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1 year ago in Approximation Theory , Mathematical Analysis By Buchiramulu Batta

Is the following version of the Weierstrass approximation theorem valid if convergence is pointwise instead of uniform?

This isn't just a pedantic technicality. It gets to the heart of what we mean by "approximation" in analysis and its utility. Uniform convergence preserves continuity and enables term-wise integration, which is vital for applications in numerical methods and PDEs. I'm trying to clarify for my students and for my own research on function spaces whether pointwise approximation is a trivial corollary or if it introduces new, fundamental constraints.

WeierstrassTheorem Pointwise Convergence

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By Prajwal Sharma Answered 1 year ago

Having lectured on this exact nuance, I can clarify that the answer is yes, but it's a significantly weaker and often less useful result. The classical Weierstrass theorem, via Bernstein polynomials or the Stone-Weierstrass theorem, inherently provides uniform convergence. If you only demand pointwise approximation, you can construct sequences of polynomials that converge pointwise to any given continuous function (and indeed, to any arbitrary function, by a separate diagonal argument). However, I've seen students misunderstand the practical loss: you forfeit the guarantee that the approximating polynomials themselves are well-behaved or that you can integrate them to approximate the integral, which is where the theorem's real power lies.

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Is the following version of the Weierstrass approximation theorem valid if convergence is pointwise instead of uniform?

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