We show that the graded algebra of elliptic modular forms and their conjugates comprises a uniformly dense subspace of the space of all continuous functions on the compactification of the fundamental domain for the action of SL2(Z) on the complex upper half-plane by fractional linear transformations.
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“Uniform Approximation of Continuous Functions on a Compact Riemann Surface by Elliptic Modular Forms,” JP Journal of Algebra, Number Theory, and Applications, 3(3), December 2003, 373-376.