#No, this is not possible

The Investigator feat (from [Unearthed Arcana: Feats for Skills][1]) either gives you proficiency in Investigation, or doubles your proficiency bonus for Investigation checks if you were already proficient.

However, the [Prodigy][2] feat (*Xanathar's Guide to Everything*, p. 75) specifies:

> Choose one skill in which you have proficiency. You gain expertise
> with that skill, which means your proficiency bonus is doubled for any
> ability check you make with it. **The skill you choose must be one
> that isn’t already benefiting from a feature, such as Expertise, that
> doubles your proficiency bonus.**

(Emphasis mine.)

So I'm afraid you cannot also pick Investigation for the Prodigy feat if you already have the Investigation feat.


  [1]: https://media.wizards.com/2017/dnd/downloads/UA-SkillFeats.pdf
  [2]: https://www.dndbeyond.com/characters/feats/prodigy