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Generalizing a result of Roe [ 14 ], Strichartz proved in [ 16 ] that if a doubly infinite sequence { f   k  } of functions onsatisfies  f   k +1 = Δf   k   and | f   k  ( x )|≤ M  for all  k =0,±1,±2,… and, then  Δf  0 ( x )=− f  0 . Strichartz also showed that the result fails for hyperbolic 3-space. This negative result can indeed be extended to any Riemannian symmetric space of noncompact type. Taking this into account, we shall prove that for all Riemannian symmetric spaces of noncompact type the theorem actually holds true when uniform boundedness is modified suitably.

A Theorem of Roe and Strichartz for Riemannian Symmetric Spaces of Noncompact Type