# General form of dilations and translations of wavelet function

In some papers, the dilations and translations of a wavelet function is written as follows:

$$\psi_{j,k}=\frac {1}{\sqrt {2^j}} \psi\left({2^{-j}t-k}\right),\quad\text{where j and k are integers}$$

However, in some literature, the above formula is written in a different way:

$$\psi_{j,k}=\ {\sqrt {2^j}} \psi\left({2^{j}x-k}\right), \quad\text{where j and k are integers}$$

Are these two formula equivalent?

Yes, they are equivalent, since $j\in\mathbb{Z}$. Note that your first definition becomes equal to the second when $j$ is substituded by $-j$. It is just a convention, if you consider a positive $j$ to be a up- or downscaling.
They are equivalent, as long as $$j_1$$ (first formula) and $$-j_2$$ (second formula) span the same integer set. They are, if $$j_1\in\mathbb{Z}$$ and $$j_2\in\mathbb{Z}$$, or if $$j_1\in\mathbb{N}^+$$ (positive integers) and $$j_2\in\mathbb{N}^-$$ (negative integers).
They are, under the change of variable: $$t\to 2^{2j}x$$. Let us remind that if $$\psi(u)$$ is of unit norm(in $$L_2$$ norm), $$\frac{1}{\sqrt{a}}\psi(u/a)$$ is of unit norm as well, and the translation $$u\to u-k$$ does not change it.