We study the following nonlinear method of approximation by trigonometric polynomials. For a periodic function f we take as an approximant a trigonometric polynomial of the form $Gₘ\left(f\right):={\sum }_{k\in \Lambda }f̂\left(k\right)e^{i(k,x)}$, where $\Lambda \subset {\mathbb{Z}}^d$ is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients f̂(k) of the function f. Note that Gₘ(f) gives the best m-term approximant in the L₂-norm, and therefore, for each f ∈ L₂, ||f-Gₘ(f)||₂ → 0 as m → ∞. It is known from previous results that in the case of...

For a sequence $\left(A_j\right)$ of mutually orthogonal projections in a Banach space, we discuss all possible limits of the sums $S_n={\sum }_{j=1}^n{A}_j$ in a “strong” sense. Those limits turn out to be some special idempotent operators (unbounded, in general). In the case of X = L₂(Ω,μ), an arbitrary unbounded closed and densely defined operator A in X may be the μ-almost sure limit of $S_n$ (i.e. $S_{n}f\to Af$ μ-a.e. for all f ∈ (A)).

The convergence of Rothe’s method in Hölder spaces is discussed. The obtained results are based on uniform boundedness of Rothe’s approximate solutions in Hölder spaces recently achieved by the first author. The convergence and its rate are derived inside a parabolic cylinder assuming an additional compatibility conditions.

Given a complete and separable metric space $X$, we study the weak convergence of sequences of measures defined on the space $𝒮(X)$ of all real-valued lower semicontinuous functions on $X$ as well as on the space $ℱ(X)$ of all closed subsets of $X$.

Assuming the continuum hypothesis, we show that (i) there is a compact convex subset L of $\Sigma \left({ℝ}^{\omega ₁}\right)$, and a probability Radon measure on L which has no separable support; (ii) there is a Corson compact space K, and a convex weak*-compact set M of Radon probability measures on K which has no $G_{\delta }$-points.

In this paper, we generalize to sub-Riemannian Carnot groups some classical results in the theory of minimal submanifolds. Our main results are for step 2 Carnot groups. In this case, we will prove the convex hull property and some “exclosure theorems” for H-minimal hypersurfaces of class C2 satisfying a Hörmander-type condition.

Let a and b be fixed real numbers such that 0 < mina,b < 1 < a + b. We prove that every function f:(0,∞) → ℝ satisfying f(as + bt) ≤ af(s) + bf(t), s,t > 0, and such that $limsu{p}_{t\to 0+}f\left(t\right)\le 0$ must be of the form f(t) = f(1)t, t > 0. This improves an earlier result in [5] where, in particular, f is assumed to be nonnegative. Some generalizations for functions defined on cones in linear spaces are given. We apply these results to give a new characterization of the $L^p$-norm.

Let X be a rearrangement-invariant space of Lebesgue-measurable functions on ${ℝ}^n$, such as the classical Lebesgue, Lorentz or Orlicz spaces. Given a nonnegative, measurable (weight) function on ${ℝ}^n$, define $X\left(w\right)=F:{ℝ}^n\to ℂ:\infty >\parallel F{\parallel }_{X\left(w\right)}:=\parallel Fw{\parallel }_X$. We investigate conditions on such a weight w that guarantee X(w) is an algebra under the convolution product F∗G defined at $x\in {ℝ}^n$ by $\left(F\ast G\right)\left(x\right)={ʃ}_{{ℝ}^n}F(x-y)G\left(y\right)dy$; more precisely, when $\parallel F\ast G{\parallel }_{X\left(w\right)}\le \parallel F{\parallel }_{X\left(w\right)}\parallel G{\parallel }_{X\left(w\right)}$ for all F,G ∈ X(w).

We generalize the classical coorbit space theory developed by Feichtinger and Gröchenig to quasi-Banach spaces. As a main result we provide atomic decompositions for coorbit spaces defined with respect to quasi-Banach spaces. These atomic decompositions are used to prove fast convergence rates of best n-term approximation schemes. We apply the abstract theory to time-frequency analysis of modulation spaces $M_m^{p,q}$, 0 < p,q ≤ ∞.