MATH 247B : Fourier analysis


  1. First homework (due Monday, Jan 29): Notes 5, Q5, 6.  Errata: in Q6, the implied constant should depend on delta (or equivalently, on s).  In Q5, there are several approaches to solve the problem.  One is to do the integer s case first and then interpolate.  The other is to exploit the fractional integral formulation of |nabla|^{-s}.  A third is to use Littlewood-Paley theory.  In the latter case, you may find my other notes on Littlewood-Paley theory (see here and also the appendix to this book) to be useful; see also Stein’s “Singular integrals” for more on Sobolev spaces.  You may also find various PDE texts (e.g. Taylor) to be useful.
  2. Second homework (due Monday, Feb 12): Notes 6, Q2, 3
  3. Third homework (due Monday, Feb 26): Notes 6, Q7; Notes 7, Q1.  (Hint: for Notes 7, Q1, use the T(1) theorem.)  Errata: in Q7, 2^{-j \alpha} should be 2^{j\alpha}.
  4. Fourth homework (due Monday, Mar 12): Notes 8, Q2, Q3.  Errata: in Q2, the disk example given only works when d=2; for d>2 one should use balls instead.