By a Markov-type 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 Markov-type 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?
By a Markov-type 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 $U_1 \subset U_2 \subset U_3,\ldots$ possessing Markov-type 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?