4

I suspect this question is rather naive, but here goes. I have a set of basis functions that I suspect are eigenfunctions of an unknown differential operator (due to some results I've seen in some of my research). I have no idea what differential operator they correspond to.

Is there some general procedure one can follow to work out what the differential operator is, given the suspected eigenfunctions? If not, is there some procedure one can follow to figure out some properties of the differential operator of which I suspect my basis functions are eigenfunctions?

I'd appreciate some suggestions on where to start. Thank you.

VarunShankar
  • 331
  • 2
    Look at the zeroth order basis function first and see if you can find a differential equation that corresponds to it. Then look at the first order basis function and use the form of the differential equation you get from the zeroth order and apply it to this. You'll have some corrective term. Doing this a few times should give you a clear idea of what the general ODE looks like. – Cameron Williams Jul 22 '13 at 02:11
  • @vergere6: it would be helpful if you gave specific examples of your suspected eigenfunctions; not only would it help folks think about your question, who knows, someone might recognize the equation of which they are eigenfunctions! – Robert Lewis Jul 22 '13 at 02:41
  • @RobertLewis: It's a research question. I'm pretty sure people don't know what differential operators they correspond to. My question was worded so as to elicit answers on the procedure, not to directly get an answer as to whether they're eigenfunctions. – VarunShankar Jul 22 '13 at 03:21
  • @vergere6 Nonetheless I am sure, people here (including me) would be interested in seeing which eigenfunctions you are talking about. And you never know someone might be familiar with them. That's one thing the internet has taught me. :-) – Fixed Point Jul 22 '13 at 06:51
  • @FixedPoint: I understand :) I will be sure to tell people what these functions are if/when I manage to prove that they are, in fact, eigenfunctions of something. This might be publishable research, and I'm loath to spill the beans before I even start working on the problem. Is this the appropriate way to go about this? – VarunShankar Jul 22 '13 at 08:52
  • @vergere6 Fair enough, the question is interesting in its own right. Waiting to see what kind of answers get posted. General techniques are useful but there could be some specific technique for your specific eigenfunctions...just sayin'. – Fixed Point Jul 22 '13 at 09:03
  • @FixedPoint: I doubt it. I just went to a conference where I met a few of the approximation theorists who work in this area studying these functions, and they don't have any clue what these things are eigenfunctions of, or if they are indeed eigenfunctions of something. – VarunShankar Jul 22 '13 at 23:24
  • @CameronWilliams: I think I didn't tag you right the first time. Could you explain what you mean? Perhaps as an answer? – VarunShankar Jul 23 '13 at 05:18

0 Answers0