On discretizing integral norms of exponential sums
The author obtains new Marcinkiewicz-Zygmund type inequalities for exponential sums \[ g(\mathbf{x})=\sum_{1\leq j\leq n}a_je^{\langle \lambda_j,\mathbf{x}\rangle},\,a_j\in\mathbb{R},\,\mathbf{x}\in\mathbb{R}^d.\tag{1} \] One of the main results is Theorem 1: Let \(1\leq p<\infty,[a,b]\subset\mathbb{R}, 0<\varepsilon\leq 1, n\in\mathbb{N},\Lambda>1, d=1.\) Then there exist discrete point sets \(Y_N=\{x_1<\ldots<x_N\}\subset (a,b)\) of cardinality \[ N\leq cn\ln^{\frac{1}{p}+1}\frac{\Lambda}{\varepsilon},\tag{2} \] where \(c>0\) is an absolute constant, and positive weights \(b_1,\ldots, b_N\) so that for every exponential sum (1) with arbitrary \(\lambda_j\in\mathbb{R}\) satisfying \[ \lambda_{j+1}-\lambda_j\geq\frac{\varepsilon}{b-a}, 1\leq j\leq n-1,\quad \max_{1\leq j\leq n}|\lambda_j|\leq\Lambda\tag{3} \] we have \[ \Vert g\Vert_{L_p([a,b])}^p\sim\sum_{1\leq j\leq N}b_j|g(x_j)|^p\tag{4}. \] (In fact (4) here is given in explicit form with \(b_j=x_{j+1}-x_j\) and constants 1/2,2). This result is extended onto the multivariate exponential sums on the unit cube \(I^d=[0,1]^d\) (Theorem 2). Here conditions (2) and (3) are replaced by \[ N\leq c(d,p)n^d\ln^{\frac{d}{p}+d}\frac{\Lambda}{\delta} \] and \[ |\lambda_j-\lambda_k|\geq\delta>0,\, j\neq k, \quad \max_{1\leq j\leq n}|\lambda_j|\leq\Lambda \] respectively. Generalizations for univariate exponential sums with nonnegative coefficients in weighted norms (with weight \(w(x)=(1-x)^{\alpha}x^{\beta},0<\alpha<\beta\)) and for multivariate exponential sums with nonnegative coefficients on polytopes are given.
Bernstein and Markov type inequalities for general exponential sums
exponential sums with nonnegative coefficients
scientific article; zbMATH DE number 7442641
scientific article