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                <forename type="first">Eusebio</forename>
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              <p>The Fourier transform is a very useful tool in many areas. This transform has been defined in many contexts, as complex numbers, quaternions, and Clifford algebras. This paper deals with a one$-$dimensional Fourier Transform defined for quaternion-valued functions. Some of its fundamental properties are proved, particularly, the inversion formula, which will be use to prove another important inversion formula for a two$-$dimensional quaternion Fourier transform. Using this Fourier transform, we also solve a partial differential equation.</p>
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