Let $A$ be a finite subset of an abelian group $G$ and let $P$ be a closed half-plane of the complex plane, containing zero. We show that (unless $A$ possesses a special, explicitly indicated structure) there exists a non-trivial Fourier coefficient of the indicator function of $A$ which belongs to $P$. In other words, there exists a non-trivial character $\chi \in \widehat{G}$ such that ${\sum }_{a\in A}\chi \left(a\right)\in P$.

We study the Complex Unconditional Metric Approximation Property for translation invariant spaces $C_{\Lambda }\left(\right)$ of continuous functions on the circle group. We show that although some “tiny” (Sidon) sets do not have this property, there are “big” sets Λ for which $C_{\Lambda }\left(\right)$ has (ℂ-UMAP); though these sets are such that $L_{\Lambda }^{\infty }\left(\right)$ contains functions which are not continuous, we show that there is a linear invariant lifting from these $L_{\Lambda }^{\infty }\left(\right)$ spaces into the Baire class 1 functions.

The problem of nonparametric regression function estimation is considered using the complete orthonormal system of trigonometric functions or Legendre polynomials $e_k$, k=0,1,..., for the observation model $y_i=f\left(x_i\right)+{\eta }_i$, i=1,...,n, where the ${\eta }_i$ are independent random variables with zero mean value and finite variance, and the observation points $x_i\in [a,b]$, i=1,...,n, form a random sample from a distribution with density $\varrho \in L^1[a,b]$. Sufficient and necessary conditions are obtained for consistency in the sense of the errors $\parallel f-{\widehat{f}}_N\parallel ,|f\left(x\right){-}_N\left(x\right)|$, $x\in [a,b]$,...