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Proof of continuity: Spherical mean function

  1. Mar 11, 2012 #1
    This is one of my homework problems.

    If h(x) is continuous in x, show that the spherical mean:

    M[itex]_{h}[/itex](x,r) = [itex]\frac{1}{w_{n}}[/itex][itex]\int[/itex][itex]_{|\xi|=1}[/itex] h(x+r[itex]\xi[/itex]) dS[itex]_{\xi}[/itex]

    is continuous for all x and r [itex]\geq[/itex] 0.

    A lot of PDE textbooks state this fact (in regards to the wave equation in 3 dimensions) - like Evans, McOwen, etc. But none ive looked at provide a proof of continuity.

    This should be simply enough but my analysis skills lack. The strategy one would need to take is first take r to be constant, prove continuity in x. Then fix x, and prove continuity in r.

    Please help me
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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