Best constants in norm inequalities for the difference operator

Abstract:

Let $\xi = {({\xi _m})_{m \in {\mathbf {Z}}}}$ be an arbitrary element of the sequence space ${l^\infty }({\mathbf {Z}})$, and let $\Delta$ be the difference operator on ${l^\infty }({\mathbf {Z}}):\Delta \xi = {({\xi _{m + 1}} - {\xi _m})_{m \in {\mathbf {Z}}}}$. The object of this investigation is the best possible value \[ C(n, k) = {\operatorname {sup}}\{ {Q_{n,k}}(\xi ):\xi \in {l^\infty }({\mathbf {Z}}), {\Delta ^n}\xi \ne 0\} \] of the quotient \[ {Q_{n,k}}(\xi ) = \frac {{\left \| {{\Delta ^k}\xi } \right \|}} {{{{\left \| \xi \right \|}^{(n - k)/n}}{{\left \| {{\Delta ^n}\xi } \right \|}^{k/n}}}}\], where $n = 2,\:3, \ldots$; $k = 1, \ldots , n - 1$. It is shown that $C(n, k)$ is at least equal to the corresponding constant $K(n, k)$, determined by Kolmogorov [Moscov. Gos. Univ. Uchen. Zap. Mat. 30 (1939), 3-13; Amer. Math. Soc. Transl. (1) 2 (1962), 233-243] for the differential operator $D$ on ${L^\infty }({\mathbf {R}})$, and exactly equal to $K(n, k)$ if $k = n - 1$. Lower bounds for $C(n, k)$ are derived that show that $C(n, k)$ is generally greater than $K(n, k)$. The values of $C(n, k)$, $k = 1, \ldots , n - 1$, are computed for $n = 2, \ldots ,5$.