Was the Louisiana Purchase an example of Jeffersonian democracy?
Yes, the Louisiana Purchase was an example of something that was done to further Jeffersonian democracy. The reason for this has to do with Jefferson's vision of what America should be.
Jefferson believed that America should be made up of small farmers. Such people would be self-sufficient, making all that they needed (or almost all). Jefferson thought that only self-sufficient people could be good citizens of a democracy because no one could ever tell them what to do. (If you can get your hands on Farmer Boy, by Laura Ingalls Wilder, you should look towards the end of the book where Almanzo is offered an aprenticeship with a prosperous carriage maker. His mother flips out because she wants him to be a small farmer so he will never have to rely on anyone else. It's my favorite statement of Jefferson's ideas...)
Anyway, back to the Louisiana Purchase... Jefferson thought that all that land would allow many more Americans to become small farmers. That would prevent the US from becoming some big industrial nation with lots of people working for other people (and depending on them).
So, the Louisiana Purchase was an example of Jeffersonian democracy because it was supposed to allow Americans to be self-sufficient small farmers rather than dependent wage laborers. This was Jefferson's hope.
Actually, the Louisiana Purchase was a violation of all of Jefferson's previously avowed principles. The Jeffersonian Republicans had staunchly argued for a strict interpretation of the Constitution. This was the basis of his opposition to the Bank of the United States and also his support for the Virginia and Kentucky Resolutions. Jefferson had constitutional qualms about the Louisiana Purchase, as Congress had no power under the Constitution to acquire property. He justified the purchase on the basis of his power as President to make treaties in the name of the United States. This was obviously a broad interpretation of the Constitution's language, the very thing which Jefferson had always opposed. He explained this violation of his own principles by stating:
the good sense of our country will correct the evil of loose construction when it shall produce good effects.