Let$G$denote a totally disconnected locally compact metric abelian group with translation invariant metric$d$and character group$Γ$_{$G$}. The Lipschitz spaces are defined by $$Lip\left( {\alpha ;p} \right) = \left\{ {f \in L^p \left( G \right):\left\| {\tau _a f - f} \right\|_p = O\left( {d\left( {a,0} \right)^\alpha } \right),a \to 0} \right\},$$ where$τ$_{$a$}$f$:$x$→$f$($x-a$) and α∈(0,1). For a suitable choice of metric it is shown that Lip (α;$p$)⊂$L$^{$r$}($Γ$_{$G$}), where α>1/$p$+1/$r$−1≧0 and 1≦$p$≦2. In the case$G$is compact the corresponding result holds for α>1/$r$−1/2 and$p$>2. In addition for$G$non-discrete the above result is shown to be sharp, in the sense that the range of values of α cannot be extended. The results include classical theorems of S. N. Bernstein, O. Szász and E. C. Titchmarsh.

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

The spectral order on$R$^{$n$}induces a natural partial ordering on the manifold $\mathcal{H}_{n}$ of monic hyperbolic polynomials of degree$n$. We show that all differential operators of Laguerre–Pólya type preserve the spectral order. We also establish a global monotony property for infinite families of deformations of these operators parametrized by the space ℓ^{∞}of real bounded sequences. As a consequence, we deduce that the monoid $\mathcal{A}^{\prime}$ of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre–Pólya type which are both extensive and isotonic. In particular, these results imply that any hyperbolic polynomial is the global minimum of its $\mathcal{A}^{\prime}$ -orbit and that Appell polynomials are characterized by a global minimum property with respect to the spectral order.