Necessary and sufficient conditions for the existence of compactly supported $L^p$-solutions for the two-dimensional two-scale dilation equations are given.

The $L^1$ norm of a trigonometric polynomial with $N$ non zero coefficients of absolute value not less than 1 exceeds a fixed positive multiple of $C\left(\log N\right)/(\log \log N{)}^2.$

We consider the question of whether the trigonometric system can be equivalent to some rearrangement of the Walsh system in $L_p$ for some p ≠ 2. We show that this question is closely related to a combinatorial problem. This enables us to prove non-equivalence for a number of rearrangements. Previously this was known for the Walsh-Paley order only.

Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality $\parallel p^{\text{'}}{\parallel }_{[-1,1]}\le \frac{1}{2}{\parallel p\parallel }_{[-1,1]}$ for a constrained polynomial $p$ of degree at most $n$, initially claimed by P. Erdős, which is different from the one in the paper of T. Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval $(-1,1)$ and establish a new asymptotically sharp inequality.