function of order $q'$ has only $N_c=\lfloor q'/2 \rfloor + 1$ nonzero coefficients. Second, we note that we will evaluate the Hermite functions for order $0 \le \beta \le q'$ with the same argument. Therefore, we can calculate the powers of the argument initially with $q'$ operations. Evaluating each Hermite function requires only $N_c$ additional operations. Therefore, to compute all Hermite functions up to order $q'$ requires approximately: $$\begin{aligned}
N_{\rm ops, hermite} & = & N_{\rm cexp} + q' + 2 \sum_{\beta = 0}^{N_c} N_c \nonumber \\
& = & N_{\rm cexp} + q' + 2 \sum_{\beta = 0}^{N_c} \lfloor \beta/2 \rfloor + 1 \nonumber \\
& \simeq & N_{\rm cexp} + q' + N_C (N_C + 1)\end{aligned}$$

The multi-index powers of $t_\beta = (\delta/\delta_j)^{|\beta|/2}$ can be computed recursively, by computing $t_n = (\delta/\delta_j) t_{n-2}$ with $t_0=(\delta/\delta_j)^{1/2}$, and $t_1=(\delta/\delta_j)^1$. This requires $2(q'+1)$ operations $0 \le n \le 2q'$.

The weight $v_j$ can be folded into the power values also, with $2(q'+1)$ operations.

Finally, the full sum to update $C_\beta$ over two dimensions requires the multiplication of Hermite functions for 2 dimensions, and the powers, plus the accumulation, which requires $3 (q'+1)^2$ operations in total.

The total number of operations to 