A sharp discrete Sobolev inequality and left inverses of higher order differences

Abstract

For integers $\alpha ,d⩾1$, we determine the best constants in the inequalities $||(a_j{)}_{j=1}^d|{|}_{{\ell }^1}⩽C_{\alpha ,d}||(({\Delta }^{\alpha }a{)}_j{)}_{j=1}^{d+\alpha }|{|}_{{\ell }^2},$ where $\Delta$ is the difference operator. We find that these constants correspond to the ${\ell }^2$ norm of the discrete Chebyshev polynomials, and show that the extremal sequences may be obtained by appropriate sampling of the discrete Landau polynomials. From beginning to end, this project was carried out without use of AI.

18 pages