An estimate for the coefficients of polynomials of given length
Let \(0\leq n_ 1<n_ 2<...<n_ m\) be real numbers and \(P(x)=\sum^{m}_{k=1}a_{n_ k}x^{n_ k},\) P is called a Dirichlet polynomial, \(a_ p(P)\) denotes the coefficient of \(x^ n\) in P(x), and the length of P is the number of nonzero coefficients in P. The following result is proved: If P is a Dirichlet polynomial of length m and if \(| n_ z-n_ s| \geq 1\) for \(k=1,2,...,m\), \(k\neq s\) then \[ | a_{n_ s}(P)| \leq C(n_ s+1)^{m-1/q\| P\|}L_ p(c,1) \] where \(C=C(c,m)\) depends neither on \(n_ s\) nor on P, \(0\leq c<1\), \(1/p+1/q=1\). This improves a result of the first author [Ill. J. Math. 27, 449-458 (1983; Zbl 0535.41028)].
scientific article; zbMATH DE number 4000624