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 $\parallel f-{\widehat{f}}_N\parallel ,|f\left(x\right){-}_N\left(x\right)|$, $x\in [a,b]$,...

Using characteristic functions of polyhedra, we construct radial p-multipliers which are continuous over ${ℝ}^n$ but not continuously differentiable through $S^{n-1}$ and give a p-multiplier criterion for homogeneous functions over ${ℝ}^2$. We also exhibit fractal p-multipliers over the real line.

We determine the convolution operators $T_{\mu }:=\mu *$ on the real analytic functions in one variable which admit a continuous linear right inverse. The characterization is given by means of a slowly decreasing condition of Ehrenpreis type and a restriction of hyperbolic type on the location of zeros of the Fourier transform μ̂(z).

On montre que si $T$ est une contraction à spectre dénombrable et telle que, pour tout $ϵ\>0$$$\parallel T^{-1}...$$

Generalized conjugate partial sums of Fourier series are used to find jumps of functions. The rate of convergence is studied and sharp results are obtained.

We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as $T_{\epsilon }\left(x\right):={\sum }_{j\in D_{\epsilon }\left(x\right)}e{*}_j\left(x\right)e_j$, where $D_{\epsilon }\left(x\right):=j:|e{*}_j\left(x\right)|\ge \epsilon$. We study a generalized version of $T_{\epsilon }$ that we call the weak thresholding approximation. We modify the $T_{\epsilon }\left(x\right)$ in the following way. For ε > 0, t ∈ (0,1) we set $D_{t,\epsilon }\left(x\right):=j:t\epsilon \le |e{*}_j\left(x\right)|<\epsilon$ and consider the weak...