Wavelet

A function $$\psi(x)$$ is called a wavelet if it is capable of becoming the base function of what I call a wavelet system in $$L^(\mathbb R)$$. More precisely and following the book definition (Def. 8.1.1 ), a wavelet is defined as below.

Let $$\psi(x) \in L^2(\mathbb R)$$. For $$j,k \in \mathbb Z$$ define the function $$\psi_{j,k}$$ by

$$\psi_{j,k}(x):=2^{j/2} = D^j T_k \psi = \psi (2^jx-k),\,\,x\in \mathbb R$$