Continuously differentiable interpolation

Question

I have real values $y_i$ given on uniform grid. I want to build interpolating function $f(x)$ such that:

1. $f(x) = y_i$, when $x=i$,
2. $f$ is continuously differentiable.

Instead of using famous cubic-splines, I would like to find a technique meeting the following requirement: $f$ is linear in $y_i$ values and depends only on $y_j$ where $j \le x$, i.e. has form $$f(x) = \sum_{j=1}^{\lfloor x\rfloor} a_j\left(x\right) \cdot y_j$$

Answer 1

You can certainly try Lagrange interpolation polynomial. It's polynomial therefore continuously differentiable.