Sinc interpolation
The theoretically perfect way to compute values between the samples of a band-limited signal, and the thing every serious resampler approximates.
The sampling theorem says a signal below the Nyquist frequency can be rebuilt exactly from its samples. The recipe is the Whittaker–Shannon interpolation formula: centre a sinc function, sin(πx)/(πx), on every sample and sum them. In the time domain that is an ideal low-pass filter.
The trouble is that sinc never ends, so nobody can use it as written. Real implementations truncate it and taper it with a window, often Kaiser, and the length decides the trade-off between sharpness, CPU and pre-ringing. Libraries such as libsamplerate and SoX expose exactly that knob. Linear interpolation is cheaper but leaks aliasing like a sieve, which goes some way to explaining how old game engines sounded.