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Let $c=(C_n)_{n\ge 0}$ be the Catalan sequence and $T$ a linear and bounded operator on a Banach space $X$ such $4T$ is a power-bounded operator. The Catalan generating function is defined by the following Taylor series, $$ C(T):=\sum_{n=0}^\infty C_nT^n. $$ Note that the operator $C(T)$ is a solution of the quadratic equation $TY^2-Y+I=0.$ In this paper we define powers of the Catalan generating function $C(T)$ in terms of the Catalan triangle numbers. We obtain new formulae which involve Catalan triangle numbers; the spectrum of $c^{\ast j}$ and the expression of $c^{-\ast j}$ for $j\ge 1$ in terms of Catalan polynomials ($\ast$ is the usual convolution product in sequences). In the last section, we give some particular examples to illustrate our results and some ideas to continue this research in the future., Comment: pp 18. arXiv admin note: text overlap with arXiv:2401.16415

In this paper we study the solution of the quadratic equation $TY^2-Y+I=0$ where $T$ is a linear and bounded operator on a Banach space $X$. We describe the spectrum set and the resolvent operator of $Y$ in terms of operator $T$. In the case that $ 4T$ is a power-bounded operator, we show that a solution (named Catalan generating function) is given by the Taylor series $$ C(T):=\sum_{n=0}^\infty C_nT^n, $$ where the sequence $(C_n)_{n\ge 0}$ is the well-known Catalan numbers. We express $C(T)$ by means of an integral representation which involves the resolvent operator $(\lambda-T)^{-1}$. Some particular examples to illustrate our results are given, in particular an iterative method defined for square matrices $T$ which involves Catalan numbers., Comment: pp 18

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 mass conservation principle; that is reflected on the large time behaviour of solutions. We prove asymptotic pointwise and $\ell^p$ decay results for the fundamental solution. We use that estimates to get rates on the $\ell^p$ decay and large time behaviour of solutions. For the $\ell^2$ case, we get optimal decay by use of Fourier techniques., Comment: pp 25

In this paper we deal with a scale of reproducing kernel Hilbert spaces $H^{(n)}_2$, $n\ge 0$, which are linear subspaces of the classical Hilbertian Hardy space on the right-hand half-plane $\mathbb{C}^+$. 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. An explicit integral formula is given for the reproducing kernel $K_{z,n}$ of $H^{(n)}_2$, from which we can find the estimate $\Vert K_{z,n}\Vert\sim\vert z\vert^{-1/2}$ for $z\in\mathbb{C}^+$. Then composition operators $C_\varphi :H_2^{(n)} \to H_2^{(n)}$, $C_\varphi f=f\circ \varphi $, on these spaces are discussed, giving some necessary and some sufficient conditions for analytic maps $\varphi: \mathbb{C}^+\to \mathbb{C}^+$ to induce bounded composition operators., Comment: 27 pp

We study reproducing kernel Hilbert spaces introduced as ranges of generalized Ces\`aro-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 given which connects the real setting with the complex one. These spaces are related with fractional operations in the context of integrated Brownian processes. We give estimates of the norms of the corresponding reproducing kernels., Comment: 32 pp

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 $p\ge 1$) are studied in detail. If $0 < \beta - \pp < \mu$, then operators $\mathcal{S}_{\beta,\mu}$ are bounded (and we compute their operator norms which depend on $p$); commute and factorize with generalized Ces\'{a}ro operator on $\mathcal{T}_p^{(\alpha)}(t^\alpha)$ . We calculate and represent explicitly their spectrum set $\sigma (\mathcal{S}_{\beta,\mu})$. The main technique is to subordinate these operators in terms of $C_0$-groups and transfer new properties from some special functions to Stieltjes operators. We also prove some similar results for generalized Stieltjes operators $ \mathcal{S}_{\beta,\mu}$ in the Sobolev-Lebesgue $\mathcal{T}_p^{(\alpha)}(\vert t\vert^\alpha)$ defined on the real line $\R$. We show connections with the Fourier and the Hilbert transform and a convolution product defined by the Hilbert transform., Comment: 45 pages, 6 figures

Mar\'ia Andresa Casamayor de la Coma, 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 her true name., Comment: in Spanish. Version with pictures

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 mathematicians of high level and to exchange ideas, works and collaborations. This work of associationism has been unnoticed in biographical and scientific studies up to now and helps to complete the picture of Garc\'ia de Galdeano as an international mathematician and as a wise importer of the main mathematical achievements of his time., Comment: In Spanish

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 properties similar to classical Lebesgue sequence spaces. In order to show the main results about fractional finite differences, we state equalities involving sums of quotients of Euler's Gamma functions. Finally, we display some graphical representations of the spectra of generalized Ces\`aro operators., Comment: 32 pages, 7 Figures

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 $A$ is an closed linear operator defined on a Banach space $X$. Our ideas are inspired on the Poisson distribution as a tool to sampling fractional differential operators into fractional differences. Using our abstract approach, we are able to show existence and uniqueness of solutions for the problem (*) on a distinguished class of weighted Lebesgue spaces of sequences, under mild conditions on strongly continuous sequences of bounded operators generated by $A,$ and natural restrictions on the nonlinearity $f$. Finally we present some original examples to illustrate our results.

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 is $C_{2n,n-1}=C_{2n+1,n}=C_n$. We present new identities for recurrence relations, linear sums and alternating sum of $C_{m,k}$. After that, we check sums (and alternating sums) of squares and cubes of $C_{m,k}$ and, consequently, for $ B_{n, k}$ and $ A_{n, k}$. In particular, one of these equalities solves an open problem posed in \cite{[GHMN]}. We also present some linear identities involving harmonic numbers $H_n$ and Catalan triangles numbers $C_{m,k}$. Finally, in the last section new open problems and identities involving $C_n$ are conjectured.

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 solutions, which are defined by new versions of functional equations defining solution families of bounded operators. The convolution product and the double Laplace transform for functions of two variables are useful tools which we apply also to extend these solutions. Finally we illustrate our results with different concrete examples.

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 characterized - by such an algebra homomorphism. Our method is based on some sequence kernels, Weyl fractional difference calculus and convolution Banach algebras that are introduced and deeply examined. To illustrate our results, improvements to bounds for Abel means, new insights on the $(C,\alpha)$ boundedness of the resolvent operator for temperated $\alpha$-times integrated semigroups, and examples of bounded homomorphisms are given in the last section.

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 related to Mittag-Leffler functions, the fractional calculus and stable L\'evy processes. We study some interesting properties of these functions such as subordination (in the sense of Bochner), convolution properties, and their Laplace transforms. Finally we present some examples where we apply these results.

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 examples show.

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 examples of $C_0$-groups and cosine functions and their Hermite expansions., Comment: 22 pages. This is a new version of our work about Hermite expansions, in which we complement some of the existing results, and we have added other relevant theorems related to the problem of expansions of groups and cosine funcions through Hermite polynomials

In this paper we introduce Laguerre expansions to approximate vector-valued functions expanding on the well-known scalar theorem. We apply this result to approximate $C_0$-semi\-groups and resolvent operators in abstract Banach spaces. We study certain Laguerre functions, its Laplace transforms and the convergence of Laguerre series in Lebesgue spaces. The concluding section of this paper is devote to consider some examples of $C_0$-semigroups: shift, convolution and holomorphic semigroups where some of these results are improved., Comment: We have shortened and improved a previuous version

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 derivation and are embedded in $L^p(\RR^+)$ and $L^p(\RR)$ respectively) are studied. We prove that if $p>1$, then $\mathcal{C}_\beta$ and $\mathcal{C}_\beta^*$ are bounded operators and commute on $\mathcal{T}_p^{(\alpha)}(t^\alpha)$ and $\mathcal{T}_p^{(\alpha)}(| t|^\alpha)$. We show explicitly the spectra $\sigma (\mathcal{C}_\beta)$ and $\sigma (\mathcal{C}_\beta^*)$ and its operator norms (which depend on $p$). For $1< p\le 2$, we prove that $ \hat{{\mathcal C}_\beta(f)}={\mathcal C}_\beta^*(\hat{f})$ and $\hat{{\mathcal C}_\beta^*(f)}={\mathcal C}_\beta(\hat{f})$ where $\hat{f}$ is the Fourier transform of a function $f\in L^p(\RR)$., Comment: 24 pages

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 $\ast_c$, from a certain set of test-functions and apply our results to concrete examples of abstract wave equations., Comment: 20 pages

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 algebraic structure (for usual convolution product $\ast$) of these solutions which are defined by a version of the Duhamel formula. We define algebra homomorphisms from a new class of test-functions and apply our results to concrete operators. Finally, we introduce the notion of $k$-distribution semigroups to extend previous concepts of distribution semigroups., Comment: 24 pages

We prove that a sectorial operator admits an H-infty - functional calculus if and only if it has a functional model of Nagy-Foias 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 McIntosh square function norm). We also show that this quadratic norm is close to the original one, in the sense that there is only a logarithmic gap between them., Comment: 26 pages