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Alberto Lastra

Publications and source records attributed to Alberto Lastra.

At least 37 records · Page 2Linked to original sources
arXiv2023

A dual watermaking scheme based on Sobolev type orthogonal moments for document authentication

A dual watermarking scheme based on Sobolev type orthogonal moments, Charlier and Meixner, is proposed based on different discrete measures. The existing relation through the connection formulas allows to provide with structure and recurrence relations, together with two difference equations satisfied by such families. Weighted polynomials derived from them are being applied in an embedding and extraction watermarking algorithm, comparing the results obtained in imperceptibly and robustness tests with other families of polynomials.

cs.MM↗
arXiv2023

Solutions of linear systems of moment differential equations via generalized matrix exponentials

A generalized exponential matrix based on the construction of kernel operators for generalized summability is defined and analyzing its main properties, generalizing the classical exponential matrix and fractional exponential matrix. This object serves as a practical tool to express the solutions of linear systems of moment differential equations in a compact manner, in the spirit of the classical exponential matrix.

math.CA↗
arXiv2023

On sequences preserving q-Gevrey asymptotic expansions

The modification of the coefficients of formal power series is analyzed in order that such variation preserves q-Gevrey asymptotic properties, in particular q-Gevrey asymptotic expansions. A characterization of such sequences is determined, providing a handy tool in practice. The sequence of q-factorials is proved to preserve q-Gevrey asymptotic expansions.

math.CV↗
arXiv2022

Extension operators for some ultraholomorphic classes defined by sequences of rapid growth

While the asymptotic Borel mapping, sending a function into its series of asymptotic expansion in a sector, is known to be surjective for arbitrary openings in the framework of ultraholomorphic classes associated with sequences of rapid growth, there is no general procedure to construct extension operators in this case. We do provide such operators in complex sectors for some particular classes considered by S.~Pilipovi{ć}, N.~Teofanov and F.~Tomi{ć} in the ultradifferentiable setting. Although these classes are, in their words, "beyond Gevrey regularity", in some cases they keep the property of stability under differentiation, which is crucial for our technique, based on formal Borel- and truncated Laplace-like transforms with suitable kernels.

math.FA↗
arXiv2022

Higher-order recurrence relations, Sobolev-type inner products and matrix factorizations

It is well known that Sobolev-type orthogonal polynomials with respect to measures supported on the real line satisfy higher-order recurrence relations and these can be expressed as a (2N+1)-banded symmetric semi-infinite matrix. In this paper we state the connection between these (2N+1)-banded matrices and the Jacobi matrices associated with the three-term recurrence relation satisfied by the standard sequence of orthonormal polynomials with respect to the 2-iterated Christoffel transformation of the measure.

math.CA↗
arXiv2022

Multisummability of formal solutions for a family of generalized singularly perturbed moment differential equations

The notion of moment differentiation is extended to the set of generalized multisums of formal power series via an appropriate integral representation and accurate estimates of the moment derivatives. The main result is applied to characterize generalized multisummability of the formal solution to a family of singularly perturbed moment differential equations in the complex domain, broadening widely the range of singularly perturbed functional equations to be considered in practice, such as singularly perturbed differential equations and singularly perturbed fractional differential equations.

math.CA↗
arXiv2022

Formal Gevrey solutions -- in analytic germs -- for higher order holomorphic PDEs

We consider a family of holomorphic PDEs whose singular locus is given by the zero set of an analytic map $P$ with $P(0)=0$. Our goal is to establish conditions for the existence and uniqueness of formal power series solutions and to determine their divergence rate. In fact, we prove that the solution is Gevrey in $P$, giving new information on divergency while compared to the classical Gevrey classes. If $P$ is not singular at $0$, we also provide Poincaré conditions to recover convergent solutions. Our strategy is to extend the dimension and lift the given PDE to a problem where results of singular PDEs can be applied. Finally, examples where the Gevrey class in $P$ is optimal are included.

math.AP↗
arXiv2021

Entire solutions of linear systems of moment differential equations and related asymptotic growth at infinity

The general entire solution to a linear system of moment differential equations is obtained in terms of a moment kernel function for generalized summability, and the Jordan decomposition of the matrix defining the problem. The growth at infinity of any solution of the system is also determined, both globally and also following rays to infinity, determining the order and type of such solutions.

math.CA↗
arXiv2021

On second order q-difference equations for high-order Sobolev-type q-Hermite orthogonal polynomials

The q-Hermite I-Sobolev type polynomials of higher order are consider for their study. Their hypergeometric representation is provided together with further useful properties such as several structure relations which give rise to a three-term recurrence relation of their elements. Two different q-difference equations satisfied by the q-Hermite I-Sobolev type polynomials of higher order are also established.

math.CA↗
arXiv2021

On the multiple-scale analysis for some linear partial $q$-difference and differential equations with holomorphic coefficients

The analytic and formal solutions of certain family of $q$-difference-differential equations under the action of a complex perturbation parameter is considered. The previous study of the last two authors provides information in the case when the main equation under study is factorizable, as a product of two equations in the so-called normal form. Each of them gives rise to a single level of $q$-Gevrey asymptotic expansion. In the present work, the main problem under study does not suffer any factorization, and a different approach is followed. More precisely, we lean on the technique developed in a paper, where the first author makes distinction among the different $q$-Gevrey asymptotic levels by successive applications of two $q$-Borel-Laplace transforms of different orders both to the same initial problem and which can be described by means of a Newton polygon.

math.CA↗
arXiv2020

Summability of formal solutions for a family of generalized moment integro-differential equations

Generalized summability results are obtained regarding formal solutions of certain families of linear moment integro-differential equations with time variable coefficients. The main result leans on the knowledge of the behavior of the moment derivatives of the elements involved in the problem. A refinement of the main result is also provided giving rise to more accurate results which remain valid in wide families of problems of high interest in practice, such as fractional integro-differential equations.

math.AP↗
arXiv2020

On the convergence of generalized power series solutions of $q$-difference equations

A sufficient condition for the convergence of a generalized formal power series solution to an algebraic $q$-difference equation is provided. The main result leans on a geometric property related to the semi-group of (complex) power exponents of such a series. This property corresponds to the situation in which the small divisors phenomenon does not arise. Some examples illustrating the cases where the obtained sufficient condition can be or cannot be applied are also depicted.

math.CA↗
arXiv2020

On second order q-difference equations satisfied by Al-Salam-Carlitz I-Sobolev type polynomials of higher order

This contribution deals with the sequence $\{\mathbb{U}_{n}^{(a)}(x;q,j)\}_{n\geq 0}$ of monic polynomials, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam--Carlitz I orthogonal polynomials, and involving an arbitrary number of $q$-derivatives on the two boundaries of the corresponding orthogonality interval. We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order $q$-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by $\mathbb{U}_{n}^{(a)}(x;q,j)$, which paves the way to establish an appealing generalization of the so-called $J$-fractions to the framework of Sobolev-type orthogonality.

math.CA↗
arXiv2020

Summability of formal solutions for some generalized moment partial differential equations

The concept of moment differentiation is extended to the class of moment summable functions, giving rise to moment differential properties. The main result leans on accurate upper estimates for the integral representation of the moment derivatives of functions under exponential-like growth at infinity, and appropriate deformation of the integration paths. The theory is applied to obtain summability results of certain family of generalized linear moment partial differential equations with variable coefficients.

math.CV↗
arXiv2019

Estimates of formal solutions for some generalized moment partial differential equations

Using increasing sequences of real numbers, we generalize the idea of formal moment differentiation first introduced by W. Balser and M. Yoshino. Slight departure from the concept of Gevrey sequences enables us to include a wide variety of operators in our study. Basing our approach on tools such as the Newton polygon and divergent formal norms, we obtain estimates for formal solutions of certain families of generalized linear moment partial differential equations with constant and time variable coefficients.

math.AP↗
arXiv2019

On a $q-$analog of a singularly perturbed problem of irregular type with two complex time variables

Analytic solutions and their formal asymptotic expansions for a family of the singularly perturbed $q-$difference-differential equations in the complex domain are constructed. They stand for a $q-$analog of the singularly perturbed partial differential equations considered in our recent work [A. Lastra, S. Malek, Boundary layer expansions for initial value problems with two complex time variables, submitted 2019]. In the present work, we construct outer and inner analytic solutions of the main equation, each of them showing asymptotic expansions of essentially different nature with respect to the perturbation parameter. The appearance of the $-1$-branch of Lambert $W$ function will be crucial in this respect.

math.CV↗
arXiv2019

Boundary layer expansions for initial value problems with two complex time variables

We study a family of partial differential equations in the complex domain, under the action of a complex perturbation parameter $ε$. We construct inner and outer solutions of the problem and relate them to asymptotic representations via Gevrey asymptotic expansions with respect to $ε$, in adequate domains. The construction of such analytic solutions is closely related to the procedure of summation with respect to an analytic germ, put forward in[J. Mozo-Fernández, R. Schäfke, Asymptotic expansions and summability with respect to an analytic germ, Publ. Math. 63 (2019), no. 1, 3--79.], whilst the asymptotic representation leans on the cohomological approach determined by Ramis-Sibuya Theorem.

math.CV↗