This is the 806th most frequent Danish word.
italienske
"Italienske" means "Italian" in English.
Here, "italienske" is used as an adjective describing the origin of the paintings (Italian). It agrees in form with "malerier," which is plural.
De italienske malerier på galleriet er meget smukke.
The Italian paintings in the gallery are very beautiful.
In this sentence, "italienske" modifies "retter" (dishes), indicating that the dishes are characteristic of Italy. It reflects agreement with "retter," which is plural as well.
Vi nød nogle italienske retter på restauranten i går.
We enjoyed some Italian dishes at the restaurant yesterday.
Here, "italienske" qualifies "kultur," indicating it pertains to Italy. Since "kultur" is a singular common-gender noun, "italienske" matches in gender and number.