A Danish form of the name Katherine, from the Greek name Aikaterine.
The etymology of the name Aikaterine is debated: it could derive from the earlier Greek name Hekaterine, which came from hekateros "each of the two"; it could derive from the name of the goddess Hecate (a goddess associated with witchcraft, tombs, demons and the underworld); it could be related to Greek aikia "torture"; or it could be from a Coptic name meaning "my consecration of your name". The Romans falsely derived it from Greek katharos "pure" and changed their spelling from Katerina to Katharina to reflect this.
Source: "Behind the Name: The Etymology and History of First Names"