Some probabilistic properties of B-splines and an application to dimensional tolerance (Q1195771)

In this short note the author discusses the density interpretation of the normalized \(B\)-splines \(M_{i,k}(t)\), defined by the knot sequence \(t_ i\leq\dots\leq t_{k+1}\bigl(\int_{t_ 0}^{t_ k} M_{i,k}(t)dt=1\bigr)\). In the only theorem he finds the variance \(\sigma^ 2:=E\{(t-\mu)^ 2\}\), \(\mu\) is the expectation of \(M_{i,k}(t)\) in the case of equispaced knots. (Reviewer's remark: The result follows easily from the well-known relation \(f[0,1,\dots,k]=(1/k) \int_ 0^ k M_{0,k}(t)f^{(k)}(t)dt\), applied to \(f(t)=t^{k+2}\)).