My understanding of interpolation specific to resampling applications is limited to the concept of inserting zeros, then designing a filter to minimize distortion in the passband and reject the images the zero-insert creates (to desired performance levels), such as what is depicted in a simple interpolate by 4 zero-insert shown below (showing the resulting spectrum before filtering).
I am noting that in creating the filter, we are creating the high order polynomial used for interpolating the input samples. I then understand that with a Spline interpolation method (as a user, and perhaps naively as I haven't really dug deeply into the underlying mathematics) we are doing piecewise interpolations using limited polynomials (therefore low order filters) over a smaller span of the sequence, and in many cases due to decreased ripple this will provide better performance.
My question then is if such "adaptive" resampling filters have been designed either using Spline approaches directly, or otherwise piecewise similar in process to a spline, but in a fashion that is best suited for high rate FPGA processing with a FIR based interpolation filter (I say that to avoid the response "just do a spline", but perhaps that would actually result in the solution even in an FPGA FIR based interpolator...). Before digging into that further has anyone seen such an approach, is it common, or are there known pitfalls why this would be a BAD idea? Thanks for the thoughts and input!