Shifted Jacobi multiplier sequences and transplantation between sine and cosine series

It is shown that the multiplier norm of a shifted Jacobi multiplier sequence can be estimated by the (same) multiplier norm of the original sequence uniformly with respect to the shift. Muckenhoupt’s transplantation theorem for Jacobi series is used essentially, for which also a functional analytic understanding is given in terms of the minimality of the Jacobi system in weighted L ^p -spaces.      

Inverse Jacobi multipliers

Inverse Jacobi multipliers are a natural generalization of inverse integrating factors ton-dimensional dynamical systems. In this paper, the corresponding theory is developed from its beginning in the formal methods of integration of ordinary differential equations and the “last multiplier” of K. G. Jacobi. We explore to what extent the nice properties of the vanishing set of inverse integrating factors are preserved in then -dimensional case. In particular, vanishing on limit cycles (in restricted sense) of an inverse Jacobi multiplier is proved by resorting to integral invariants. Extensions of known constructions of inverse integrating factors by means of power series, local Lie Groups and algebraic solutions are provided for inverse Jacobi multipliers as well as a suitable generalization of the concept to systems with discontinuous right-hand side.     

λ-Symmetries for the Reduction of Continuous and Discrete Equations

In this work we present results on the construction of λ-symmetries for ordinary differential equation using ideas derived from the notion of nonlocal symmetries and Jacobi last multiplier. We then apply the results obtained to the case of ordinary difference equations.     

On Lagrangians and Hamiltonians of some fourth-order nonlinear Kudryashov ODEs

We derive the Lagrangians of the reduced fourth-order ordinary differential equations studied by Kudryashov, when they satisfy the conditions stated by Fels [Fels ME, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations. Trans Am Math Soc 1996;348:5007–29] using Jacobi’s last multiplier technique. In addition the Hamiltonians of these equations are derived via Jacobi–Ostrogradski’s theory. In particular, we compute the Lagrangians and Hamiltonians of fourth-order Kudryashov equations which pass the Painlevé test.     

Operators associated with the Jacobi semigroup

The aim of this paper is to introduce some operators induced by the Jacobi differential operator and associated with the Jacobi semigroup, where the Jacobi measure is considered in the multidimensional case.In this context, we introduce potential operators, fractional integrals, fractional derivates, Bessel potentials and give a version of Carleson measures.We establish a version of Meyer’s multiplier theorem and by means of this theorem, we study fractional integrals and fractional derivates.Potential spaces related to Jacobi expansions are introduced and using fractional derivates, we give a characterization of these spaces. A version of Calderon’s Reproduction Formula and a version of Fefferman’s theorem are given.Finally, we present a definition of Triebel–Lizorkin spaces and Besov spaces in the Jacobi setting.     

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.     

Multipliers on homogeneous Banach spaces with respect to Jacobi polynomials

A bounded linear operator is called multiplier with respect to Jacobi polynomials if and only if it commutes with all Jacobi translation operators on $[-1,1]$ . Multipliers on homogeneous Banach spaces on $[-1,1]$ determined by the Jacobi translation operator are introduced and studied. First we prove four equivalent characterizations of a multiplier for an arbitrary homogeneous Banach spaces $B$ on $[-1,1]$ . One of them implies the existence of an algebra isomorphism from the set of all multipliers on $B$ into the set of all pseudomeasures. Further, we study multipliers on specific examples of homogeneous Banach spaces on $[-1,1]$ . Amongst others, multipliers on the Wiener algebra, on the Beurling space and on Sobolev spaces are analyzed. We obtain that the multiplier spaces of the Wiener algebra, the Beurling space and of all Sobolev spaces are isometric isomorphic to each other. Furthermore, these multiplier spaces are all isometric isomorphic to the set of all pseudomeasures.     

The Integrability Conditions in the Inverse Problem of the Calculus of Variations for Second-Order Ordinary Differential Equations

A novel approach to a coordinate-free analysis of the multiplier question in the inverseproblem of the calculus of variations, initiated in a previous publication, is completed in thefollowing sense: under quite general circumstances, the complete set of passivity or integrabilityconditions is computed for systems with arbitrary dimension n. The results are appliedto prove that the problem is always solvable in the case that the Jacobi endomorphism of thesystem is a multiple of the identity. This generalizes to arbitrary n a result derived byDouglas for n=2.     

Inverse problem of the calculus of variations on Lie groups

This article studies the inverse problem of the calculus of variations for the special case of the geodesic flow associated to the canonical symmetric bi-invariant connection of a Lie group. Necessary background on the differential geometric structure of the tangent bundle of a manifold as well as the Fröhlicher-Nijenhuis theory of derivations is introduced briefly. The first obstructions to the inverse problem are considered in general and then as they appear in the special case of the Lie group connection. Thereafter, higher order obstructions are studied in a way that is impossible in general. As a result a new algebraic condition on the variational multiplier is derived, that involves the Nijenhuis torsion of the Jacobi endomorphism. The Euclidean group of the plane is considered as a working example of the theory and it is shown that the geodesic system is variational by applying the Cartan-Kähler theorem. The same system is then reconsidered locally and a closed form solution for the variational multiplier is obtained. Finally some more examples are considered that point up the strengths and weaknesses of the theory.     

Travelling Wave Solutions and Conservation Laws of the (2+1)-dimensional Broer-Kaup-Kupershmidt Equatio

The travelling wave solutions and conservation laws of the (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK) equation are considered in this paper. Under the travelling wave frame, the BKK equation is transformed to a system of ordinary differential equations (ODEs) with two dependent variables. Therefore, it happens that one dependent variable $u$ can be decoupled into a second order ODE that corresponds to a Hamiltonian planar dynamical system involving three arbitrary constants. By using the bifurcation analysis, we obtain the bounded travelling wave solutions $u$, which include the kink, anti-kink and periodic wave solutions. Finally, the conservation laws of the BBK equation are derived by employing the multiplier approach.     

Under which conditions is the Jacobi space L_{w^{(a,b)} }^p [ - 1,1] subset of L_{w^{(\alpha ,\beta )} }^1 [ - 1,1] ?

Exact conditions for α, β, a, b > ?1 and 1 ≤ p ≤ ∞ are determined under which the inclusion property $L_{w^{(a,b)} }^p [ - 1,1]$ ? $L_{w^{(\alpha ,\beta )} }^1 [ - 1,1]$ is valid. It is shown that the conditions characterize the inclusion property. The paper concludes with some results, in which the inclusion property can be detected in relation with estimates of Jacobi differential operators and with Muckenhoupt’s transplantation theorems and multiplier theorems for Jacobi series.     

Two-weight Hilbert transform and Lipschitz property of Jacobi matrices associated to hyperbolic polynomials

We are going to prove a Lipschitz property of Jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical Perron-Frobenius-Ruelle operators associated to hyperbolic polynomial dynamics. This Lipschitz estimate will not depend on the dimension of the Jacobi matrix. It is obtained using some sufficient conditions for two-weight boundedness of the Hilbert transform. It has been proved in [F. Peherstorfer, A. Volberg, P. Yuditskii, Limit periodic Jacobi matrices with prescribed p-adic hull and a singular continuous spectrum, Math. Res. Lett. 13 (2-3) (2006) 215-230] for all polynomials with sufficiently big hyperbolicity and in the most symmetric case t=0 that the Lipschitz estimate becomes exponentially better when the dimension of the Jacobi matrix grows. This allows us to get for such polynomials the solution of a problem of Bellissard, in other words, to prove the limit periodicity of the limit Jacobi matrix. We suggest a scheme how to approach Bellissard's problem for all hyperbolic dynamics by uniting the methods of the present paper and those of [F. Peherstorfer, A. Volberg, P. Yuditskii, Limit periodic Jacobi matrices with prescribed p-adic hull and a singular continuous spectrum, Math. Res. Lett. 13 (2-3) (2006) 215-230]. On the other hand, the nearness of Jacobi matrices under consideration in operator norm implies a certain nearness of their canonical spectral measures. One can notice that this last claim just gives us the classical commutative Perron-Frobenius-Ruelle theorem (it is concerned exactly with the nearness of such measures). In particular, in many situations we can see that the classical Perron-Frobenius-Ruelle theorem is a corollary of a certain non-commutative observation concerning the quantitative nearness of pertinent Jacobi matrices in operator norm.     

Reduction of fourth order ordinary differential equations to second and third order Lie linearizable forms

Meleshko presented a new method for reducing third order autonomous ordinary differential equations (ODEs) to Lie linearizable second order ODEs. We extended his work by reducing fourth order autonomous ODEs to second and third order linearizable ODEs and then applying the Ibragimov and Meleshko linearization test for the obtained ODEs. The application of the algorithm to several ODEs is also presented.     

A Note on Parallel Preconditioning for the All-at-Once Solution of Riesz Fractional Diffusion Equations

The $p$-step backward difference formula (BDF) for solving systems of ODEs can be formulated as all-at-once linear systems that are solved by parallel-in-time preconditioned Krylov subspace solvers (see McDonald et al. [36] and Lin and Ng [32]). However, when the BDF$p$ (2 ≤ $p$ ≤ 6) method is used to solve time-dependent PDEs, the generalization of these studies is not straightforward as $p$-step BDF is not selfstarting for $p$ ≥ 2. In this note, we focus on the 2-step BDF which is often superior to the trapezoidal rule for solving the Riesz fractional diffusion equations, and show that it results into an all-at-once discretized system that is a low-rank perturbation of a block triangular Toeplitz system. We first give an estimation of the condition number of the all-at-once systems and then, capitalizing on previous work, we propose two block circulant (BC) preconditioners. Both the invertibility of these two BC preconditioners and the eigenvalue distributions of preconditioned matrices are discussed in details. An efficient implementation of these BC preconditioners is also presented, including the fast computation of dense structured Jacobi matrices. Finally, numerical experiments involving both the one- and two-dimensional Riesz fractional diffusion equations are reported to support our theoretical findings.     

Optimum Modified Extrapolated Jacobi Method for Consistently Ordered Matrices

1.IlltroductionandPreliminariesWeconsiderthelineaxsystemAx=b,(1.1)wheteAERn,",bERnanddet(A)/O.WeaJ8oassumethatAhasthef0rmwhereA11,A22axesquarenonsingular(usuallydiagonal)matrices-Asisknow[61,AisaconSisentlyordered2-cyclicmatris.Forsolving(1.1)weintendtousethef0lfowngsimpleiterativemethod:In(1.4)and(1.6),w1,w2arenonzeroparameters(extraP0lati0nparameters)andI1,I2areidelltitymatricesofthesamesizesasA11anA22respectively.Theconstructionofmeth0d(1.3)isbasedonthesplittingA=M-N,whereM=Dfl-1…     

Moments of the Hermitian Matrix Jacobi Process

In this paper, we compute the expectation of traces of powers of the Hermitian matrix Jacobi process for a large enough but fixed size. To proceed, we first derive the semi-group density of its eigenvalues process as a bilinear series of symmetric Jacobi polynomials. Next, we use the expansion of power sums in the Schur polynomial basis and the integral Cauchy–Binet formula in order to determine the partitions having nonzero contributions after integration. It turns out that these are hooks of bounded weight and the sought expectation results from the integral of a product of two Schur functions with respect to a generalized beta distribution. For special values of the parameters on which the matrix Jacobi process depends, the last integral reduces to the Cauchy determinant and we close the paper with the investigation of the asymptotic behavior of the resulting formula as the matrix size tends to infinity.     

Improved relaxations for the parametric solutions of ODEs using differential inequalities

A new method is described for computing nonlinear convex and concave relaxations of the solutions of parametric ordinary differential equations (ODEs). Such relaxations enable deterministic global optimization algorithms to be applied to problems with ODEs embedded, which arise in a wide variety of engineering applications. The proposed method computes relaxations as the solutions of an auxiliary system of ODEs, and a method for automatically constructing and numerically solving appropriate auxiliary ODEs is presented. This approach is similar to two existing methods, which are analyzed and shown to have undesirable properties that are avoided by the new method. Two numerical examples demonstrate that these improvements lead to significantly tighter relaxations than previous methods.     

Eichler integrals, period relations and Jacobi forms

This paper contains three main results: the first one is to derive two “period relations” and the second one is a complete characterization of period functions of Jacobi forms in terms of period relations. These are done by introducing a concept of “Jacobi integrals” on the full Jacobi group. The last one is to show, for the given holomorphic function P(τ, z) having two period relations, there exists a unique Jacobi integral, up to Jacobi forms, with a given function P(τ, z) as its period function. This is done by constructing a generalized Jacobi Poincaré series explicitly. This is to say that every holomorphic function with “period relations” is coming from a Jacobi integral. It is an analogy of Eichler cohomology theory studied in Knopp (Bull Am Math Soc 80:607–632, 1974) for the functions with elliptic and modular variables. It explains the functional equations satisfied by the “Mordell integrals” associated with the Lerch sums (Zwegers in Mock theta functions, PhD thesis, Universiteit Utrecht, 2002) or, more generally, with the higher Appell functions (Semikhatov et?al. in Commun Math Phys 255(2):469–512, 2005). Developing theories of Jacobi integrals with elliptic and modular variables in this paper is a natural extension of the Eichler integral with modular variable. Period functions can be explained in terms of the parabolic cohomology group as well.