Pointwise Remez- and Nikolskii-type inequalities for exponential sums

Let So is the collection of all n + 1 term exponential sums with constant first term. We prove the following two theorems. Theorem 1 (Remez-type inequality for $E_n$ at 0). Let $s in left( 0, frac 12 right],.$ There are absolute constants $c_1 > 0$ and $c_2 > 0$ such that where the supremum is taken for all $f in E_n$ satisfying Theorem 2 (Nikolskii-type inequality for $E_n$ ). There are absolute constants $c_1 > 0$ and $c_2 > 0$ such that for every $a < y < b$ and $q > 0,.$ It is quite remarkable that, in the above Remez- and Nikolskii-type inequalities, behaves like , where denotes the collection of all algebraic polynomials of degree at most n with real coefficients. Received: 4 November 1998 / in final form: 2 March 1999