Los conceptos de lineabilidad, espaciabilidad y algebrabilidad son novedosos y estan rapidamente atrayendo la atencion de diferentes grupos de investigacion en matematicas en Europa y en los EEUU, El tema de la hiperciclicidad lleva... more
Los conceptos de lineabilidad, espaciabilidad y algebrabilidad son novedosos y estan rapidamente atrayendo la atencion de diferentes grupos de investigacion en matematicas en Europa y en los EEUU, El tema de la hiperciclicidad lleva siendo estudiado en profundidad desde los ultimos quince anos. La tesis se compone de cuatro capitulos. El principal objetivo de los capitulos 1 y 2 es el hecho de que algunos fenomenos que, aparentemente, son poco comunes, ocurren mas de los que podriamos pensar, en el sentido de que existen espacios vectoriales de dimension infinita de funciones (sucesiones) verificando propiedades que podriamos llamar patologicas, tales como ser sobreyectivo en todas partes o no ser monotono en ningun punto y diferenciable en todo punto. Por lo que algunas de estas propiedades ocurren de forma lineal, en forma de espacios vectoriales, espacios vectoriales cerrados, espacios de Banach, o algebras infinitamente generadas, lo cual no deja de ser muy sorprendente. Los capitulos 3 y 4 tratan sobre teoria de aproximacion, en particular sobre hiperciclicidad y el caos en espacios de Banach. Lo innovador del tema es la forma de abordar algunos problemas clasicos dentro de esta toria. Las conclusiones obtenidas en los capitulos 3 y 4 dejan claro que es posible construir subespacios vectoriales hiperciclicos (y densos) sin necesidad de hacer uso alguno del teorema de la categoria de Baire o tecnicas de existencia, que es la herramiento clasica y mas comun siempre utilizada. Algunos elementos pueden construirse explicitamente, como se demuestra en dichos capitulos. Las tecnicas desarrolladas representan un punto de vista completamente nuevo, y una nueva forma de abordar este tipo de problema del area de hiperciclicidad y caos, materia estudiada en profundidad en los ultimos quince anos. El capitulo 4 tambien trat sobre formas de construir vectores ciclicos e hiperciclicos para determinadas familias de operadores en espacios de Banach separables. Condiciones necesarias de caos son dadas, y utilizadas mas adelante para probar resultados relativos a la caoticidad de ciertos operadores en espacios de Banach. En resumen la tesis aporta nuevas tecnicas para abordar problemas clasicos de analisis real y complejo, analisis funcional y teoria de operadores, asi como nuevos resultados en la materia estudiada, da una nueva perspectivas constructiva a resultados ya conocidos.
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A mapping phi : [-1, 1] -> [0, infinity) is a curved majorant for a polynomial p in one real variable if vertical bar p(x)vertical bar <= phi(x) for all x is an element of [-1, 1]. If P(n)(phi)(R) is the set of all one real variable... more
A mapping phi : [-1, 1] -> [0, infinity) is a curved majorant for a polynomial p in one real variable if vertical bar p(x)vertical bar <= phi(x) for all x is an element of [-1, 1]. If P(n)(phi)(R) is the set of all one real variable polynomials of degree at most n having the curved majorant phi, then we study the problem of determining, explicitly, the best possible constant M(n)(phi)(x) in the inequality vertical bar p'(x)vertical bar <= M(n)(phi)(x)parallel to p parallel to, for each fixed x is an element of [-1, 1], where p is an element of p(n)(phi)(R) and parallel to p parallel to is the sup norm of p over the interval [-1, 1]. These types of estimates are known as Bernstein type inequalities for polynomials with a curved majorant. The cases treated in this manuscript, namely phi(x) = root 1 - x(2) or phi(x) = vertical bar x vertical bar for an x is an element of [-1, 1] (circular and linear majorant respectively), were first studied by Rahman in [10]. In that reference the author provided, for each n is an element of N, the maximum of M(n)(phi)(x) over [-1, 1] as well as an upper bound for M(n)(phi)(x) for each x is an element of [-1, 1], where phi is either a circular or a linear majorant. Here we provide sharp Bernstein inequalities for some specific families of polynomials having a linear or circular majorant by means of classical convex analysis techniques (in particular we use the Krein-Milman approach).
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1 Departamento de Matemáticas, Universidad de Cádiz, Avda. de la Universidad s/n. 11405, Jerez de la Frontera, Cádiz, Spain 2 Departamento de Matemáticas, Universidad de Cádiz, Sacramento, 82, 11002 Cádiz, Spain 3 Departamento de... more
1 Departamento de Matemáticas, Universidad de Cádiz, Avda. de la Universidad s/n. 11405, Jerez de la Frontera, Cádiz, Spain 2 Departamento de Matemáticas, Universidad de Cádiz, Sacramento, 82, 11002 Cádiz, Spain 3 Departamento de Análisis Matemático, Facultad ...
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We find large algebraic structures inside the following sets of pathological functions: (i) perfectly everywhere surjective functions, (ii) differentiable functions with almost nowhere continuous derivatives, (iii) differentiable nowhere... more
We find large algebraic structures inside the following sets of pathological functions: (i) perfectly everywhere surjective functions, (ii) differentiable functions with almost nowhere continuous derivatives, (iii) differentiable nowhere monotone functions, and (iv) Sierpinski-Zygmund functions. The conclusions obtained on (i) and (iii) are improvements of some already known results
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If $\Delta$ stands for the region enclosed by the triangle in ${\mathsf R}^2$ of vertices $(0,0)$, $(0,1)$ and $(1,0)$ (or simplex for short), we consider the space ${\mathcal P}(^2\Delta)$ of the 2-homogeneous polynomials on ${\mathsf... more
If $\Delta$ stands for the region enclosed by the triangle in ${\mathsf R}^2$ of vertices $(0,0)$, $(0,1)$ and $(1,0)$ (or simplex for short), we consider the space ${\mathcal P}(^2\Delta)$ of the 2-homogeneous polynomials on ${\mathsf R}^2$ endowed with the norm given by $\|ax^2+bxy+cy^2\|_\Delta:=\sup\{|ax^2+bxy+cy^2|:(x,y)\in\Delta\}$ for every $a,b,c\in{\mathsf R}$. We investigate some geometrical properties of this norm. We provide an explicit formula for $\|\cdot\|_\Delta$, a full description of the extreme points of the corresponding unit ball and a parametrization and a plot of its unit sphere. Using this geometrical information we also find sharp Bernstein and Markov inequalities for ${\mathcal P}(^2\Delta)$ and show that a classical inequality of Martin does not remain true for homogeneous polynomials on non symmetric convex bodies.
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K. Ball has proved the “complex plank problem”: if ( x k ) k = 1 n \left (x_{k}\right )_{k=1}^{n} is a sequence of norm 1 1 vectors in a complex Hilbert space ( H , ⟨ ⋅ , ⋅ ⟩ ) \left (H, \, \langle \cdot ,\cdot \rangle \right ) , then... more
K. Ball has proved the “complex plank problem”: if ( x k ) k = 1 n \left (x_{k}\right )_{k=1}^{n} is a sequence of norm 1 1 vectors in a complex Hilbert space ( H , ⟨ ⋅ , ⋅ ⟩ ) \left (H, \, \langle \cdot ,\cdot \rangle \right ) , then there exists a unit vector x x for which \[ | ⟨ x , x k ⟩ | ≥ 1 / n , k = 1 , … , n . \left |\langle {x}, x_{k}\rangle \right |\geq 1/\sqrt {n}\,,\quad k=1, \ldots , n\,. \] In general, this result is not true on real Hilbert spaces. However, in special cases we prove that the same result holds true. In general, for some unit vector x x we have derived the estimate \[ | ⟨ x , x k ⟩ | ≥ max { λ 1 / n , 1 / λ n n } , \left |\langle {x}, x_{k}\rangle \right | \geq \max \left \{\sqrt {\lambda _{1}/n},\, 1/\sqrt {\lambda _{n}n}\right \}\,, \] where λ 1 \lambda _{1} is the smallest and λ n \lambda _{n} is the largest eigenvalue of the Hermitian matrix A = [ ⟨ x j , x k ⟩ ] A=\left [\langle {x_{j}}, x_{k}\rangle \right ] , j , k = 1 , … , n j, k=1, \ldots , n...
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Let (Omega, Sigma, mu) be a measure space and 1 < p < infinity. We show that, under quite general conditions, the set L-p(Omega) - boolean OR(1 <= q<p) L-q(Omega) is maximal spaceable, that is, it contains (except for the null... more
Let (Omega, Sigma, mu) be a measure space and 1 < p < infinity. We show that, under quite general conditions, the set L-p(Omega) - boolean OR(1 <= q<p) L-q(Omega) is maximal spaceable, that is, it contains (except for the null vector) a closed subspace F of L-p(Omega) such that dim (F) = dim (L-p (Omega)) This result is so general that we had to develop a hybridization technique for measure spaces in order to construct a space such that the set L-p(Omega) - L-q (Omega),1 <= q < p, fails to be maximal spaceable. In proving these results we have computed the dimension of L-p(Omega) for arbitrary measure spaces (Omega, Sigma, mu). The aim of the results presented here is, among others, to generalize all the previous work (since the 1960's) related to the linear structure of the sets L-p(Omega) - L-q(Omega) with q < p and L-p(Omega) - boolean OR(1 <= q<p) L-q(Omega).