Square normalized Splines

Question

The M-Spline basis functions have the neat property that they integrate to one $$ \int M_i(x, k, t) dx = 1 $$ and therefore any M-Spline $$ \int \sum_i a_i M_i(x,k,t) dx = 1 $$ if the coefficients $a_i$ sum to one. I am wondering if there is a square normalized version of this, so a version where we get: $$ \int \left(\sum_i a_i M^*_i(x,k,t)\right)^2 dx = 1 $$ We could of course just use any normal M or B spline, square and normalize it, but is there some work done on this and are there some nice recurrence relations known?