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## Publications

Publications (88)

In this paper, we treat some fractional differential equations on the sequence Lebesgue spaces ℓp(N0) with p≥1. The Caputo fractional calculus extends the usual derivation. The operator, associated to the Cauchy problem, is defined by a convolution with a sequence of compact support and belongs to the Banach algebra ℓ1(Z). We treat in detail some o...

In this paper we study solutions of the quadratic equation $$AY^2-Y+I=0$$ A Y 2 - Y + I = 0 where A is the generator of a one parameter family of operator ( $$C_0$$ C 0 -semigroup or cosine functions) on a Banach space X with growth bound $$w_0 \le \frac{1}{4}$$ w 0 ≤ 1 4 . In the case of $$C_0$$ C 0 -semigroups, we show that a solution, which we c...

En este trabajo presentamos avances relacionados con el estudio de la intensa actividad epistolar de Zoel García de Galdeano con destacados matemáticos extranjeros de su época. En particular aportamos un listado y una revisión general de todas las cartas localizadas hasta el momento y abordamos un estudio más detallado del contenido de algunas de e...

We study reproducing kernel Hilbert spaces introduced as ranges of generalized Cesàro–Hardy operators, in one real variable and in one complex variable. Such spaces can be seen as formed by absolutely continuous functions on the positive half-line (or paths of infinite length) of fractional order, in the real case. A theorem of Paley–Wiener type is...

For \(\mu , \beta \in {\mathbb {R}}\), we introduce and study in detail the generalized Stieltjes operators $$\begin{aligned} {\mathcal {S}}_{\beta ,\mu } f(t):={t^{\mu -\beta }}\int _0^\infty {s^{\beta -1}\over (s+t)^{\mu }}f(s)\mathrm{d}s, \qquad t>0, \end{aligned}$$on Sobolev spaces \({{\mathcal{T}}_{p}^{{(\alpha )}}} (t^{\alpha })\) (where \(\a...

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these poly...

We give representations for solutions of time-fractional differential equations that involve operators on Lebesgue spaces of sequences defined by discrete convolutions involving kernels through the discrete Fourier transform. We consider finite difference operators of first and second orders, which are generators of uniformly continuous semigroups...

Banach algebras is a multilayered area in mathematics with many ramifications. With a diverse coverage of different schools working on the subject, this proceedings volume reflects recent achievements in areas such as Banach algebras over groups, abstract harmonic analysis, group actions, amenability, topological homology, Arens irregularity, C*-al...

We study spectral synthesis properties in convolution Sobolev algebras on the real line. Mainly, a description of primary closed ideals in such algebras is given. Then we address an approximation problem involving bounded representations of such Sobolev algebras, which arises naturally in relation with the asymptotic behavior of integrated semigrou...

We prove that the class of positive operators from L∞(μ) to L1(ν) has the Bishop-Phelps-Bollobás property for any positive measures μ and ν. The same result also holds for the pair (c0, ℓ1). We also provide an example showing that not every pair of Banach lattices satisfies the Bishop-Phelps-Bollobás property for positive operators.

For α, β, μ > 0, the following integral operators, that generalize the Poisson transform, are studied in detail on Lebesgue spaces Lp(ℝ⁺) for 1 ≤ p < ∞. If 0 < β−1/p < αμ, then these operators Pα,β,μ are bounded (and we compute their operator norms which depend on p); and commute on their range. We calculate and represent explicitly their spectra σ...

The paper is devoted to understand the large time behaviour and decay of the solution of the discrete heat equation in the one dimensional mesh Z on $\ell^p$ spaces, and its analogies with the continuous-space case. We do a deep study of the moments of the discrete gaussian kernel (which is given in terms of Bessel functions), in particular the mas...

In this chapter, we consider the Catalan numbers, C n = 1 n + 1 2 n n , and two of their generalizations, Catalan triangle numbers, B n , k and A n , k , for n , k ∈ N . They are combinatorial numbers and present interesting properties as recursive formulae, generating functions and combinatorial interpretations. We treat the moments of these Catal...

In this paper we deal with a scale of reproducing kernel Hilbert spaces H2(n), n≥0, which are linear subspaces of the classical Hilbertian Hardy space on the right-hand half-plane C+. They are obtained as ranges of the Laplace transform in extended versions of the Paley-Wiener theorem which involve absolutely continuous functions of higher degree....

For $\mu>\beta>0$, the generalized Stieltjes operators $$ \mathcal{S}_{\beta,\mu} f(t):={t^{\mu-\beta}}\int_0^\infty {s^{\beta-1}\over (s+t)^{\mu}}f(s)ds, \qquad t>0, $$ defined on Sobolev spaces $\mathcal{T}_p^{(\alpha)}(t^\alpha)$ (where $\alpha\ge 0$ is the fractional order of derivation and these spaces are embedded in $L^p(\RR^+)$ for $p\ge 1$...

In this paper we treat the following partial differential equation, the quasigeostrophic equation: ∂/∂t+u·∇f=-σ-Aαf, 0≤α≤1 , where (A,D(A)) is the infinitesimal generator of a convolution C0 -semigroup of positive kernel on Lp(Rn), with 1≤p<∞. Firstly, we give remarkable pointwise and integral inequalities involving the fractional powers (-A)α for...

Mar\'ia Andresa Casamayor, born in Zaragoza, is known as the first woman who published a scientific book in Spain. In this paper we provide answers to several of the most important questions about her unknown biography such as her birth day, the origins of her family, the houses where she and her close family lived, her job as a teacher and even he...

In this paper, we present the work of international mathematical associationism carried out by the mathematician from Navarra (Spain) Zoel Garc\'ia de Galdeano (1846-1924) during over 30 years. Garcia de Galdeano was a member of many of the most important mathematical associations of his time, thus being able to be in touch with foreign mathematici...

In this paper, we present a complete spectral research of generalized Ces\`aro operators on Sobolev-Lebesgue sequence spaces. The main idea is to subordinate such operators to suitable $C_0$-semigroups on these sequence spaces. We introduce that family of sequence spaces using the fractional finite differences and we prove some structural propertie...

In this paper we give new relations between the Weyl fractional calculus, the usual convolution product and the Laplace transform. To express them, we consider and study in detail integrated exponential functions, a particular class of Kummer functions. We also extend a equality due to S. Goldstein in the last section.

We develop an operator-theoretical method for the analysis on well posedness of partial differential equations that can be modeled in the form \begin{equation*} \left\{ \begin{array}{rll} \Delta^{\alpha} u(n) &= Au(n+2) + f(n,u(n)), \quad n \in \mathbb{N}_0, \,\, 1< \alpha \leq 2; u(0) &= u_0; u(1) &= u_1, \end{array} \right. \end{equation*} where...

The existence of distributional chaos and distributional irregular vectors has been recently considered in the study of linear dynamics of operators and C 0-semigroups. In this paper we extend some previous results on both notions to sequences of operators, C 0-semigroups, C-regularized semigroups, and α-times integrated semigroups on Fréchet space...

We discuss the behaviour at infinity of $n$-times integrated semigroups with nonquasianalytic growth and invertible generator. The results obtained extend in this setting a theorem of O. El Mennaoui on stability of bounded once integrated semigroups, and (partially) a theorem of Q. P. V$\tilde{\rm u}$ on stability of $C_0$-semigroups.

In this paper we consider combinatorial numbers $C_{m, k}$ for $m\ge 1$ and $k\ge 0$ which unifies the entries of the Catalan triangles $ B_{n, k}$ and $ A_{n, k}$ for appropriate values of parameters $m$ and $k$, i.e., $B_{n, k}=C_{2n,n-k}$ and $A_{n, k}=C_{2n+1,n+1-k}$. In fact, some of these numbers are the well-known Catalan numbers $C_n$ that...

The existence of distributional chaos and distributional irregular vectors has been recently considered in the study of linear dynamics of operators and C 0-semigroups. In this paper we extend some previous results on both notions to sequences of operators, C 0-semigroups, C-regularized semigroups, and α-times integrated semigroups on Fréchet space...

In this paper we show the unexpected property that extension from local to
global without loss of regularity holds for the solutions of a wide class of
vector-valued differential equations, in particular for the class of fractional
abstract Cauchy problems in the subdiffusive case. The main technique is the
use of the algebraic structure of these s...

Let $X$ be a complex Banach space. The connection between algebra
homomorphisms defined on subalgebras of the Banach algebra
$\ell^{1}(\mathbb{N}_0)$ and the algebraic structure of Ces\`{a}ro sums of a
linear operator $T\in \mathcal{B}(X)$ is established. In particular, we show
that every $(C, \alpha)$-bounded operator $T$ induces - and is in fact...

We obtain a vector-valued subordination principle for $(g_{\alpha},
g_{\alpha})$-regularized resolvent families which unified and improves various
previous results in the literature. As a consequence we establish new relations
between solutions of different fractional Cauchy problems. To do that, we
consider scaled Wright functions which are relate...

In this paper we obtain $L^1$-weighted norms of classical orthogonal
polynomials (Hermite, Laguerre and Jacobi polynomials) in terms of the zeros of
these orthogonal polynomials; these expressions are usually known as quadrature
rules. In particular these new formulae are useful to calculate directly some
positive defined integrals as several examp...

We characterize operator-valued Riesz means via an algebraic law of composition and establish their functional calculus accordingly. With this aim,
we give a new integral expression of the Leibniz derivation rule for smooth functions.

In this paper we introduce vector-valued Hermite expansions to approximate
one-parameter operator families such as $C_0$-groups and cosine functions. In
both cases we estimate the rate of convergence of these Hermite expansions to
the related family and compare with other known approximations. Finally we
illustrate our results with particular examp...

In this paper we introduce Laguerre expansions to approximate
$C_0$-semigroups and resolvent operators. We give the rate of convergence of
Laguerre expansion to the $C_0$-semigroup and compare with other known
approximations. To do that, we need to study Laguerre functions and the
convergence of Laguerre series in Lebesgue spaces. To finish, we con...

For $\beta>0$ and $p\ge 1$, the generalized Ces\`aro operator $$
\mathcal{C}_\beta f(t):=\frac{\beta}{t^\beta}\int_0^t (t-s)^{\beta-1}f(s)ds $$
and its companion operator $\mathcal{C}_\beta^*$ defined on Sobolev spaces
$\mathcal{T}_p^{(\alpha)}(t^\alpha)$ and $\mathcal{T}_p^{(\alpha)}(|
t|^\alpha)$ (where $\alpha\ge 0$ is the fractional order of de...

In this paper we give sharp extension results for convoluted solutions of abstract Cauchy problems in Banach spaces. The main technique is the use of the algebraic structure (for the usual convolution product *) of these solutions which are defined by a version of the Duhamel formula. We define algebra homomorphisms from a new class of test-functio...

In this paper we give sharp extensions of convoluted solutions of wave
equations in abstract Banach spaces. The main technique is to use the
algebraic structure, for convolution products $\ast$ and $\ast_c$, of
these solutions which are defined by a version of the Duhamel's formula.
We define algebra homomorphisms, for the convolution product $\ast...

Y. Katznelson and L. Tzafriri proved that if T is a power-bounded operator and f is an analytic function, in the Wiener algebra, of spectral synthesis with respect to its peripheral spectrum then lim n→∞ ∥T n f(T)∥=0. Here f(T) is given by the usual functional calculus associated with T. The analogous version for bounded C 0 -semigroups of operator...

In this article we study the uniform stability of an (a,k)-regularized family {S(t)}t≥0 generated by a closed operator A. We give sufficient conditions, on the scalar kernels a, k and the operator A, to ensure the uniform stability of the family {S(t)}t≥0 in Hilbert spaces. Our main result is a generalization of Theorem 1 in [Proc. Amer. Math. Soc....

The main aim of this paper is to show that certain Banach spaces, defined via integral kernel operators, are Banach modules (with respect to some known Banach algebras and convolution products on ℝ + ). To do this, we consider some suitable kernels such that the Hardy-type operator is bounded in weighted Lebesgue spaces L ω p (ℝ + ) for p≥1. We als...

Various Lp form Poincaré and Opial inequalities are given for vector-valued convolution products. We apply our results to infinitesimal generators of C0-semigroups and cosine functions. Typical examples of these operators are differential operators in Lebesgue spaces.

We extend results of Caffarelli--Silvestre and Stinga--Torrea regarding a
characterization of fractional powers of differential operators via an
extension problem. Our results apply to generators of integrated families of
operators, in particular to infinitesimal generators of bounded $C_0$
semigroups and operators with purely imaginary symbol. We...

We study the range of the Laplace transform on convolution Banach algebras T((alpha))(t(alpha)), alpha > 0, defined by fractional derivation. We introduce Banach algebras A(0)((alpha)) (C(+)) of holomorphic functions in the right hand half-plane which are defined using complex fractional derivation along rays leaving the origin. We prove that the r...

We characterize the dense ideals of certain convolution Sobolev algebras, on the positive half-line, as those ideals I which satisfy the Nyman conditions Z(I)=∅ and γ(I)=0. Here Z(I) is the hull of I and γ(I):=inf {inf supp (f):f∈I}.
KeywordsConvolution–Sobolev algebra–Dense ideal–Nyman theorem–Fractional derivation

We prove that a sectorial operator admits an H
∞-functional calculus if and only if it has a functional model of Nagy–Foiaş type. Furthermore, we give a concrete formula for the characteristic function (in a generalized sense) of such an operator. More generally, this approach applies to any sectorial operator by passing to a different norm (the Mc...

We establish an inversion formula of Post-Widder type for λα-multiplied vector-valued Laplace transforms (α > 0). This result implies an inversion theorem for resolvents of generators of α-times integrated families (semigroups and cosine functions) which, in particular, provides a unified proof of previously known inversion formulae for α-times int...

A Landau–Kolmogorov type inequality for generators of a wide class of strongly continuous families of bounded and linear operators defined on a Banach space is shown. Our approach allows us to recover (in a unified way) known results about uniformly bounded C0-semigroups and cosine functions as well as to prove new results for other families of ope...

In this paper we obtain the moments {Φm}m⩾0 defined by where is the usual combinatorial number. We also provide the moments in the Catalan triangle whose (n,p) entry is defined by and, in particular, new identities involving the well-known Catalan numbers.

In this paper we use the resolvent semigroup associated to a C 0 -semigroup to introduce the vector-valued Stieltjes transform defined by a C 0 -semigroup. We give new results which extend known ones in the case of scalar generalized Stieltjes transform. We work with the vector-valued Weyl fractional calculus to present a deep connection between bo...

The main aim of this paper is to extend definitions of Hilbert transform, Dirichlet and Fejér operators (defined by convolution
with suitable kernels in Lebesgue spaces) in arbitrary Banach spaces. We present a self-contained theory which includes different
approaches of other authors whose starting points were usually C
0-groups or cosine function...

Transform methods are used to establish algebra homomorphisms related to convoluted semigroups and convoluted cosine functions. Such families are now basic in the study of the abstract Cauchy problem. The framework they provide is flexible enough to encompass most of the concepts used up to now to treat Cauchy problems of the first- and second-orde...

In this paper, we present new results about the space Lp(ν) for ν being a vector measure defined in the Borel σ-algebra of a compact abelian group G and satisfying certain property concerning translation of simple functions. Namely, we show that Lp(ν) is a translation invariant space which can be endowed with an algebra structure via usual convolut...

Let T be a sectorial operator. It is known that the existence of a bounded (suitably scaled) H∞ calculus for T, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra Λ∞.la (R + ). Such an algebra includes functions defined by Mikhlin-type conditions and so the Besov c...

In this paper we deal with the weighted Banach algebra L
ω
1(ℝ+, *c
), where *c is the cosine convolution product. We describe its character space and its multiplier algebra. Our main results concern bounded algebra homomorphisms from L
ω
1(ℝ+, *c
). We give a variant of Kisyński’s theorem for such homomorphisms and characterize them in terms of in...

In this paper we prove new identities in the Catalan triangle whose (n, p) entry is defined by B(n,p) := p/n (2n n - p), n,p is an element of N, p <= n. In fact, we show some new identities involving the well-known Catalan numbers, and specially the identity Sigma(i)(p=1) B(n,p)B(n,n+p-i) (n + 2p - i) = (n + 1)C(n) (2(n - 1) i - 1), i <= n, that ap...

In this paper we study the local well-posed integrated Cauchy problem, v 0 (t) = Av(t) + t fi ¡(fi + 1) x; v(0) = 0; t 2 (0;•); with • > 0, fi ‚ 0, and x 2 X where X is a Banach space and A a closed operator on X. We extend solutions increasing the regularity in fi. The global case (• = 1) is also treated in detail. Growths of solutions are given i...

We characterize algebra homomorphisms from the Lebesgue algebra $L^1_\omega(\mathbb{R})$ into a Banach algebra $\mathcal{A}$. As a consequence of this result, every bounded algebra homomorphism $\varPhi:L^1_\omega(\mathbb{R})\to\mathcal{A}$ is approached through a uniformly bounded family of fractional homomorphisms, and the Hille–Yosida theorem fo...

In this paper, we give connections between distribution cosine func-tions (defined in [10]) and almost-distribution cosine functions (introduced in [13]). We prove several equalities involving trigonometric convolution prod-ucts and distribution cosine functions as well as some relations between dis-tribution cosine functions and ultradistribution...

Some sequences of matrix polynomials have been introduced recently as solutions of certain second-order differential equations,
which can be seen as appropriate generalizations, to the matrix setting, of classical orthogonal polynomials. In this paper,
we consider families (in a complex parameter) of matrix-valued special functions of Hermite type,...

In this paper, we prove directly that α-times integrated groups define algebra homomorphisms. We also give a theorem of equivalence
between smooth distribution groups and α-times integrated groups.

We consider some extensions of well-boundedness and Cm-scalarity by using fractional calculus, and prove some theorems accordingly. These results are applied to the usual Laplacian
on Rn and sub-Laplacians on nilpotent Lie groups.

Our first aim in this paper is to give sufficient conditions for the hypercyclicity and topological mixing of a strongly continuous cosine function. We apply these results to study the cosine function associated to translation groups. We also prove that every separable infinite dimensional complex Banach space admits a topologically mixing uniforml...

In this paper we prove new polynomial identities in the Catalan triangle using the WZ theory. In fact, we check new polynomial identities in the general expression P(n,i,k):=∑ p=1 i B n,p B n,n+p-i (n+2p-i) k ,i≤n, for some i≤n and k∈ℕ∪{0}, being B n,p :=p n2n n-p,n,p∈ℕ,p≤n, the (n,p) entry in the Catalan triangle. We show that P(n,i,k) involves th...

In this paper new equalities between two different convolution products in cancellative naturally ordered semigroups (but
not in groups) are given. We also give several applications in particular cases
\Bbb N*{\Bbb N}^*
and
\Bbb R+*.{\Bbb R}^{+*}.

Groups of unbounded operators are approached in the setting of the Esterle quasimultiplier theory. We introduce groups of regular quasimultipliers of growth ω, or ω-groups for short, where ω is a continuous weight on the real line. We study the relationship of ω-groups with families of operators and homomorphisms such as regularized, distribution a...

Let T be a sectorial operator. It is known that the existence of a bounded (suitably scaled) H1 calculus for T, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra ¿® 1;1(R+) [4]. Such an algebra includes functions de¯ned by Mikhlin-type conditions and so the Besov...

We use a dual convolution to the classical convolution in L (ℝ) to find other expressions to Fourier trigonometric convolutions. These convolutions are used to get trigonometric equalities and formulae with special functions.

The aim of this paper is to characterize cosine and sine functions in terms
of vector-valued cosine transforms on a Banach space X. We also introduce
vector-valued sine transforms.

We show that a bounded homomorphism T : L 1 ω (R +) → A is equivalent to a uniformly bounded family of fractional homomorphisms T α : AC (α) ω (R +) → A for any α > 0. We add this characterization to the Widder-Arendt-Kisyski theorem and relate it to α-times integrated semigroups.

We investigate the weak spectral mapping property (WSMP) ̂μ(σ(A)) = σ(̂μ(A)), where A is the generator of a C 0-semigroup in a Banach space X, μ is a measure, and ̂μ(A) is defined by the Phillips functional calculus. We consider the special case when X is a Banach algebra and the operators e At, t ≥ 0, are multipliers.

We introduce the notion of almost-distribution cosine functions in a setting similar to that of distribution semigroups defined by Lions. We prove general results on equivalence between almost-distribution cosine functions and α-times integrated cosine functions.

We derive two expansions of the Randles–Sevcik function \(\sqrt \pi X(x)\): an asymptotic expansion of \(\sqrt \pi X(x)\) for x → ∞ and its Taylor expansion at any x
0 ∈ \(\mathbb{R}\). These expansions are accompanied by error bounds for the remainder at any order of the approximation.

We show that the Davies functional calculus and the AC( )- calculus coincide under common hypotheses. Then we apply the calculus to operators on Banach spaces, to investigate spectral invariance and norm estimates linked to abstract Cauchy equations. This extends some previous results in the area, and unifies diverse approaches. The theory is appli...

We show that the Davies functional calculus and the AC((nu))-calculus coincide under common hypotheses. Then we apply the calculus to operators on Banach spaces, to investigate spectral invariance and norm estimates linked to abstract Cauchy equations. This extends some previous results in the area, and unifies diverse approaches. The theory is app...

By applying fractional integration and derivation to the vector-valued Laplace transform, a functional calculus for α-times integrated semigroups is obtained. This functional calculus is related to smooth distribution semigroups. As application, fractional powers of its infinitesimal generator are defined.

Given (A, D(A)) a closed (not necessarily densely defined) linear operator in a Banach space X, a new family of bounded and linear operators, the α-times integrated trigonometric sine function (with α0) is introduced in order to find the link between the α-times integrated cosine function (generated by-A2) and the α-times integrated group (generate...

In this paper, we study the connection between Weyl fractional calculus of the gaussian function and Hermite functions. This relationship appears in a natural way to treat integrated families. Some particular cases are considered.

We define gamma operators and use them to give solutions of the equation u(s+1)=(s+A)u(s) where (A,D(A)) is the infinitesimal generator of a C 0 -group. Two applications to differential equations are also considered.