arXiv subjects

J. M. Almira

Publications and source records attributed to J. M. Almira.

44 records · Page 3Linked to original sources

arXiv2011

Shapiro's Theorem for subspaces

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.

math.CA↗

arXiv2011

Riemannian Geometry and the Fundamental Theorem of Algebra

If a (non-constant) polynomial has no zero, then a certain Riemannian metric is constructed on the two dimensional sphere. Several geometric arguments are then shown to contradict this fact.

math.DG↗

arXiv2011

A note on classical and p-adic Fréchet functional equations with restrictions

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 $Δ_{h_1h_2... h_s}f(x)=0$ for all x in X and h_k in B_k, k=1,2,...,s, then $Δ_{h_1h_2... h_s}f(x)=0$ for all (x,h_1,...,h_s) in X^{s+1}.

math.CA↗

arXiv2011

On strict inclusion relations between approximation and interpolation spaces

We study strict inclusion relations between approximation and interpolation spaces.

math.CA↗

arXiv2010

Approximation schemes satisfying Shapiro's Theorem

An approximation scheme is a family of homogeneous subsets $(A_n)$ of a quasi-Banach space $X$, such that $A_1 \subsetneq A_2 \subsetneq ... \subsetneq X$, $A_n + A_n \subset A_{K(n)}$, and $\bar{\cup_n A_n} = X$. Continuing the line of research originating at a classical paper by S.N. Bernstein (in 1938), we give several characterizations of the approximation schemes with the property that, for every sequence $\{ε_n\}\searrow 0$, there exists $x\in X$ such that $dist(x,A_n)\neq \mathbf{O}(ε_n)$ (in this case we say that $(X,\{A_n\})$ satisfies Shapiro's Theorem). If $X$ is a Banach space, $x \in X$ as above exists if and only if, for every sequence $\{δ_n\} \searrow 0$, there exists $y \in X$ such that $dist(y,A_n) \geq δ_n$. We give numerous examples of approximation schemes satisfying Shapiro's Theorem.

math.CA↗

arXiv2010

A simple observation about compactness and fast decay of Fourier coefficients

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.

math.CA↗

arXiv2010

Characterization of approximation schemes satisfying Shapiro's Theorem

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.

math.CA↗

arXiv2007

Muntz type Theorems I

In this paper, we concentrate our attention on the Muntz problem in the univariate setting and for the uniform norm.

math.CA↗