Can a progressive marginal taxation be regressive in terms of average tax rate? Or does progressive marginal taxation imply that it is a progressive average taxation?

Here, I mean that if $T(Y)$ is a tax function of income, marginal tax rate would be $dT/dY$ and average tax rate would be $T(Y)/Y$.
$$
\frac{\partial\frac{T(Y)}{Y}}{\partial Y}= \frac{T'(Y)}{Y} - \frac{T(Y)}{Y^2}
$$ 

This can only be smaller than 0 if 

$$ T'(Y) &lt; \frac{T(Y)}{Y} $$

In other words, the marginal tax rate needs to be smaller than the average tax rate. This can't happen because you start with an average tax rate of zero and a positive marginal tax rate. From then on, in a progressive scheme, the tax due increases faster than taxable income. Progressive taxation is actually characterized by the fact that the marginal tax rate exceeds the average one. See here, p.27.