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On Eigenfunction Expansions of the Hermite Differential Operator on {\open R}^{n}

The eigenfunction expansions play a great role in many branches. Important applications occur in Physics, Quantum Mechanics, Approximation Theory, etc. The classical eigenfunctions which correspond to some differential operators have beautiful properties in Hilbert space L_{2}(\Omega) . One of these properties is that each function f \in L_{2}(\Omega) may be expanded in a Fourier series (with respect to these eigenfunctions) converging to f in the L_{2} -norm. Similar expansions can sometimes be generalized in some ways to Banach spaces L_{p}(\Omega) for suitable values of p , not only for p = 2 . In this work, we deal with the Hermite functions and their corresponding differential operator. We generalize the definition of the Hermite functions from {\open R} to {\open R}^{n} for n \geq 2 and define an appropriate differential operator whose eigenfunctions are the (generalized) Hermite functions. Then, similar ideas to [3] and [12] as well as [4] are used to introduce a space of test functions {\cal H} and its dual space {\cal H}^{\prime} which are comparable with the well-known spaces {\cal S} and {\cal S}^{\prime} , respectively. The elements treated in those spaces are determined via the Fourier coefficients with respect to the Hermite functions mentioned before. Afterwards, the results of [6] and [8] will be used to solve some differential equations.