This is an attempt of a comprehensive treatment of the results concerning estimates of the $L^1$-norms of linear means of multiple Fourier series, the Lebesgue constants. Most of them are obtained by estimating the Fourier transform of a function generating such a method. Frequently the properties of the support of this function affect distinctive features in behavior of these norms. By this geometry enters and works hand-in-hand with analysis; moreover, the results are classified mostly in accordance with their geometrical nature. Not rarely Number Theory tools are brought in. We deal only with the trigonometric case – no generalizations for other orthogonal systems are discussed nor are applications to approximation. Several open problems are posed.

Let $G$ be a finite abelian group and $E$ a subset of it. Suppose that we know for all subsets $T$ of $G$ of size up to $k$ for how many $x\in G$ the translate $x+T$ is contained in $E$. This information is collectively called the $k$-deck of $E$. One can naturally extend the domain of definition of the $k$-deck to include functions on $G$. Given the group $G$, when is the $k$-deck of a set in $G$ sufficient to determine the set up to translation? The $2$-deck is not sufficient (even when we allow for reflection of the set, which does not change the $2$-deck) and the first interesting case is $k=3$. We further restrict $G$ to be cyclic and determine the values of $n$ for which the $3$-deck of a subset of ${\mathbb{Z}}_n$ is sufficient to determine the set up to translation. This completes the work begun by Grünbaum and Moore [GM] as far as the $3$-deck is concerned. We additionally estimate from above the probability that for a random subset of ${\mathbb{Z}}_n$, there exists another subset, not a translate of the first, with the same $3$-deck. We give an exponentially small upper bound when the previously known one was $O(1/\sqrt{n})$.

Bourgain’s theorem says that under certain conditions a function $f:\{0,1{\}}^n\to \{0,1\}$ can be approximated by a function $g$ which depends only on a small number of variables. By following his proof we obtain a generalization for the case that there is a nonuniform product measure on the domain of $f$.

Given integers $s,t$, define a function ${\varphi }_{s,t}$ on the space of all formal series expansions by ${\varphi }_{s,t}(\sum a_n{x}^n)=\sum a_{sn+t}{x}^n.$ For each function ${\varphi }_{s,t}$, we determine the collection of all rational functions whose Taylor expansions at zero are fixed by ${\varphi }_{s,t}$. This collection can be described as a subspace of rational functions whose basis elements correspond to certain $s$-cyclotomic cosets associated with the pair $(s,t)$.

In this note we use the theory of theta functions to discover formulas for the number of representations of N as a sum of three squares and for the number of representations of N as a sum of three triangular numbers. We discover various new relations between these functions and short, motivated proofs of well known formulas of related combinatorial and number-theoretic interest.