The following list of Ottoman imperial titles appears in the 1574 Münşeâtü's-Selâtîn of Feridun Ahmed Bey, a collection of correspondence from Ottoman sultans from (allegedly) the earliest days of the Ottoman beylik to Feridun Bey's own time:

“His majesty [the second] Iskandar, the banner of the Possessor of knowledge, the Saturn of sublimity of the sphere, [the one who belongs to] the rank of Khusrau, the palace of supremacy and power of Nushirwan, the throne of justice, the rule of the Caesar, Jamshid of the realm (lit. – place), Khurshid of the world, the khaqan, the one who distributes the crowns, the possessor of the countries, the sultan, the one who sits on the throne, the ṣāḥibqirān (lit. - the one who is under the lucky combination of stars), the padishah of the kingdom, the successful one”

The title you're probably asking about, and which is most often pointed to as evidence for the Ottoman claims to Byzantine succession, is that of Caesar—in Ottoman Turkish, qayser. While this certainly represents a broad claim to Roman-Byzantine imperial legitimacy, it is perhaps somewhat misleading as a sign of continuity with the previous inhabitants of Constantinople: Ottoman scholars and historians usually used the term to refer to the Ottoman sultan, only rarely using it for the Byzantine emperor! (One scholar has suggested that this is because qayser is a generally positive imperial title, and the Ottomans had a vested interest in portraying the late Palaiologos as a decadent, decaying shadow of the empire's former glories.) Moreover, it's a title used almost (if not entirely) exclusively in Arabic, Persian, and Turkish texts—neither the Ottomans nor their Greek subjects referred to the sultan as καῖσαρ when writing in Greek.

So what titles were preferred in Greek? In official documents addressed to European courts, Mehmed II refers to himself as (μέγας) αύθέντης and (μέγας) ἀμηρᾶς: ("great") "master" and "emir," respectively. Neither of these terms is particularly Byzantine, with amiras/emir being a notably Arab/Islamic title. While Mehmed was willing to don the Constantinian purple for Eastern audiences, he seems to have been somewhat cagier when it came to Europeans and his Greek subjects. The same, however, cannot be said of the Greek-language scholars and panegyrists who attached themselves to his court.

Men like Michael Kritoboulos, George Amiroutzes, and George of Trebizond addressed their new Ottoman ruler with the title of βασιλε´υς—basileus. This is a title with strong Byzantine connotations, though historically it had also been used to refer to figures like the Sasanian emperor. (In fact, at one point George of Trebizond refers to Mehmed as βασιλεύσ βασιλέων—the "king of kings," a title reminiscent of the Persian shahanshah as well as the titulature of Jesus in Revelation. As George wrote a short treatise on Mehmed as a divine ruler, he probably had both senses in mind.) Patriarchal documents from the 1470s refer to the sultan as basileus as well. It should, however, be noted that the patriarchy of Constantinople was by this point entirely subordinate to the sultan, and that the writers enjoyed or sought court patronage—none of these examples can really be said to be from a neutral viewpoint, much less that of the "average Greek."

[A partial exception to this trend is the historian Laonikos Chalkokondyles, who uses the term basileus for both the Byzantine and Ottoman emperors in his Demonstrations of Histories (c. 1464). But Chalkokondyles rather self-consciously distanced himself from Byzantine imperial ideologies in his writing, so it's difficult to say whether or not he was using the title to mark continuity.]

Greek courtiers seeking to legitimate Ottoman rule after the Byzantine pattern did not, of course, limit themselves to maintaining imperial titles. Perhaps the most striking demonstration of the continuity of Byzantine-Ottoman rule may be found in the late sixteenth-century Chronicle of Pseudo-Sphrantzes. This text—which may draw from Italian predecessors—describes the Ottoman dynasty's descent from John Komnenos, the brother of the emperor Andronikos I (r. 1183-1185)! Thus the Ottomans are presented as carrying on the Byzantine legacy not only in government, but in blood as well. Obviously, this example is somewhat exceptional, but I think it makes it rather clear the lengths to which Greek admirers of the Ottomans were willing to go in making the case for imperial continuity.

READING

See: Konstantinos Moustakas, "Byzantine 'Visions' of Ottoman Empire: Theories of Ottoman Legitimacy by Byzantine Scholars after the Fall of Constantinople" [academia.edu]; Moustakas, "The Myth of the Byzantine Origins of the Osmanlis. An Essay in Interpretation" [Byzantine and Modern Greek Studies—requires institutional access]; Hasan Çolak, "Tekfur, fasiliyus, and kayser: Disdain, Negligence, and Appropriation of Byzantine Imperial Titulature in the Ottoman World" [academia.edu]