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Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
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Construct smooth functions with prescribed derivatives
To be specific, suppose we are given a sequence of smooth functions $\{f_k\}_{k\geq 0}$ on flat torus $\mathbf{T}^2$(you may think of it as doubly periodic functions on $\mathbf{R}^2$ and smooth).
Qu …