More analysis questions from a high school student
What's new 2021-09-15
A few months ago I posted a question about analytic functions that I received from a bright high school student, which turned out to be studied and resolved by de Bruijn. Based on this positive resolution, I thought I might try my luck again and list three further questions that this student asked which do not seem to be trivially resolvable.
- Does there exist a smooth function which is nowhere analytic, but is such that the Taylor series converges for every ? (Of course, this series would not then converge to , but instead to some analytic function for each .) I have a vague feeling that perhaps the Baire category theorem should be able to resolve this question, but it seems to require a bit of effort.
- Is there a function which meets every polynomial to infinite order in the following sense: for every polynomial , there exists such that for all ? Such a function would be rather pathological, perhaps resembling a space-filling curve.
- Is there a power series that diverges everywhere (except at ), but which becomes pointwise convergent after dividing each of the monomials into pieces for some summing absolutely to , and then rearranging, i.e., there is some rearrangement of that is pointwise convergent for every ?
Feel free to post answers or other thoughts on these questions in the comments.