$L_p$-Inequalities for Differences and Derivatives of Positive Order for Functions with Spectrum in the Ball or Spherical Layer

We prove inequalities of Riesz, Bernstein, and Bohr–Favard type in the metric of the spaces $L_p$, $0<p<\infty$, for functions whose spectrum is contained in a closed ball or a closed spherical layer. As an application, a discrete description of Lizorkin–Triebel spaces in terms of coordinate differences of positive order is given.