On averaging sets

AbstractFor any set Φ={f1,f2,...,fs} ofC3-functions on the interval [−1, 1], and for any weight functionw(x) satisfyingL1≥w(x)≥L2(1−|x|)β(L1,L2>0, β≥0) and $$\int_{ - 1}^1 {w(x)dx = 1} $$ , we give a constructive proof for the existence of quadrature formulas of the type $$\frac{1}{n}\sum\limits_{j = 1}^n {f_\mu (x_j )} \int_{ - 1}^1 {f_\mu (x)w(x)dx} {\text{ }}(\mu = 1,2,{\text{ }} \ldots ,s)$$ for sufficiently largen, −1<x1<x2<...<xn<1. Assuming the orthonormality of the derivativesf′1,f′2,...,f′s with respect to the weight functionw(x), we obtain explicit bounds for the numbern of interpolation points for which such formulas exist. As an application to combinatorics, we prove the existence ofd-dimensional sphericalt-designs of sizen for eachn>cd·t12d4,cd>0 a constant.