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Finding an analytic function (f(t): ℝ⁺ ∪ {0} → ℂ) given its values at the natural numbers (f(n)=zₙ , n∈ℕ)?

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You cannot do this since sin(pi*x), constant 0 function, x^31 sin(pi*x) are three functions that have the same values at the positive integers (0 for all of them) and they are all different functions.

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