Note that BL interpolation is a special case of linear filtering. (Proof: Convolution representation above.)
Consider a filter which delays its input by samples:
after critically sampling in the time domain.
The sinc function is an infinite-impulse-response (IIR) digital filter with no recursive form non-realizable.
To obtain a finite impulse response (FIR) interpolating filter, let's formulate a least-squares filter-design problem:
Desired Interpolator Frequency Response
FIR Frequency Response, Zero-Phase Alignment
Error to Minimize
By Parseval's Theorem
Optimal Least-Squares FIR Interpolator