Interpolation with an FIR filter
Let's view this from a slightly different angle:


If the signal is properly sampled, i.e. in accordance with the Shannon/Nyquist criterion, then the samples contain all information about the original signal. If not, all bets are off, so we'll skip this for now.
Interpolation is then equivalent to sampling the signal at a non-integer time. In your case you want to know what is x(t=2.1) by using the information x[1], x[2], x[3] ...
There is only one "correct" answer to "what is x(t=2.1). In your example it happens to be 1.05061339.... It can be calculated using the Whittaker Shannon interpolation (see Wikipedia) as $$x(t)=\sum_{n=-\infty }^{\infty }x[n]\cdot \mbox{sinc}(\frac{t-nT}{T})$$
This can be interpreted as the convolution with an impulse response. However, it's not "finite".
Unfortunately The Whitaker Shannon interpolation is not very practical: The impulse response is infinite, the number of zeros is infinite and it's falloff with time is slow, so it's not a great candidate for windowing to get a time limited impulse response.
The Whittaker Shannon interpolation is equivalent to convolution with the impulse response of an ideal low pass filter. All practical interpolation methods will also involve  a low pass filter. Even linear interpolation can be interpreted as a low pass filter, it's just a very bad one.
Interpolation filters are made "practical" by adjusting filter parameters to the requirements of the application: how much precision do you need, what's the spectral content of the signal, what latency can you tolerate, etc.
Filter parameters to adjust are filter order, cutoff frequency, rolloff, filter design method (equiripple, least squares, etc), linear or minimum phase, side lobe height, stop band attenuation, pass band ripple, etc.