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Let $X$ be a complete, simply connected harmonic manifold with sectionalcurvatures $K$ satisfying $K \leq -1$, and let $\partial X$ denote the boundaryat infinity of $X$. Let $h > 0$ denote the mean curvature of horospheres in$X$, and let $\rho = h/2$. Fixing a basepoint $o \in X$, for $\xi \in \partialX$, let $B_{\xi}$ denote the Busemann function at $\xi$ such that $B_{\xi}(o) =0$, then for $\lambda \in \mathbb{C}$ the function $e^{(i\lambda -\rho)B_{\xi}}$ is an eigenfunction of the Laplace-Beltrami operator witheigenvalue $-(\lambda^2 + \rho^2)$.  For a function $f$ on $X$, we define the Fourier transform of $f$ by$$\tilde{f}(\lambda, \xi) := \int_X f(x) e^{(-i\lambda - \rho)B_{\xi}(x)}dvol(x)$$ for all $\lambda \in \mathbb{C}, \xi \in \partial X$ for which theintegral converges. We prove a Fourier inversion formula $$f(x) = C_0\int_{0}^{\infty} \int_{\partial X} \tilde{f}(\lambda, \xi) e^{(i\lambda -\rho)B_{\xi}(x)} d\lambda_o(\xi) |c(\lambda)|^{-2} d\lambda$$ for $f \inC^{\infty}_c(X)$, where $c$ is a certain function on $\mathbb{R} - \{0\}$,$\lambda_o$ is the visibility measure on $\partial X$ with respect to thebasepoint $o \in X$ and $C_0 > 0$ is a constant. We also prove a Planchereltheorem. This generalizes the corresponding results for rank one symmetricspaces of noncompact type and negatively curved harmonic $NA$ groups (orDamek-Ricci spaces).

The Fourier transform on negatively curved harmonic manifolds