A BASIS FOR THE LOCAL SOLUTIONS OF AN ELLIPTIC EQUATION
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With a view toward applications to eigenfunction expansions and spectral asymptotics for partial differential operators with continuous spectra, the authors study the problem of characterizing a solution of a given second-order elliptic linear differential equation by its behavior at a given point. When the elliptic operator is the Laplacian plus lower-order terms, and the coefficients of those terms are sufficiently smooth in the angular directions about the chosen point, the classification of harmonic functions by their local behavior (via spherical harmonics) can be carried over intact to the solutions of the more general equation, because local solutions of the two equations can be placed in one-to-one correspondence. Under this correspondence, the images of the standard basis of harmonic polynomials constitute a basis for expanding every solution that is definable in some neighborhood (no matter how small) of the point. 1982.