#No, this is not possible
The Investigator feat (from Unearthed Arcana: Feats for Skills) either gives you proficiency in Investigation, or doubles your proficiency bonus for Investigation checks if you were already proficient.
However, the Prodigy 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.
