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This is an extension of a previous work of the first author [1], on monomial density functions, to study the approximation features over [ a, b ] of, the based on  α x   exponential density functions  h α  ( x ) when  α  &gt; 0 is discrete and/or fractional. If  α  is discrete, a random variable of  h n  ( x ) is proved to form a martingale over a reversed filtration    and is compared with a similar situation that happens to hold with respective monomial density functions. In the case of fractional  α , we advance a new stochastic operator which generates, via a nonlinear technique, unique exponential and monomial spline approximants over [ a, b ] to functions   ( x ) ∈   , a certain noncommutative inner product space.

On martingales invoked by stochastic exponential and monomial densities