On Asymptotic Properties of Certain $B$-Splines in Terms of Theta-like Functions

The asymptotic behavior of the Mellin transform of the associated $B$-splines $B_N^*(t) :=t^{-N}B_N(t)$ with special knots in terms of theta-like functions is found. The proof is based on polynomial interpolation of power functions and properties of certain theta-like functions. Pointwise asymptotics of $B_N^*$ and $B_N$ are discussed as well.