In this work, we construct a class of Hermite-type interpolation basis functions based on the sixth-order ordinary differential equation \({S^{(6)}}(\mathrm{{t}}) - {\tau }^4{S^{(2)}}(t) = 0\). Using them, we propose a kind of \(C^2\) tension interpolation splines with a local tension parameter \(\tau _i\). For \(C^2\) interpolation, the given interpolant has \(O(h^2)\) convergence. Some applications of the \(C^2\) tension interpolation splines on the construction of interest rate term structure in Chinese financial market are given. Moreover, a kind of generalized non-uniform B-splines of the space spanned by \(\text {span}\left\{ {1,t, \ldots ,{t^{n - 4}},\sin (\tau t),\cos (\tau t),\sinh (\tau t),\cosh (\tau t)} \right\} \) is constructed.