Jacobi Expansions Research Articles

Sharp Hardy’s Inequality for Orthogonal Expansions in Hp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^p$$\\end{document} Spaces

Hardy’s inequality on Hp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^p$$\\end{document} spaces, p∈(0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in (0,1]$$\\end{document}, in the context of orthogonal expansions is investigated for general bases on a wide class of domains in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^d$$\\end{document} with Lebesgue measure. The obtained result is applied to various Hermite, Laguerre, and Jacobi expansions. For that purpose some delicate estimates of the higher order derivatives for the underlying functions and of the associated heat or Poison kernels are proved. Moreover, sharpness of studied Hardy’s inequalities is justified by a construction of an explicit counterexample, which is adjusted to all considered settings.

Discrete harmonic analysis associated with Jacobi expansions III: The Littlewood–Paley–Stein [formula omitted]-functions and the Laplace type multipliers

The research about harmonic analysis associated with Jacobi expansions carried out in Arenas et al. (2020) and Arenas et al. (2022) is continued in this paper. Given the operator J(α,β)=J(α,β)−I, where J(α,β) is the three-term recurrence relation for the normalized Jacobi polynomials and I is the identity operator, we define the corresponding Littlewood–Paley–Stein gk(α,β)-functions associated with it and we prove an equivalence of norms with weights for them. As a consequence, we deduce a result for Laplace type multipliers.

Coefficient multipliers in the Hardy space associated with Jacobi expansions

In this paper a multiplier theorem in the Hardy space H 1 ( T ) H^1(\mathbb {T}) associated with Jacobi expansions of exponential type is proved, that is, a bilateral sequence { λ n } n = − ∞ ∞ \left \{\lambda _n\right \}_{n=-\infty }^{\infty } is a multiplier from H 1 ( T ) H^1(\mathbb {T}) into the sequence space ℓ 1 ( Z ) \ell ^1(\mathbb {Z}) associated with Jacobi expansions of exponential type, if \[ sup N ∑ k = 1 ∞ ( ∑ k N > | j | ≤ ( k + 1 ) N | λ j | ) 2 > ∞ . \sup _N\sum _{k=1}^{\infty }\left (\sum _{kN>|j|\le (k+1)N}|\lambda _j|\right )^2>\infty . \] This is a generalization of a multiplier theorem on usual Fourier expansions in the Hardy space H 1 ( T ) H^1(\mathbb {T}) , and for λ n = ( | n | + 1 ) − 1 \lambda _n=(|n|+1)^{-1} , a Hardy type inequality for Jacobi expansions is immediate which has ever been proved by Kanjin and Sato [Math. Inequal. Appl. 7 (2004), pp. 551–555].

Refined multilayered beam, plate and shell elements based on Jacobi polynomials

In this paper, theories of structures based on hierarchical Jacobi expansions are explored for the static analysis of multilayered beams, plates and shells. They belong to the family of classical orthogonal polynomials. This expansion is employed in the framework of the Carrera Unified Formulation (CUF), which allows to generate finite element stiffness matrices in a straightforward way. CUF allows also to employ both layer-wise and equivalent single layer approaches in order to obtain the desired degree of precision and computational cost. In this work, CUF is exploited for the analysis of one-dimensional beams and two-dimensional plates and shells, and several case studies from the literature are analysed. Displacements, in-plane, transverse and shear stresses are shown. In particular, for some benchmarks, the shear stresses are calculated using the constitutive relations and the stress recovery technique. The obtained results clearly show the convenience of using equivalent single layer models when calculating displacements, in-plane stresses and shear stresses recovered by three-dimensional indefinite equilibrium equations. On the other hand, layer-wise models are able to accurately predict the structural behaviour, even though higher degrees of freedom are needed.

Space-time fractional diffusion equation associated with Jacobi expansions

We consider the operator , where is the three term recurrence relation for the Jacobi polynomial normalized at the point x = 1 and I is the identity operator acting on . Knowing that for , these polynomials satisfy a product formula which provides a hypergroup structure on , we use the tools of harmonic analysis on this hypergroup to solve the space-time fractional diffusion equation Here, , is the Caputo time fractional derivative of order σ. The solution is given by , being the fundamental solution expressed in terms of Mittag-Leffler function and is the convolution on the hypergroup . We prove that, for , is nonnegative and we give some of its properties. Next, we study the one parameter operators associated with this fundamental solution.

On some free algebras of orthogonal modular forms II

In this paper we construct 16 free algebras of modular forms on type IV symmetric domains for some reflection groups related to the eight rescaled root lattices \(A_1(2)\), \(A_1(3)\), \(A_1(4)\), \(2A_1(2)\), \(A_2(2)\), \(A_2(3)\), \(A_3(2)\), \(D_4(2)\). As a corollary, we prove the modularity of formal Fourier–Jacobi expansions for these reflection groups.

Genuinely sharp heat kernel estimates on compact rank-one symmetric spaces, for Jacobi expansions, on a ball and on a simplex

We prove genuinely sharp two-sided global estimates for heat kernels on all compact rank-one symmetric spaces. This generalizes the authors' recent result obtained for a Euclidean sphere of arbitrary dimension. Furthermore, similar heat kernel bounds are shown in the context of classical Jacobi expansions, on a ball and on a simplex. These results are more precise than the qualitatively sharp Gaussian estimates proved recently by several authors.

Discrete Harmonic Analysis Associated with Jacobi Expansions II: the Riesz Transform

The present work is the continuation of our study (Arenas et al. J. Math. Anal. Appl. 490(123996), 21, 2020) on discrete harmonic analysis related to Jacobi expansions. The role of a Laplacian is played by the operator J(α,β) defined by the three-term recurrence relation for the normalised Jacobi polynomials. The main interest is to establish weighted inequalities for the Riesz transform associated with J(α,β). We make use of an appropriate discrete vector-valued local Calderon-Zygmund theory.

Molecular fingerprints based on Jacobi expansions of electron densities

Molecular fingerprint indices are derived from Jacobi expansion of molecular electron density in a ball. Electron density computed in the linear combination of atomic orbitals framework, at any computational level, is expanded in terms of suitable Jacobi polynomials times regular spherical harmonics. The procedure is applicable to calculations with either Gaussian or Slater-type orbitals. A very efficient algorithm previously reported for Canterakis–Zernike expansions is shown to be also applicable for this type of expansions. The procedure has been implemented in the DAMQT suite for the analysis of electron density, which facilitates the application to densities obtained with standard packages for molecular structure calculations. Fingerprints derived from Jacobi expansions are compared with other derived from Canterakis–Zernike expansions in some pharmacological molecules.

Transference of Lp bounds between symmetrized Jacobi expansions and Dunkl transform

We prove two theorems showing connections between multipliers for the symmetrized Jacobi expansions and multipliers for the Dunkl transform on $$\mathbb{R}$$ . These results can be seen as generalizations of the classical Igari’s result relating Jacobi and Hankel multipliers. Moreover, we show a transference theorem relating Jacobi and Dunkl transplantations.

Discrete harmonic analysis associated with Jacobi expansions I: The heat semigroup

In this paper we commence the study of discrete harmonic analysis associated with Jacobi orthogonal polynomials of order (α,β). Particularly, we analyse the heat semigroup Wt(α,β), t≥0, related to the operator J(α,β)−I, where J(α,β) is the three-term recurrence relation for the normalised Jacobi polynomials and I is the identity operator. First, we prove the positivity of the operator Wt(α,β) under some suitable restrictions on the parameters α and β. In our main result, we investigate the mapping properties of the maximal operator defined by the heat semigroup in weighted ℓp-spaces using discrete vector-valued local Calderón-Zygmund theory. Moreover, we treat the Poisson semigroup by means of an appropriate subordination identity.

Multipliers for Jacobi expansions in the Hardy spaces

The purpose of the paper is to study the coefficient multipliers of the Hardy spaces H^{p} associated with Jacobi expansions of exponential type. The main results are about the boundedness from H^{p} to ell ^{q} of the multiplier operators in terms of Jacobi expansions of exponential type for (i) p=1, 2le q<infty ; (ii) gamma ( alpha ,beta )^{-1}< p<1le q<infty , under appropriate conditions, where gamma (alpha ,beta )in (1,infty ] is a number depending on the parameters of the Jacobi system.

Fast algorithms for Jacobi expansions via nonoscillatory phase functions

AbstractWe describe a suite of fast algorithms for evaluating Jacobi polynomials, applying the corresponding discrete Sturm–Liouville eigentransforms and calculating Gauss–Jacobi quadrature rules. Our approach, which applies in the case in which both of the parameters $\alpha $ and $\beta $ in Jacobi’s differential equation are of magnitude less than $1/2$, is based on the well-known fact that in this regime Jacobi’s differential equation admits a nonoscillatory phase function that can be loosely approximated via an affine function over much of its domain. We illustrate this with several numerical experiments, the source code for which is publicly available.

Generalized Jacobi\u2013Weierstrass operators and Jacobi expansions

We present a realization for some K-functionals associated with Jacobi expansions in terms of generalized Jacobi–Weierstrass operators. Fractional powers of the operators as well as results concerning simultaneous approximation and Nikolskii–Stechkin type inequalities are also considered.

Approaching Bilinear Multipliers via a Functional Calculus

We propose a framework for bilinear multiplier operators defined via the (bivariate) spectral theorem. Under this framework, we prove Coifman–Meyer type multiplier theorems and fractional Leibniz rules. Our theory applies to bilinear multipliers associated with the discrete Laplacian on mathbb {Z}^d, general bi-radial bilinear Dunkl multipliers, and to bilinear multipliers associated with the Jacobi expansions.

On the use of Hahn’s asymptotic formula and stabilized recurrence for a fast, simple and stable Chebyshev–Jacobi transform

We describe a fast, simple, and stable transform of Chebyshev expansion coefficients to Jacobi expansion coefficients and its inverse based on the numerical evaluation of Jacobi expansions at the Chebyshev--Lobatto points. This is achieved via a decomposition of Hahn's interior asymptotic formula into a small sum of diagonally scaled discrete sine and cosine transforms and the use of stable recurrence relations. It is known that the Clenshaw--Smith algorithm is not uniformly stable on the entire interval of orthogonality. Therefore, Reinsch's modification is extended for Jacobi polynomials and employed near the endpoints to improve numerical stability.

Variable exponent Sobolev spaces associated with Jacobi expansions

In this paper we define variable exponent Sobolev spaces associated with Jacobi expansions. We prove that our generalized Sobolev spaces can be characterized as variable exponent potential spaces and as variable exponent Triebel-Lizorkin type spaces.

Fully Discrete Approximation of Parametric and Stochastic Elliptic PDEs

It has recently been demonstrated that locality of spatial supports in the parametrization of coefficients in elliptic PDEs can lead to improved convergence rates of sparse polynomial expansions of the corresponding parameter-dependent solutions. These results by themselves do not yield practically realizable approximations, since they do not cover the approximation of the arising expansion coefficients, which are functions of the spatial variable. In this work, we study the combined spatial and parametric approximability for elliptic PDEs with affine or lognormal parametrizations of the diffusion coefficients and corresponding Taylor, Jacobi, and Hermite expansions, to obtain fully discrete approximations. Our analysis yields convergence rates of the fully discrete approximation in terms of the total number of degrees of freedom. The main vehicle consists of $\ell^p$ summability results for the coefficient sequences measured in higher-order Hilbertian Sobolev norms. We also discuss similar results for no...

Fractal Jacobi Systems and Convergence of Fourier–Jacobi Expansions of Fractal Interpolation Functions

The fractal interpolation function (FIF) is a special type of continuous function on a compact subset of $${\mathbb{R}}$$ interpolating a given data set. They have been proved to be a very important tool in the study of irregular curves arising from financial series, electrocardiograms and bioelectric recording in general as an alternative to the classical methods. It is well known that Jacobi polynomials form an orthonormal system in $${\mathcal{L}^{2}(-1,1)}$$ with respect to the weight function $${\rho^{(r,s)}(x)=(1-x)^{r} (1+x)^{s}}$$ , $${r > -1}$$ and $${s > -1}$$ . In this paper, a fractal Jacobi system which is fractal analogous of Jacobi polynomials is defined. The Weierstrass type theorem providing an approximation for square integrable function in terms of $${\alpha}$$ -fractal Jacobi sum is derived. A fractal basis for the space of weighted square integrable functions $${\mathcal{L}_{\rho}^{2}(-1,1)}$$ is found. The Fourier–Jacobi expansion corresponding to an affine FIF (AFIF) interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is established. The closeness of the original function to the Fourier–Jacobi expansion of the AFIF is proved for certain scale vector. Finally, the Fourier–Jacobi expansion corresponding to a non-affine smooth FIF interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is investigated as well.

Riesz–Jacobi Transforms as Principal Value Integrals

We establish an integral representation for the Riesz transforms naturally associated with classical Jacobi expansions. We prove that the Riesz–Jacobi transforms of odd orders express as principal value integrals against kernels having non-integrable singularities on the diagonal. On the other hand, we show that the Riesz–Jacobi transforms of even orders are not singular operators. In fact they are given as usual integrals against integrable kernels plus or minus, depending on the order, the identity operator. Our analysis indicates that similar results, existing in the literature and corresponding to several other settings related to classical discrete and continuous orthogonal expansions, should be reinvestigated so as to be refined and in some cases also corrected.