What an excellent question! The Ottomans to a degree catalyzed the early Renaissance in Europe, and at all times, were an active partner in the trade of ideas in Europe throughout the 14th, 15th, and early 16th century, facilitated by Italian and Greek merchants sailing between population centers in Italy and the Levant.

In general, many elements of the Renaissance were 'catching up', or reaching parity, to levels of scientific and cultural advancement in the Muslim world and Ottomans were at the head of that world.

Renaissance art was heavily influenced by Greek-speaking Byzantine refugees fleeing to Venice and Italy after the 1453 sack of Constantinople. In the high Renaissance, western art would fixate on the Orient and Turks as a fascinating 'other', inspiring masterpieces including Paolo Veronese's portrait of Sultan Bayezid I.

Renaissance science began with the 'rediscovery' of Greek works on physics, astronomy, and math. Most of that was never 'lost' to the Islamic world, which continued to refine and add to Platonic understandings of the universe. Arab and Ottoman doctors had been using versions of Galenos’ works as medical textbooks, translated in to Arabic, for centuries. These works were re-translated to Greek and then Italian during the 1500s. The Ottomans would go on to introduce firearms and heavy artillery as well, revolutionizing warfare in Europe and ending the age of castles.

This exchange should not be confused for friendly relations. During most of the 15th and 16th century, the Turks were perceived as a menacing threat to the borders of Christendom. There was not much direct cultural interaction that did not involve Italian or Greek merchants, who were the bridges that linked East and West in this era. It's hard to overstate how important they were to the cultural, economic, and scientific expansion of Europe during this time. Of secondary, but still significant, importance were Jews in Spain, Portugal, and Greece, who also converted trade routes to cultural highways between civilizations.

As an interesting bookend, the Ottomans also helped close the Renaissance and open the Age of Exploration, as traders sought a sea route to India that avoided taxes levied by Ottoman and Arab middlemen.