. Then $g_n=\frac{\chi_{E_n}}{|E_n|}$ forms a bounded approximate identity for $L^1(G)\cap L^2(G)$. Using this point of view, one may construct a lot of approximate identities. However, I am interested in those items that have been considered (raised) by some practical reasons in physics, electrical engineering or ... .
In general, Good kernels serve as a bridge between the continuous and the discontinuous, and even the discrete.
The mathematical tools of analysis are generally defined for functions that are at least continuous, and often smooth. But when you model a physical object, that model is generally discontinuous. For example, density suddenly drops from a non-zero value to zero as you cross the boundary of your model. In the "real world", this doesn't occur. In fact, below a certain level, the very concept of the boundary of a physical object breaks down. Which atoms are part of the object vs not? How far out from each atom in the object does the space inside the object vs outside extend? "Volume", "Density", "Surface Area", "Length", etc are not even definable, much less measurable, beyond certain limits of accuracy. To apply mathematical methods to "real world objects" - i.e., to do physics and engineering - is to pretend that this limit of definition does not exist.
For convenience, we represent particles as points, and extended objects as volumes of space with well-defined boundaries. But doing so engenders discontinuities, particularly in densities. The density function in the vicinity of a particle is not even representable by a function. It must somehow integrate to give the mass of the particle while being $0$ everywhere except at that one point.
So to apply our continuous mathematical tools to these discontinuous models, we have to approximate the discontinuous functions with continuous ones. And that exactly is what these good kernels are doing. They approximate a density "function" of a particle of mass $1$ located at the identity. Operations that require a continuous density can be applied to the kernels, then taking the limit as $n \to \infty$ provides the answer for the actual particle.