Gauss's Interpolation Formula

$$f(x)\approx t_n(x)=\sum_{k=0}^{2n} f_k\zeta_k(x),$$

where $t_n(x)$ is a trigonometric Polynomial of degree $n$ such that $t_n(x_k)=f_k$ for $k=0$, ..., $2n$, and

$$\zeta_k(x)={\sin[{\textstyle{1\over 2}}(x-x_0)]\cdots\sin[{\... ...(x_k-x_{k+1})]\cdots\sin[{\textstyle{1\over 2}}(x_k-x_{2n})]}.$$

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 881, 1972.

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 442-443, 1987.