DEPR: resample with PeriodIndex
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Based on https://github.com/pandas-dev/pandas/issues/57033 and https://github.com/pandas-dev/pandas/issues/56588 should we consider waiting on this deprecation? I dont think #56588 is relevant, will take a closer look at #57033 In line with my comments on those issues, I think it would be great if we could slow down these deprecations.  There are several people who are interested in this issue - including me - and so I wonder if we could step back and see if there is now a path forward to trying to fix period resampling rather that removing it (and without just putting the burden on you to fix it).

I would like to start getting in a place to help by understanding more about what @jbrockmendel was referring to by the downsampler being very broken.

For example, in #53481 it is pointed out that `nunique` gives wrong answers when doing a downsampling (or even when passing the same frequency):

```
pi = pd.period_range(start="2000", periods=3, freq="M")
ser= pd.Series(["foo", "bar", "baz"], index=pi)
rs = ser.resample("M")  # should be a no-op!

>>> rs.nunique()
2000-01    foo
2000-02    bar
2000-03    baz
Freq: M, dtype: object
```

but nonetheless the actual resample operation works as I expect it to:

```
>>> rs.first()
2000-01    a
2000-02    b
2000-03    c
Freq: M, dtype: object
```

So I was thinking it would be helpful if we could start by collecting the cases where the resampler is not acting correctly, and then define what the behavior ought to be. As we do this, I can try to get up to speed with the current resampling code to help add some person-hours to fixing the problems, rather than removing the feature. > So I was thinking it would be helpful if we could start by collecting the cases where the resampler is not acting correctly, and then define what the behavior ought to be. As we do this, I can try to get up to speed with the current resampling code to help add some person-hours to fixing the problems, rather than removing the feature.

To be a little more explicit here, I use period indexes and resampling pretty frequently, and I do not recall a case where the resampling operation produced incorrect results.  As a result, I am having a hard time getting a handle on what is so broken and therefore needs to be fixed, and I think that seeing the examples of incorrect behavior will help me "dig in" to help fix the code. I'm totally fine on holding off on enforcing this deprecation until we get resolution in the linked discussions.  LMK if you have a preference over where to continue discussion.

> I think that seeing the examples of incorrect behavior

Easiest to point you toward xfailed tests test_resample_nat_index_series (reason="Don't know why this fails") and test_monthly_convention_span (reason="Commented out for more than 3 years. Should this work?")

I suspect that the problem is in PeriodResampler._downsample where we `return self.asfreq()` in several cases, discarding "how" and **kwargs that looks really sketchy.
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