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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2004 • Question 4

Question

Math 6010 • Linear Models 2004 • Question 4

Choose and fix an integer \(n\ge 1\). For all continuous functions \(f:[0,1]\to \mathbb {R}\) define \[ B_nf(p)=\sum _{k=0}^{n}\binom {n}{k}p^k(1-p)^{n-k}f(k/n), \qquad \text {for all }p\in [0,1]. \] [ \(B_nf\) is the Bernstein polynomial of \(f\).] Prove that if \(f\) is increasing, then so is \(B_nf\).

Raw LaTeX

Choose and fix an integer \(n\ge 1\). For all continuous functions
\(f:[0,1]\to\mathbb{R}\) define
\[
  B_nf(p)=\sum_{k=0}^{n}\binom{n}{k}p^k(1-p)^{n-k}f(k/n),
  \qquad \text{for all }p\in[0,1].
\]
[ \(B_nf\) is the Bernstein polynomial of \(f\).] Prove that if \(f\) is increasing, then
so is \(B_nf\).

Topic Tags

Classical Probability and Inclusion-Exclusion Uniform Integrability and Truncation Methods