Let such that . Prove that .
Let be an infinitely differentiable function. Does there existe also infinitely differentiable such that for all ?
Let be continuous functions. We call the -th moment of . Prove that if and have identical -th moments for all , then .
Let be a continous and bounded function such that for all
Then is constant.
If are sufficiently differentiable, their Wronskian is defined as
Obviously, if are linearly dependent, then their Wronskian vanishes identically on , and the converse is true if the functions are analytic. Find non-analytic functions for which the converse fails.
Assume is a nonincreasing sequence of positive integers whose sum diverges and assume signs are chosen so that the modified series converges. Then
Assume now only the signed series converges, with no hypothesis on the unsigned sequence. Show
Suppose is such that for each , is eventually zero. Then is a polynomial.