For some reason I wanted to construct polynomials approximating this piecewise constant function :

Of course approximation cannot be uniform, since the function is not continuous. But it can be achieved in the sense of convergence of graphs in the Hausdorff metric: their limit should be the “graph” shown above, with the vertical line included. In concrete terms, this means for every there is
such that for
the polynomial
satisfies
and also
How to get such explicitly? I started with the functions
when
is large. The idea is that as
, the limit of
is what is wanted:
when
,
when
. Also, for each
there is a Taylor polynomial
that approximates
uniformly on
. Since the Taylor series is alternating, it is not hard to find suitable
. Let’s shoot for
in the Taylor remainder and see where this leads:
- Degree
polynomial for
- Degree
polynomial for
- Degree
polynomial for
- Degree
polynomial for
- Degree
polynomial for
The results are unimpressive, though:

To get within of the desired square-ness, we need
. This means
. Then, to have the Taylor remainder bounded by
at
, we need
. Instead of messing with Stirling’s formula, just observe that
does not even begin to decrease until
exceeds
, which is more than
. That’s a … high degree polynomial. I would not try to ask a computer algebra system to plot it.
Bernstein polynomials turn out to work better. On the interval they are given by
To avoid dealing with , it is better to use odd degrees. For comparison, I used the same or smaller degrees as above:
.

Looks good. But I don’t know of a way to estimate the degree of Bernstein polynomial required to obtain Hausdorff distance less than a given (say,
) from the square function.