Let $c>0$, $$\varphi(x):=e^{-(cx)^2}\;\;\;\text{for }x\in[0,1)$$ and $$\varphi_x(y):=\sum_{h\in\mathbb Z}\varphi(x+h-y)\;\;\;\text{for }x,y\in\mathbb R.$$ Say we have $k\in\mathbb N$ points $x_1,\ldots,x_k\in[0,1)$ with equidistant spacing. If $\gamma\ge1$, how do we need to choose $c$ such that $$\int_0^1\left|1-\sum_{i=1}^k\varphi_{x_i}(y)\right|^\gamma\:{\rm d}y\tag1$$ is minimized? Can we solve this problem analytically? At least for $\gamma=1$ for $\gamma=2$?
EDIT: I would also be interested in the solution in the (maybe simpler) case where $$\varphi_x(y):=\varphi(x-y)\;\;\;\text{for }x,y\in[0,1)$$ instead.
EDIT 2: I've tried to find the optimal choice numerically. I've set $\gamma=2$ and increased $c^2$ from $0.01$ successively by $0.01$.
Taking $k=2,4,8,16,32$, I've always obtained a minimal error $\approx.00340322$ with $c^2=12.15,48.61,194.45,777.8,3111.2$, respectively.
Maybe someone sees what the formula for the optimal $c^2$ in terms of $k$ and $\gamma$ is from that ...