Methods for bounding the range of a polynomial
Let \(p(x)= \sum_{s=0}^n a_s x^s\) be a real polynomial. Then bounds of \(p(x)\) over the interval \([0, 1]\) are determined as follows: Let \(A_j (x)\), \(j\in J\), be nonnegative basis functions with a finite index set \(J\), such that \(\sum_{j\in J} A_j (x)=1\) and \(x^s= \sum_{j\in J} \mu_{js} A_j (x)\) for \(s=0, \dots, n\) and appropriate coefficients \(\mu_{js}\). Then \(\min_{j\in J} \xi_j\) and \(\max_{j\in J} \xi_j\) are a lower and an upper bound, respectively. -- Practical applications arise at the development of polynomials by Bernstein polynomials and by \(B\)-splines.
scientific article; zbMATH DE number 804736