Bandlimited interpolation of discrete-time signals is a basic tool having extensive application in digital signal processing. In general, the problem is to correctly compute signal values at arbitrary continuous times from a set of discrete-time samples of the signal amplitude. In other words, we must be able to interpolate the signal between samples. Since the original signal is always assumed to be bandlimited to half the sampling rate, (otherwise aliasing distortion would occur upon sampling), Shannon's sampling theorem tells us the signal can be exactly and uniquely reconstructed for all time from its samples by bandlimited interpolation.
There are many methods for interpolating discrete points. For example, Lagrange interpolation is the classical technique of finding an order N polynomial which passes through N+1 given points.
The technique known as cubic splines fits a third-order polynomial through two points so as to achieve a certain slope at one of the points. (This allows for a smooth chain of third-order polynomial passing through a set of points.)
You may also have heard of Bezier splines which interpolate a set of points using smooth curves which don't necessarily pass through the points. (Bezier curves are commonly used in graphics and drawing programs, such as Adobe Illustrator.)
The above methods are suitable for graphics and other uses, but they are not ideal for digital audio. In digital audio, what matters is the audibility of interpolation error between samples. Since Shannon's sampling theorem says it is possible to restore an audio signal exactly from its samples, it makes sense that the best digital audio interpolators would be based on that theory. Such ``ideal'' interpolation is called bandlimited interpolation.
A bandlimited interpolation algorithm designed along these lines is described in the theory of operation tutorial. There is also free open-source software available in the C programming language. An excellent online tool can be used to compare various sampling-rate conversion implementations, both commercial and FOSS.