2023/07/21 by Tiangang Cui, Cui, Tiangang, Friedrich Pillichshammer +1
Mathematics · #Approximation Theory and Sequence Spaces #FOS: Mathematics #Mathematical Analysis and Transform Methods #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2307.11533
openalex publication_date 2023/07/21 · openalex created_date 2023/07/25 · openalex updated_date 2026/07/28
Bernstein polynomials provide a constructive proof for the Weierstrass approximation theorem, which states that every continuous function on a closed bounded interval can be uniformly approximated by polynomials with arbitrary accuracy. Interestingly the proof of this result can be done using elementary probability theory. This way one can even get error bounds for Lipschitz functions. In this note, we present these techniques and show how the method can be extended naturally to other interesting situations. As examples, we obtain in an elementary way results for the Szász-Mirakjan operator and the Baskakov operator.