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I am interested in functions of the form: \sum_{j=1}^\infty a_j x^{p_j} where p_j can be any non-negative real number. Wikipedia has informed me that this is a subset of the signomial functions, but I have not found any references that describe their properties. For example, how general is the class of function that this includes? Are they closed under composition?

Any references would be great!

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2 Answers 2

You'll have trouble with compositions like f(g(x)), where f(x) = x^(sqrt(2)) and g(x) = -1. Sums like f(x) = \sum_{j>=1} x^(q(j)), where q(1), q(2), ... is an enumeration of the positive rationals, also seem pretty bad (what is the coefficient of x in f(x)^2?). As long as you stick to functions with positive leading term and exponents converging to +infinity, though, I think you should be able to treat them like formal power series near x=0.

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Part of the problem with this question is that even if you were only doing a finite sum, functions of this form have bad properties if the exponents aren't rational; for example they won't be defined a lot of the time. With the infinite sum you also have a lot of nasty convergence questions to consider. Can you explain what you need such functions for? Do you need to do primarily numerical computations?

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