By a Markovtype inequality I mean an inequality of the form $$ \ p^{(k)} \ \leq \lambda_{k,n} \ p \,\quad \forall p \in U_n, $$ for some $\lambda_{k,n} > 0$, where $U_n \subset L^\infty[1,1]$ is a subspace of dimension $n$ and $$ \ g \ = \sup_{x \in [1,1]}  g(x) . $$ If $U_n$ is the space of algebraic polynomials of degree $n$, then it is well known that $$ \lambda_{k,n} = O(n^{2k}),\quad n \rightarrow \infty, $$ for each fixed $k$, whereas for trigonometric polynomials, $$ \lambda_{k,n} = O(n^{k}),\quad n \rightarrow \infty. $$ My question is as follows. Is it possible to find subspaces $U_1 \subset U_2 \subset U_3,\ldots$ with $\bigcup^{\infty}_{n=1} U_n$ dense in $L^\infty[1,1]$ possessing Markovtype inequalities with arbitrary exponents, i.e. $$ \lambda_{k,n} = O(n^{\alpha k}),\quad n \rightarrow \infty, $$ for some given $\alpha > 0$? Moreover can one find such subspaces in a constructive manner?
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