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A dual pairG andG * of smooth and generalized random variables, respectively, over the white noise probability space is studied.G is constructed by norms involving exponentials of the Ornstein-Uhlenbeck operator,G * is its dual. Sufficient criteria are proved for when a function onL(ℝ) is theL-transform of an element inG orG *.

On a dual pair of spaces of smooth and generalized random variables