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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?

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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.

So, when writing a paper or reading a book, it is always important to understand the definition of the transform used.

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They’re equivalent, but the distinction seems to usually come from which definition of multiresolution analysis one uses - ascending or descending chains of detail/approximation spaces.

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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.

The choice of notation depends on the community: maths, signal processing, functional analysis.

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