In short, I would like pointers to closed-form formulas, or efficient algorithms for computing the inverse of ether sine, cosine, or Hartley transforms of the B-Spline basis. The motivation, as explained below, comes from approximating the frequency domain of functions of bounded frequency.
Motivation
It is well known that splines, and in particular cubic splines, have good approximation properties in bounded intervals. I would like to use splines to use them in the frequency domain to approximate real-valued functions of bounded frequency, either on the positive numbers, or the entire real-line. Since splines are meant, first and foremost, for real numbers, I would like to concentrate on real (rather than complex) domains and look at things like the sine/cosine, or Hartley transforms.
Suppose that $B_1, \dots, B_n$ is a B-Spline basis defined on some real interval of the form $[0, \tau]$ for sine/cosine transforms or $[-\tau, \tau]$ for Hartley transform. The transform of the function I would like to approximate is: $$ T[f](\omega) \approx \sum_{i=1}^n a_i B_i(\omega) $$ Thus, my function becomes $$ f(x) = \sum_{i=1}^n a_i T^{-1}[B_i](x) $$