thus is ideal for modeling galaxies. Furthermore, the generating “step-up” and “step-down” operators for the Hermite polynomials are simply combinations of the $x_i$, and $\partial_i$ operators.

Refregier (2003) shows that if we decompose a source image, $f$, into shapelet coefficients, the transformation to a lensed image may be expressed quite simply as: $$f'=(1+\kappa \hat{K}+\gamma_j \hat{S}_j)f$$ where the various lensing operators are: $$\begin{aligned}
\hat{K}&=&1+\frac{1}{2}\left(\hat{a}_1^{\dagger 2}+\hat{a}_2^{\dagger
  2}-\hat{a}_1^2-\hat{a}_2^2\right)\nonumber \\
\hat{S}_1&=&\frac{1}{2}\left(\hat{a}_1^{\dagger 2}-\hat{a}_2^{\dagger
  2}-\hat{a}_1^2+\hat{a}_2^2\right)\nonumber \\
\hat{S}_2&=&\frac{1}{2}\left(\hat{a}_1^{\dagger}\hat{a}_2^{\dagger}-\hat{a}_1\hat{a}_2\right)\ ,\end{aligned}$$ $\hat{a}^{\dagger}$ and $\hat{a}$ are the normal step-up and step-down operators, and the subscript refers to the directional component of the coefficient (i.e. 1 for the first or x-component, and 2 for the second, or y-component). Note that in the weak field limit, these operators indicate that power will be transferred between coefficients with indices $|\Delta n|+|\Delta m|=2$, which preserves symmetry as well as keeping the image representation in shapelet space compact.

In Goldberg & Bacon (2005), similar (albeit more complicated) transforms were found relating the derivatives of shear. We will not reproduce the full second order operators here, as they are written in full in the earlier work, but we will point out some key features. First, some of the elements in the operators have an explicit dependence 