Suppose i have extended two d-variate functions $f$ and $g$ (two densities: positives and integrate to one) supported on $mathbb{R}_{+}^d$ into the following (tensorised) Laguerre($alpha = 0$) orthonormal basis of $L^2(mathbb R_{+}^d)$: $$left(varphi_{mathbf k}(mathbf x) = sqrt{2}^d e^{-lvert mathbf x rvert} sumlimits_{mathbf j le mathbf k} binom{mathbf k}{mathbf j} frac{(-2mathbf x)^{mathbf j}}{mathbf j !}right)_{mathbf k in mathbb N^d}.$$

I obtain coefficients $a_{mathbf p}$ for $f$ and $b_{mathbf p}$ for g, such that :

$$f(x) = sumlimits_{mathbf p in mathbb{N}^d} a_{mathbf p} varphi_{mathbf p}(mathbf x) text{ and } g(x) = sumlimits_{mathbf p in mathbb{N}^d} b_{mathbf p} varphi_{mathbf p}(mathbf x)$$

I already know that the follwoing convolution formula holds :

$$(f star g)(x) = sumlimits_{mathbf p in mathbb{N}^d} c_{mathbf p} varphi_{mathbf p}(mathbf x)$$

where the coefficients are given by:

$$c_{mathbf p} = sqrt{2}^{-d}sum_{mathbf epsilon in {0,1}^d} (-1)^{lvert mathbf epsilon rvert} sumlimits_{mathbf k le mathbf p – mathbf epsilon} a_{mathbf p} b_{mathbf p – mathbf epsilon – mathbf k}$$

Suppose now that $f$ and $g$ belong to smooth laguerre balls, i.e there exists $mathbf r(f),mathbf r(g) in mathbb R_{+}^d$ and $L(f),L(g) >0$ such that :

$$sum_limits{mathbf p in mathbb N^d} a_{mathbf p}^2 e^{langle mathbf r(f),mathbf prangle} le L(f)$$

and same thing for $g$. This ensure some bound on the truncation error, which is what I am after.

**Question:** Can I find constants $mathbf r(fstar g)$ and $L(fstar g)$ such that the same bound applies to the convolution? Moreover for the n-convolutions of $f_1,…,f_n$ ( by recursion, or better if posible).