Abstract:
In this short note we prove that, if (C[a,b],{A_n}) is an approximation scheme and (A_n) satisfies de La Vall\'ee-Poussin Theorem, there are instances of continuous functions on [a,b], real analytic on (a,b], which are poorly approximable by the elements of the approximation scheme (A_n). This illustrates the thesis that the smoothness conditions guaranteeing that a function is well approximable must be, at least in these cases, global. The failure of smoothness at endpoints may result in an arbitrarily slow rate of approximation. A result of this kind, which is highly nonconstructive, based on different arguments, and applicable to different approximation schemes, was recently proved by Almira and Oikhberg (see arXiv:1009.5535v2).

Abstract:
Let $X$ be a Banach space and suppose $Y\subseteq X$ is a Banach space compactly embedded into $X$, and $(a_k)$ is a weakly null sequence of functionals in $X^*$. Then there exists a sequence $\{\varepsilon_n\} \searrow 0$ such that $|a_n(y)| \leq \varepsilon_n \|y\|_Y$ for every $n\in\mathbb{N}$ and every $y\in Y$. We prove this result and we use it for the study of fast decay of Fourier coefficients in $L^p(\mathbb{T})$ and frame coefficients in the Hilbert setting.

Abstract:
In this paper we show a lethargy result in the non-Arquimedian context, for general ultrametric approximation schemes and, as a consequence, we prove the existence of p-adic transcendental numbers whose best approximation errors by algebraic p-adic numbers of degree less than or equal to n decays slowly.

Abstract:
We study the finite dimensional spaces $V$ which are invariant under the action of the finite differences operator $\Delta_h^m$. Concretely, we prove that if $V$ is such an space, there exists a finite dimensional translation invariant space $W$ such that $V\subseteq W$. In particular, all elements of $V$ are exponential polynomials. Furthermore, $V$ admits a decomposition $V=P\oplus E$ with $P$ a space of polynomials and $E$ a translation invariant space. As a consequence of this study, we prove a generalization of a famous result by P. Montel which states that, if $f:\mathbb{R}\to \mathbb{C}$ is a continuous function satisfying $\Delta_{h_1}^mf(t) = \Delta_{h_2}^mf(t)=0$ for all $t\in\mathbb{R}$ and certain $h_1,h_2\in\mathbb{R}\setminus\{0\}$ such that $h_1/h_2\not\in\mathbb{Q}$, then $f(t)=a_0+a_1t+\cdots+a_{m-1}t^{m-1}$ for all $t\in\mathbb{R}$ and certain complex numbers $a_0,a_1,\cdots,a_{m-1}$. We demonstrate, with quite different arguments, the same result not only for ordinary functions $f(t)$ but also for complex valued distributions. Finally, we also consider in this paper the subspaces $V$ which are $\Delta_{h_1h_2\cdots h_m}$-invariant for all $h_1,\cdots,h_m\in\mathbb{R}$.

Abstract:
In this paper we characterize the approximation schemes that satisfy Shapiro's theorem and we use this result for several classical approximation processes. In particular, we study approximation of operators by finite rank operators and n-term approximation for several dictionaries and norms. Moreover, we compare our main theorem with a classical result by Yu. Brundyi and we show two examples of approximation schemes that do not satisfy Shapiro's theorem.

Abstract:
Given X,Y two Q-vector spaces, and f:X -> Y, we study under which conditions on the sets $B_k\subseteq X$, k=1,...,s, if $\Delta_{h_1h_2... h_s}f(x)=0$ for all x in X and h_k in B_k, k=1,2,...,s, then $\Delta_{h_1h_2... h_s}f(x)=0$ for all (x,h_1,...,h_s) in X^{s+1}.

Abstract:
In a previous paper (see arXiv:1003.3411 [math.CA]), we investigated the existence of an element x of a quasi-Banach space X whose errors of best approximation by a given approximation scheme (A_n) (defined by E(x,A_n) = \inf_{a \in A_n} \|x - a_n\|) decay arbitrarily slowly. In this work, we consider the question of whether x witnessing the slowness rate of approximation can be selected in a prescribed subspace of X. In many particular cases, the answer turns out to be positive.