Vector-valued Case Research Articles

Fractional Calculus for Non-Discrete Signed Measures

In this paper, we suggest a first-ever construction of fractional integral and differential operators based on signed measures including a vector-valued case. The study focuses on constructing the fractional power of the Riemann–Stieltjes integral with a signed measure, using semigroup theory. The main result is a theorem that provides the exact form of a semigroup for the Riemann–Stieltjes integral with a measure having a countable number of extrema. This article provides examples of semigroups based on integral operators with signed measures and discusses the fractional powers of differential operators with partial derivatives.

Spectral Asymptotic Properties of Semiregular Non-commutative Harmonic Oscillators

We study here the spectral Weyl asymptotics for a semiregular system, extending to the vector-valued case results of Helffer and Robert, and more recently of Doll, Gannot and Wunsch. The class of systems considered here contains the important example of the Jaynes–Cummings system that describes light-matter interaction.

Localization in musical steelpans

The steelpan is a pitched percussion instrument that takes the form of a concave bowl with several localized dimpled regions of varying curvature. Each of these localized zones, called notes, can vibrate independently when struck, and produce a sustained tone of a well-defined pitch. While the association of the localized zones with individual notes has long been known and exploited, the relationship between the shell geometry and the strength of the mode confinement remains unclear. Here, we explore the spectral properties of the steelpan modelled as a vibrating elastic shell. To characterize the resulting eigenvalue problem, we generalize a recently developed theory of localization landscapes for scalar elliptic operators to the vector-valued case, and predict the location of confined eigenmodes by solving a Poisson problem. A finite-element discretization of the shell shows that the localization strength is determined by the difference in curvature between the note and the surrounding bowl. In addition to providing an explanation for how a steelpan operates as a two-dimensional xylophone, our study provides a geometric principle for designing localized modes in elastic shells.

Estimates for generalized Bohr radii in one and higher dimensions

AbstractIn this article, we study a generalized Bohr radius$R_{p, q}(X), p, q\in [1, \infty )$defined for a complex Banach spaceX. In particular, we determine the exact value of$R_{p, q}(\mathbb {C})$for the cases (i)$p, q\in [1, 2]$, (ii)$p\in (2, \infty ), q\in [1, 2]$, and (iii)$p, q\in [2, \infty )$. Moreover, we consider ann-variable version$R_{p, q}^n(X)$of the quantity$R_{p, q}(X)$and determine (i)$R_{p, q}^n(\mathcal {H})$for an infinite-dimensional complex Hilbert space$\mathcal {H}$and (ii) the precise asymptotic value of$R_{p, q}^n(X)$as$n\to \infty $for finite-dimensionalX. We also study the multidimensional analog of a related concept called thep-Bohr radius. To be specific, we obtain the asymptotic value of then-dimensionalp-Bohr radius for bounded complex-valued functions, and in the vector-valued case, we provide a lower estimate for the same, which is independent ofn.

Cauchy Integral and Boundary Value for Vector-Valued Tempered Distributions

Using the historically general growth condition on scalar-valued analytic functions, which have tempered distributions as boundary values, we show that vector-valued analytic functions in tubes TC=Rn+iC obtain vector-valued tempered distributions as boundary values. In a certain vector-valued case, we study the structure of this boundary value, which is shown to be the Fourier transform of the distributional derivative of a vector-valued continuous function of polynomial growth. A set of vector-valued functions used to show the structure of the boundary value is shown to have a one–one and onto relationship with a set of vector-valued distributions, which generalize the Schwartz space DL2′(Rn); the tempered distribution Fourier transform defines the relationship between these two sets. By combining the previously stated results, we obtain a Cauchy integral representation of the vector-valued analytic functions in terms of the boundary value.

Integrable and absolutely continuous vector-valued functions

The theory of integration for functions with values in a topological vector space is a bit of a messy subject. There are many different choices for the integral, most distinct from one another in general, and with no compelling reason to always adopt one definition or another as the “right” one. No attempt is made here to resolve this question of which integral is the “right” integral, but we overview a few of the common notions and prove a couple of useful properties for the notion of integrability by seminorm. For this notion of integrability, the completeness of the space of integrable functions with values in a complete locally convex space is established. This space is then easily seen to be isomorphic to the completion of the projective tensor product of the usual L1 space of scalar functions with the vector space. Absolute continuity for this notion of integrability is also considered. Here, it is shown that the familiar properties of differentiation for absolutely continuous scalar functions holds in the vector-valued case.

Weak solutions for nonlinear waves in adhesive phenomena

We discuss a notion of weak solution for a semilinear wave equation that models the interaction of an elastic body with a rigid substrate through an adhesive layer, relying on results in [2]. Our analysis embraces the vector-valued case in arbitrary dimension as well as the case of non-local operators (e.g. fractional Laplacian).

Stability and convergence of Strang splitting. Part II: Tensorial Allen-Cahn equations

We consider the second-order in time Strang-splitting approximation for tensorial (e.g. vector-valued and matrix-valued) Allen-Cahn equations. Both the linear propagator and the nonlinear propagator are computed explicitly. For the vector-valued case, we prove the maximum principle and unconditional energy dissipation for a judiciously modified energy functional. The modified energy functional is close to the classical energy up to O(τ) where τ is the splitting step. For the matrix-valued case, we prove a sharp maximum principle in the matrix Frobenius norm. We show modified energy dissipation under very mild splitting step constraints. We exhibit several numerical examples to show the efficiency of the method as well as the sharpness of the results.

On tensor products of nuclear operators in Banach spaces

The following result of G. Pisier contributed to the appearance of this paper: if a convolution operator ★f : M(G) → C(G), where $G$ is a compact Abelian group, can be factored through a Hilbert space, then f has the absolutely summable set of Fourier coefficients. We give some generalizations of the Pisier's result to the cases of factorizations of operators through the operators from the Lorentz-Schatten classes Sp,q in Hilbert spaces both in scalar and in vector-valued cases. Some applications are given.

On the Shape Fields Finiteness Principle

Abstract In this paper, we improve the finiteness constant for the finiteness principles for $C^m({\mathbb{R}}^n,{\mathbb{R}}^D)$ and $C^{m-1,1}({\mathbb{R}}^n,{\mathbb{R}}^D)$ selection proven in [ 19] and extend the more general shape fields finiteness principle to the vector-valued case.

Stability and instability of breathers in the U(1) Sasa–Satsuma and nonlinear Schrödinger models* *Partially funded by Product. CNPq Grant No. 305205/2016-1, IMUS and VI PPIT-US program reference I3C and CMM Conicyt PIA AFB170001 and Fondecyt 1150202.

We consider the Sasa–Satsuma (SS) and nonlinear Schrödinger (NLS) equations posed along the line, in 1 + 1 dimensions. Both equations are canonical integrable U(1) models, with solitons, multi-solitons and breather solutions Yang (2010 SIAM Mathematical Modeling and Computation). For these two equations, we recognize four distinct localized breather modes: the Sasa–Satsuma for SS, and for NLS the Satsuma–Yajima, Kuznetsov–Ma and Peregrine breathers. Very little is known about the stability of these solutions, mainly because of their complex structure, which does not fit into the classical soliton behavior Grillakis et al (1987 J. Funct. Anal. 74 160–97). In this paper we find the natural H 2 variational characterization for each of them. This seems to be the first known variational characterization for these solutions; in particular, the first one obtained for the famous Peregrine breather. We also prove that Sasa–Satsuma breathers are H 2 nonlinearly stable, improving the linear stability property previously proved by Pelinovsky and Yang (2005 Chaos 15 037115). Moreover, in the SS case, we provide an alternative understanding of the SS solution as a breather, and not only as an embedded soliton. The method of proof is based in the use of a H 2 based Lyapunov functional, in the spirit of Alejo and Muñoz (2013 Commun. Math. Phys. 324 233–62), extended this time to the vector-valued case. We also provide another rigorous justification of the instability of the remaining three nonlinear modes (Satsuma–Yajima, Peregrine and Kuznetsov–Ma), based in the study of their corresponding linear variational structure (as critical points of a suitable Lyapunov functional), and complementing the instability results recently proved e.g. in Muñoz (2017 Proyecciones (Antofagasta) 36 653–83).

On counting cuspidal automorphic representations for GSp(4)

Abstract We find the number s k ⁢ ( p , Ω ) s_{k}(p,\Omega) of cuspidal automorphic representations of GSp ⁢ ( 4 , A Q ) \mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}}) with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight k ≥ 3 k\geq 3 , and the non-archimedean component at 𝑝 is an Iwahori-spherical representation of type Ω and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for s k ⁢ ( p , Ω ) s_{k}(p,\Omega) generalizes to the vector-valued case and a finite number of ramified places.

A dynamical interval Newton method

In this paper, a dynamical interval Newton’s method is proposed to solve systems of nonlinear equations. Differently from other interval techniques available in the literature, the proposed method is capable of determining the roots of vector-valued time-varying functions. The design is carried out by firstly proposing an interval Newton’s method capable of determining the roots of a scalar time-varying function and then extending these results to the vector-valued case. If the function to be zeroed is polynomial, then it is shown how to couple the proposed scalar and vector Newton’s methods to determine improved estimates of its roots. Examples of application of the proposed procedure to the inverse kinematics of a robotic manipulator and to the problem of designing an observer for a nonlinear system are reported.

Homomorphisms Between Algebras of Holomorphic Functions on the Infinite Polydisk

We study the vector-valued spectrum $\mathcal{M}_\infty(B_{c_0},B_{c_0})$, that is, the set of non null algebra homomorphisms from $\mathcal H^\infty(B_{c_0})$ to $\mathcal H^\infty(B_{c_0})$ which is naturally projected onto the closed unit ball of $\mathcal H^\infty(B_{c_0}, \ell_\infty)$, likewise the scalar-valued spectrum $\mathcal M_\infty(B_{c_0})$ which is projected over $\bar{B}_{\ell_\infty}$. Our itinerary begins in the scalar-valued spectrum $\mathcal{M}_\infty(B_{c_0})$: by expanding a result by Cole, Gamelin and Johnson (1992) we prove that on each fiber there are $2^c$ disjoint analytic Gleason isometric copies of $B_{\ell_\infty}$. For the vector-valued case, building on the previous result we obtain $2^c$ disjoint analytic Gleason isometric copies of $B_{\mathcal{H}^\infty(B_{c_0},\ell_\infty)}$ on each fiber. We also take a look at the relationship between fibers and Gleason parts for both vector-valued spectra $\mathcal{M}_{u,\infty}(B_{c_0},B_{c_0})$ and $\mathcal{M}_\infty(B_{c_0},B_{c_0})$.

Vector-valued operators, optimal weighted estimates and the Cp condition

In this paper some new results concerning the Cp classes introduced by Muckenhoupt (1981) and later extended by Sawyer (1983), are provided. In particular, we extend the result to the full expected range p < 0, to the weak norm, to other operators and to their vector-valued extensions. Some of those results rely upon sparse domination, which in the vector-valued case are provided as well. We will also provide sharp weighted estimates for vector-valued extensions relying on those sparse domination results.

Fourier spectrum of Clifford Hp spaces on [formula omitted] for 1 ≤ p ≤ ∞

This article studies the Fourier spectrum characterization of functions in the Clifford algebra-valued Hardy spaces Hp(R+n+1),1≤p≤∞. Namely, for f∈Lp(Rn), Clifford algebra-valued, f is further the non-tangential boundary limit of some function in Hp(R+n+1), 1≤p≤∞, if and only if fˆ=χ+fˆ, where χ+(ξ_)=12(1+iξ_|ξ_|), the Fourier transformation and the above relation are suitably interpreted (for some cases in the distribution sense). These results further develop the relevant context of Alan McIntosh. As a particular case of our results, the vector-valued Clifford Hardy space functions are identical with the conjugate harmonic systems in the work of Stein and Weiss. The latter proved the corresponding singular integral version of the vector-valued cases for 1≤p<∞. We also obtain the generalized conjugate harmonic systems for the whole Clifford algebra-valued Hardy spaces rather than only the vector-valued cases in the Stein-Weiss setting.

On the solutions of Caputo\u2013Hadamard Pettis-type fractional differential equations

Let E be a Banach space with the topological dual E^*. The aim of this paper is two-fold. On the one hand, we prove some basic properties of Hadamard-type fractional integral operators. These results are related to earlier results about integral operators acting on different function spaces, but for the vector-valued case they are of independent interest. Note that we discuss it in a rather general setting. We study Hadamard–Pettis integral operators in both single and multivalued case. On the other hand, we apply these results to obtain the existence of solutions of the fractional-type problem dαx(t)dtα=λf(t,x(t)),α∈(0,1),t∈[1,e],x(1)+bx(e)=h\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{d^\\alpha x(t)}{d t^\\alpha }= \\lambda f(t,x(t)), \\quad \\alpha \\in (0,1),~~t \\in [1,e], \\quad x(1)+ b x(e)=h \\end{aligned}$$\\end{document}with certain constants lambda , b, where h in E and f: [1,e]times E rightarrow E is Pettis integrable function such that, for every varphi in E^*, varphi f lies in an appropriate Orlicz spaces. Here frac{d^alpha }{d t^alpha } stands the Caputo–Hadamard fractional differential operator.

Vectorial Ekeland Variational Principle and Cyclically Antimonotone Equilibrium Problems

In this article, we extend the cyclic antimonotonicity from scalar bifunctions to vector bifunctions. We find out a cyclically antimonotone vector bifunction can be regarded as a family of cyclically antimonotone scalar bifunctions. Using a pre-order principle (see Qiu, 2014), we prove a new version of Ekeland variational principle (briefly, denoted by EVP), which is quite different from the previous ones, for the objective function consists of a family of scalar functions. From the new version, we deduce several vectorial EVPs for cyclically antimonotone equilibrium problems, which extend and improve the previous results. By developing the original method proposed by Castellani and Giuli, we deduce a number of existence results (no matter scalar-valued case, or vector-valued case), when the feasible set is a sequentially compact topological space or a countably compact topological space. Finally, we propose a general coercivity condition. Combining the general coercivity condition and the obtained existence results with compactness conditions, we obtain several existence results for equilibrium problems in noncompact settings.

A note on the behavior of the Dunkl maximal operator

Abstract This note is a contribution to the Proceedings of the Conference of the Tunisian Mathematical Society CSMT 2017. After briefly revisiting the case of the standard Hardy–Littlewood maximal operator, we will discuss the behavior of the Dunkl maximal operator in both the scalar and vector-valued cases.

An Efficient Algorithm for Matrix-Valued and Vector-Valued Optimal Mass Transport

We present an efficient algorithm for recent generalizations of optimal mass transport theory to matrix-valued and vector-valued densities. These generalizations lead to several applications including diffusion tensor imaging, color image processing, and multi-modality imaging. The algorithm is based on sequential quadratic programming. By approximating the Hessian of the cost and solving each iteration in an inexact manner, we are able to solve each iteration with relatively low cost while still maintaining a fast convergence rate. The core of the algorithm is solving a weighted Poisson equation, where different efficient preconditioners may be employed. We utilize incomplete Cholesky factorization, which yields an efficient and straightforward solver for our problem. Several illustrative examples are presented for both the matrix and vector-valued cases.