Let B represent the ball of radius ρ in R n and S its boundary; consider the map , where represents the mean value of f on a sphere of radius r centered at p . We summarize and discuss the results concerning the injectivity of , the characterization of the range of , and the inversion of . There is a close connection between mean values over spheres and solutions of initial value problems for the wave equation. We also summarize the results for the corresponding wave equation problem.
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Finch et al. (2007) studied this question.
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