Jacobi matrices for sums of weight functions

Orthogonal polynomials are conveniently represented by the tridiagonal Jacobi matrix of coefficients of the recurrence relation which they satisfy. LetJ ₁ andJ ₂ be finite Jacobi matrices for the weight functionsw ₁ andw ₂, resp. Is it possible to determine a Jacobi matrix \(\tilde J\) , corresponding to the weight functions \(\tilde w\) =w ₁+w ₂ using onlyJ ₁ andJ ₂ and if so, what can be said about its dimension? Thus, it is important to clarify the connection between a finite Jacobi matrix and its corresponding weight function(s). This leads to the need for stable numerical processes that evaluate such matrices. Three newO(n ²) methods are derived that “merge” Jacobi matrices directly without using any information about the corresponding weight functions. The first can be implemented using any of the updating techniques developed earlier by the authors. The second new method, based on rotations, is the most stable. The third new method is closely related to the modified Chebyshev algorithm and, although it is the most economical of the three, suffers from instability for certain kinds of data. The concepts and the methods are illustrated by small numerical examples, the algorithms are outlined and the results of numerical tests are reported.     

Efficient Jacobi-Gauss collocation method for solving initial value problems of Bratu type

In this paper, we propose the shifted Jacobi-Gauss collocation spectral method for solving initial value problems of Bratu type, which is widely applicable in fuel ignition of the combustion theory and heat transfer. The spatial approximation is based on shifted Jacobi polynomials J _n ^(α,β) (x) with α, β ∈ (?1, ∞), x ∈ [0, 1] and n the polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes. Illustrative examples have been discussed to demonstrate the validity and applicability of the proposed technique. Comparing the numerical results of the proposed method with some well-known results show that the method is efficient and gives excellent numerical results.     

Analysis of a collocation method for integrating rapidly oscillatory functions

A collocation method for approximating integrals of rapidly oscillatory functions is analyzed. The method is efficient for integrals involving Bessel functions J_v(rx) with a large oscillation frequency parameter r, as well as for many other one- and multi-dimensional integrals of functions with rapid irregular oscillations. The analysis provides a convergence rate and it shows that the relative error of the method is even decreasing as the frequency of the oscillations increases.     

Algorithm for the computation of Bessel function integrals

A fortran subroutine is given for the computation of integrals of the form ∫^c₀f(x)J_v(αx)dx, where v = 0, 1,…,10.     

Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields

It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric familiy of number fields. In this paper we consider the infinite parametric family of simplest quartic fields K generated by a root ξ of the polynomial P _t (x) = x ⁴ ? tx ³ ? 6x ² +tx+1, assuming that t > 0, t ≠ 3 and t ² +16 has no odd square factors. In addition to generators of power integral bases we also calculate the minimal index and all elements of minimal index in all fields in this family.     

Heckman-Opdam's Jacobi polynomials for the BCn root system and generalized spherical functions

We prove that the radial part of the Laplacian on the space of generalized spherical functions on the symmetric space GL(m+n)/GL(m)×GL(n) is the Sutherland differential operator for the root system BC_n and the radial parts of the differential operators corresponding to the higher Casimirs yield the integrals of the quantum Calogero-Moser system. It allows us to give a representation theoretical construction for the three parameter family of Heckman-Opdam's Jacobi polynomials for the BC_n root system.     

A note on the numerical evaluation of finite integrals of oscillatory functions

A method is described for the numerical evaluation of integrals of the form ∫ _?1 ¹ f(x)K(m,x)dx, wheref(x) is smooth in [?1,1], whileK(m,x) is highly oscillatory for large values ofm.     

A product quadrature algorithm by Hermite interpolation

This paper is concerned with numerical integration of ∫¹₋₁f(x)k(x)dx by product integration rules based on Hermite interpolation. Special attention is given to the kernel k(x) = e^iτx, with a view to providing high precision rules for oscillatory integrals. Convergence results and error estimates are obtained in the case where the points of integration are zeros of p_n(W; x) or of (1 − x²)p_n−2(W; x), where p_n(W; x), n = 0, 1, 2…, are the orthonormal polynomials associated with a generalized Jacobi weight W. Further, examples are given that test the performance of the algorithm for oscillatory weight functions.     

Symbolic and numerical computation on Bessel functions of complex argument and large magnitude

The Lanczos τ-method, with perturbations proportional to Faber polynomials, is employed to approximate the Bessel functions of the first kind J_v(z) and the second kind Y_v(z), the Hankel functions of the first kind H_v⁽¹⁾(z) and the second kind H_v⁽²⁾(z) of integer order v for specific outer regions of the complex plane, i.e. ¦z¦ ⩾ R for some R. The scaled symbolic representation of the Faber polynomials and the appropriate automated τ-method approximation are introduced. Both symbolic and numerical computation are discussed. In addition, numerical experiments are employed to test the proposed τ-method. Computed accuracy for J₀(z) and Y₀(z) for ¦z¦ ⩾ 8 are presented. The results are compared with those obtained from the truncated Chebyshev series approximations and with those of the τ-method approximations on the inner disk ¦z¦ ⩽ 8. Some concluding remarks and suggestions on future research are given.     

On the power of the zeros of Bessel functions

For ν≥0 let c_νk be the k-th positive zero of the cylinder functionC _v(t)=J _v(t)cosα-Y _v(t)sinα, 0≤α>π whereJ _ν(t) andY _ν(t) denote the Bessel functions of the first and the second kind, respectively. We prove thatC _v,k ^1+H(x) is convex as a function of ν, ifc _νk≥x>0 and ν≥0, whereH(x) is specified in Theorem 1.1.     

Pair correlation of roots of rational functions with rational generating functions and quadratic denominators

For any rational functions with complex coefficients A(z),B(z), and C(z), where A(z), C(z) are not identically zero, we consider the sequence of rational functions H _m (z) with generating function ∑H _m (z)t ^m =1/(A(z)t ²+B(z)t+C(z)). We provide an explicit formula for the limiting pair correlation function of the roots of $\prod_{m=0}^{n}H_{m}(z)$ , as n→∞, counting multiplicities, on certain closed subarcs J of a curve $\mathcal{C}$ where the roots lie. We give an example where the limiting pair correlation function does not exist if J contains the endpoints of $\mathcal{C}$ .     

Weighted Least Square Convergence of Lagrange Interpolation on the Unit Circle

In the paper, a result of Walsh and Sharma on least square convergence of Lagrange interpolation polynomials based on the n-th roots of unity is extended to Lagrange interpolation on the sets obtained by projecting vertically the zeros of (1-x)²=P ^(a,β) _n(x),a>0,β>0,(1-x)P^(a,β) _n(x),a>0,β>-1,(1+x)P P^(a,β) _n(x),a>-1,β0 and P^(a,β) _n(x),a>-1,β>-1, respectively, onto the unit circle, where P^(a,β) _n(x),a>-1,β>-1, stands for the n-th Jacobi polynomial. Moreover, a result of Saff and Walsh is also extended.     

On the zeros of derivatives of Bessel functions

In this paper we are interested in the behaviour respect tov of thekth positive zeroc′ _vk of the derivative of the general Bessel functionC _v(x)=J _v(x)cosα?Y _v(x)sinα, 0≤α<π, whereJ _v(x) andY _v(x) indicate the Bessel functions of first and second kind respectively. It is well known that forc′ _vk>∥v∥,c′ _vk increases asv increases. Here we prove several additional properties forc′ _vk. Our main result is thatc′ _vk is concave as a function ofv, whenc′ _vk>∥v∥>0. This implies the concavity ofc′ _vk for everyk=2,3, ?. In the case of the zerosJ′ _vk of _d ^dx J _v(x) we extend this property tok=1 for everyv≥0.     

Some generating functions of Jacobi polynomials

In this paper, Weisner’s group-theoretic method of obtaining generating functions is utilized in the study of Jacobi polynomialsP> _n ^(a,ß)(x) by giving suitable interpretations to the index (n) and the parameter (β) to find out the elements for constructing a six-dimensional Lie algebra.     

Nikol’skii inequality between the uniform norm and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>-norm with jacobi weight of algebraic polynomials on an interval

We study the Nikol’skii inequality for algebraic polynomials on the interval [?1, 1] between the uniform norm and the norm of the space L _q ^(α,β) , 1 ≤ q < ∞, with the Jacobi weight ?(α,β)(x) = (1 ? x) ^α (1 + x) ^β , α ≥ β > ?1. We prove that, in the case α > β ≥ ?1/2, the polynomial with unit leading coefficient that deviates least from zero in the space L _q ^(α+1,,β) with the Jacobi weight ? ^(α+1,β)(x) = (1?x) ^α+1(1+x) ^β is the unique extremal polynomial in the Nikol’skii inequality. To prove this result, we use the generalized translation operator associated with the Jacobi weight. We describe the set of all functions at which the norm of this operator in the space L _q ^(α,β) for 1 ≤ q < ∞ and α > β ≥ ?1/2 is attained.     

Convergent expansions of the Bessel functions in terms of elementary functions

We consider the Bessel functions J _ν (z) and Y _ν (z) for R ν > ?1/2 and R z ≥ 0. We derive a convergent expansion of J _ν (z) in terms of the derivatives of \((\sin z)/z\), and a convergent expansion of Y _ν (z) in terms of derivatives of \((1-\cos z)/z\), derivatives of (1 ? e ^?z )/z and Γ(2ν, z). Both expansions hold uniformly in z in any fixed horizontal strip and are accompanied by error bounds. The accuracy of the approximations is illustrated with some numerical experiments.     

Global inverse spectral problems for rational matrix functions

Given rational matrix functions ψ₁(λ) = I_m + C₁(λI_n1 − A₁)⁻¹B₁ and ψ₂(λ) = I_m + C₂(λI_n2 − A₂)⁻¹B₂ which are analytic and invertible on the unit circle, we characterize in terms of the operators A₁,B₁,C₁,A₂,B₂,C₂ when there exists a single rational matrix function W(λ) = I_m + C(λI_n − A)⁻¹B such that WH²_m^⊥ = ψ ₁H²_m^⊥and WH²_m = ψ₂H²_m. When this is the case, we give explicit formulae for A,B,C in terms of A₁,B₁,C₁,A₂,B₂,C₂. Applications include Wiener-Hopf factorization, J- inner-outer factorization, and coprime factorization. The results on J-inner-outer factorization have application to a model reduction problem for discrete time linear systems.     

Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices

The following method for integrating the Cauchy problem for a Toda lattice on the half-line is well known: to a solution u(t), t ∈, [0, ∞), of the problem, one assigns a self-adjoint semi-infinite Jacobi matrix J(t) whose spectral measure dπ(λ; t) undergoes simple evolution in time t. The solution of the Cauchy problem goes as follows. One writes out the spectral measure dπ(λ; 0) for the initial value u(0) of the solution and the corresponding Jacobi matrix J(0) and then computes the time evolution dπ(λ; t) of this measure. Using the solution of the inverse spectral problem, one reconstructs the Jacobi matrix J(t) from dπ(λ; t) and hence finds the desired solution u(t). In the present paper, this approach is generalized to the case in which the role of J(t) is played by a block Jacobi matrix generating a normal operator in the orthogonal sum of finite-dimensional spaces with spectral measure dπ(ζ; t) defined on the complex plane. Some recent results on the spectral theory of these normal operators permit one to use the integration method described above for a rather wide class of differential-difference nonlinear equations replacing the Toda lattice.     

Meromorphic functions sharing two small functions with its derivative

In this paper, we find all the forms of meromorphic functions f(z) that share the value 0 CM^∗, and share b(z)IM^∗ with g(z)=a₁(z)f(z)+a₂(z)f^′(z). And a₁(z), a₂(z) and b(z) (a₂(z),b(z)?0) be small functions with respect to f(z). As an application, we show that some of nonlinear differential equations have no transcendental meromorphic solution.     

The heat operator and mock Jacobi forms

We study a necessary and sufficient condition for Jacobi integrals of weight -r+\fracj2-r+\frac{j}{2}, r∈ℤ≥0, and index ℳ^(j) on ℋ×ℂ ^j to have a dual Jacobi form of weight r+\fracj2+2r+\frac{j}{2}+2 and index ℳ^(j). Such a meromorphic Jacobi integral with a dual Jacobi form is called a mock Jacobi form whose concept was first introduced by Zagier in Séminaire Bourbaki, 60éme année, 2006–2007, N° 986. In fact, we show the map L^r+1_M^(j)L^{r+1}_{\mathcal{M}^{(j)}} from the space of mock Jacobi forms to that of Jacobi forms is surjective by constructing the corresponding inverse image via Eichler integral of vector valued modular forms which are coming from the theta decomposition of Jacobi forms. We discuss Lerch sums as a typical example.