Suppose I have discrete noisy signal $X = (0.096, -0.0632, 0.351, 0.531, 0.360, 0.006, -0.320)$ sampled at discrete time points $T = (1, 2, 3, 4, 5, 6, 7)$. Filtering (zero-padded) $X$ with symmetrical FIR $G = (\frac{1}{4}, \frac{1}{2}, \frac{1}{4})$ gives me a good approximation of underlying $f(t)$ but only at discrete time points $T$. Is it possible to use filter coefficients to approximate the values of $f(t)$ between discrete values, e.g. estimate $f(\sqrt 2)$?
In Interpolation with an FIR filter user Hilmar mentions Whittaker–Shannon interpolation formula, but it is sensitive to noise. Is it possible to construct smooth but more noise-robust interpolation by using FIR coefficients somehow?
Or, separate interpolation step (e.g. spline or polynomial interpolation) after FIR application is the only practical way to go?