Sunday, 22 November 2009

mp.mathematical physics - Constraints on the Fourier transform of a constant modulus function

If g happens to be in L1, then the amplitude of the Fourier transform of fg is bounded by the L1 norm of g, for any unimodular f. This is the only restriction from above since you can always choose f so that fgge0, thus bringing the (essential) supremum of widehatfg up to |g|L1.



Another part of the question is how small we can make A. I guess "arbitrarily small", but don't have a proof. (Except for special case: if g is in L1, then we can chop it into pieces with disjoint supports and small L1 norm, and then use f to move the Fourier transforms of pieces far from one another.)

No comments:

Post a Comment