Is there a name for this smoothing formula?
1) I don't have any intuition for this smoothing formula.  This seems like some kind of windowing process that uses weights that are smaller for further away values and larger for closer values ? Is there a name for it ?


The first misconception is that this is not a smoothing formula.  It's a way of representing an already smooth continuous-time signal in discrete samples.

All it's doing is as @robert says in the comments: it's assuming that the value of the signal between $\tau_i$ and $\tau_{i+1}$ is just $Y(\tau_i)$ --- it's piecewise constant.

I believe that $p(t)$ is just a "pulse".


  2) The second equality in the smoothing relation implies that $Y(\tau_{i}) \times \sqrt  \triangle = Y_{i} $. I don't follow that ?


This is just allowing us to simplify $Y(\tau_{i}) \times \sqrt  \triangle$ and allow us to talk about $Y_i$ as shorthand notation.

edit from r b-j: it's just an alternative shorthand notation for $Y(\tau)\bigg|_{\tau = \tau_i}$ if $Y_i$ was anything else (like coefficients from some other set), i can't say that it's a ZOH with a DC gain of 0 dB.  and that "$\sqrt{\triangle}$" is chaff.


  3) It looks like the weights are going to be negative and positive depending on the value of $\tau$. Is this correct ?


The "weights" are either $0$ or $\frac{1}{\sqrt{\triangle}}$. For any value of $\tau$ outside $0 \le \tau - i\triangle \le \triangle$ the "weight" (pulse) is zero.


  4) Like I said, I can send the 3 pages ( what I wrote is a part of a longer derivation of a functional relation for least squares ) to anyone who is interested.  Dr. Kalaith makes this approximation sound trivial but it's not to me..


Please give me feedback on the above answers.