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.