Function with compact support whose iterated antiderivatives also have compact support
Notation: If is continuous, let us denote its indefinite integral from 0, i.e., , and iteratively .
Remark: If f is a continuous function with support contained in the open interval ]0,1[ then If has support contained in ]0,1[ iff .
Main question: Does there exist a function f with support contained in the open interval ]0,1[ such that has support contained in ]0,1[ for every , or, equivalently, for all ?
Equivalent formulation: Does there exists a sequence of functions each with support contained in the open interval ]0,1[, such that is the derivative of ?
Weaker question: Does there at least exist a continuous function f with the properties demanded in the main question?
Stronger question: Does there exist a function f with compact support, whose Fourier transform vanishes identically on a nontrivial interval?
(A positive answer to the latter would imply a positive answer to the main question: rescale the function so its support is contained in ]0,1[, multiply it appropriately so its Fourier transform vanishes in a neighborhood of 0, and observe that the Fourier transform of is, up to constants, times that of f.)
Edit: Before someone points out that the identically zero function fits the bill, I should add that I want my functions to not vanish identically.