On the summability of Jacobi series at Lebesgue points

In the case -1/2 < <1/2 and >-1/2, conditions are obtained in terms of the matrix of a linear method of summability and, among others, an antipole condition, which ensure the convergence of the linear means of the Fourier–Jacobi means of an integrable function at the Lebesgue point x=1. In the case -1/2<<1/2 and -1 1/2, it is proved that the linear means of the Fourier–Jacobi series converge at the Lebesgue point x=1, without any additional antipole condition.     

Nonnegative Fourier–Jacobi Coefficients and Some Classes of Functions

Fourier–Jacobi series with nonnegative Fourier–Jacobi coefficients are considered. Under special restrictions on the Jacobi weight function, we establish in terms of Fourier–Jacobi coefficients a necessary and sufficient condition in order that the sum of the Fourier–Jacobi series should possess certain structural properties.     

Approximate Properties of the de la Vallee Poussin Means for the Discrete Fourier-Jacobi Sums

We consider the system of the classical Jacobi polynomials of degree at most N which generate an orthogonal system on the discrete set of the zeros of the Jacobi polynomial of degree N. Given an arbitrary continuous function on the interval [-1,1], we construct the de la Vallee Poussin-type means for discrete Fourier–Jacobi sums over the orthonormal system introduced above. We prove that, under certain relations between N and the parameters in the definition of de la Vall'ee Poussin means, the latter approximate a continuous function with the best approximation rate in the space C[-1,1] of continuous functions.     

Convergence rate of Fourier–Laplace series of L-functions

The almost everywhere convergence rates of Fourier–Laplace series are given for functions in certain subclasses of L²(Σ_n−1) defined in terms of moduli of continuity.     

Fourier–Jacobi Type Spherical Functions for Discrete Series Representations of Sp(2, R)

In this paper we define a kind of generalized spherical functions on Sp(2, R). We call it Fourier–Jacobi type, since it can be considered as a generalized Whittaker model associated with the Jacobi maximal parabolic subgroup. Also we give the multiplicity theorem and an explicit formula of these functions for discrete series representations of Sp(2, R).     

The Influence of Singular Points of the Laplace Image on the Generalized Spectrum of the Original

Integral representations of the Fourier–Jacobi coefficients are constructed in the form of Riemann–Mellin integrals, and on this basis asymptotic formulas for calculating coefficients for large-order numbers are derived.     

On the mean convergence of Fourier–Jacobi series

The convergence of Fourier–Jacobi series in the spaces L _p,A,B is studied in the case where the Lebesgue constants are unbounded.     

Fourier-Jacobi Type Spherical Functions for PJ-Principal Series Representations of Sp(2, R)

The paper studies a generalized spherical function, or a generalizedWhittaker model for generalized principal series representationsof G=Sp(2, R) induced from the Jacobi maximal parabolic subgroupP_J, which is called the Fourier–Jacobi type. In particular,a multiplicity theorem and an explicit formula via the MeijerG-functions for this function are given.     

Fourier–Bessel-type series: The fourth-order case

The structured higher-order Bessel-type linear ordinary differential equations were first discovered in 1994. There is a denumerable infinity of these higher-order equations, all of then of even-order.These differential equations possess many of the properties of the classical second-order Bessel differential equation, but these higher-order cases bring remarkable new analytic structures. In many ways it is sufficient to study the properties of the fourth-order Bessel-type differential equation to be able to assess the corresponding properties of the sixth-and higher-order cases.This paper follows a number of earlier papers devoted to the study of the fourth-order case. These publications show the connections between the special function properties of solutions of the differential equation, and the properties of linear differential operators generated by the associated linear differential expression in certain weighted Lebesgue, and Lebesgue–Stieltjes function spaces.To follow the earlier papers on the study of the fourth-order Bessel-type differential equation, this present paper determines the form of the Fourier–Bessel-type series which best extends the classical theory of the second-order Fourier–Bessel series.In fact the Fourier–Bessel-type series are based on a new orthogonal system in terms of the regular eigensolutions of the fourth-order Bessel-type equation. The corresponding eigenvalues are obtained by restricting the spectral parameter to the zeros of an analytic function arising already in the Dini boundary conditions.     

Expansion of functions in a Fourier-Chebyshev series by shifted Chebyshev polynomials of the first kind

The paper considers an algorithm that develops functions in a Fourier—Chebyshev series by shifted Chebyshev polynomials of the first kind. The relationship of the algorithm with Fourier transform and discrete cosine transform is established.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 25–27, 1992.     

Whittaker–Fourier Coefficients of Metaplectic Eisenstein Series

It is shown that the Fourier–Whittaker coefficients of Eisenstein series on the n-fold cover of GL(n) are L-functions, improving prior results of T. Suzuki.     

On a Banach space approximable by Jacobi polynomials

Let X represent either the space C[-1,1] L _p ^(α,β) (w), 1 ≦ p < ∞ on [-1, 1]. Then Xare Banach spaces under the sup or the p norms, respectively. We prove that there exists a normalized Banach subspace X ¹ _αβ of Xsuch that every f ∈ X ¹ _αβ can be represented by a linear combination of Jacobi polynomials to any degree of accuracy. Our method to prove such an approximation problem is Fourier–Jacobi analysis based on the convergence of Fourier–Jacobi expansions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Transferring Boundedness from Conjugate Operators Associated with Jacobi, Laguerre, and Fourier–Bessel Expansions to Conjugate Operators in the Hankel Setting

In this paper we establish transference results showing that the boundedness of the conjugate operator associated with Hankel transforms on Lorentz spaces can be deduced from the corresponding boundedness of the conjugate operators defined on Laguerre, Jacobi, and Fourier–Bessel settings. Our result also allows us to characterize the power weights in order that conjugation associated with Laguerre, Jacobi, and Fourier–Bessel expansions define bounded operators between the corresponding weighted L ^p spaces. This paper is partially supported by MTM2004/05878. Third and fourth authors are also partially supported by grant PI042004/067.     

Jacobi functions and Euler products for Hermitian modular forms

Hecke operators on spaces of Jacobi modular forms of the unitary group of genus n are investigated. Rational power series are constructed in terms of the Fourier-Jacobi coefficients of Hermitian forms. For modular forms of genus 2 one has obtained a representation of the nonstandard zeta function of Hermitian forms in terms of Dirichlet series, constructed from the Fourier-Jacobi coefficients, and one has proved the possibility of the analytic continuation of such series into the left half-plane.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova Akademii Nauk SSSR, Vol. 183, pp. 77–123, 1990.     

Interpolation approximations based on Gauss–Lobatto–Legendre–Birkhoff quadrature

We derive in this paper the asymptotic estimates of the nodes and weights of the Gauss–Lobatto–Legendre–Birkhoff (GLLB) quadrature formula, and obtain optimal error estimates for the associated GLLB interpolation in Jacobi weighted Sobolev spaces. We also present a user-oriented implementation of the pseudospectral methods based on the GLLB quadrature nodes for Neumann problems. This approach allows an exact imposition of Neumann boundary conditions, and is as efficient as the pseudospectral methods based on Gauss–Lobatto quadrature for PDEs with Dirichlet boundary conditions.     

Lagrange interpolation polynomials and orthogonal Fourier-Jacobi series

Let >–1 and > –1. Then a function f(x), continuous on the segment [–1; 1], exists such that the sequence of Lagrange interpolation polynomials constructed from the roots of Jacobi polynomials diverges almost everywhere on [–1; 1], and, at the same time, the Fourier-Jacobi series of function f(x) converges uniformly to f(x) on any segment [a; b] (1; 1).Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 215–226, August, 1976.     

Some integral transforms with the hypergeometric function F4(α, β, γ, δ; x,y)

We investigate one of the most efficient methods for solving differential equations and boundary-value problems —the integral transform method. The properties of the Jacobi polynomial are used to construct a new integral transform with the hypergeometric function F ₄ in the kernel.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 26–30, 1986.     

A New Family of Cyclic Difference Sets with Singer Parameters in Characteristic Three

We construct a new family of cyclic difference sets with parameters ((3 ^d – 1)/2, (3 ^{d – 1} – 1)/2, (3 ^{d – 2} – 1)/2) for each odd d. The difference sets are constructed with certain maps that form Jacobi sums. These new difference sets are similar to Maschietti's hyperoval difference sets, of the Segre type, in characteristic two. We conclude by calculating the 3-ranks of the new difference sets.     

An Expansion Formula for the Askey–Wilson Function

The Askey–Wilson function transform is a q-analogue of the Jacobi function transform with kernel given by an explicit non-polynomial eigenfunction of the Askey–Wilson second order q-difference operator. The kernel is called the Askey–Wilson function. In this paper an explicit expansion formula for the Askey–Wilson function in terms of Askey–Wilson polynomials is proven. With this expansion formula at hand, the image under the Askey–Wilson function transform of an Askey–Wilson polynomial multiplied by an analogue of the Gaussian is computed explicitly. As a special case of these formulas a q-analogue (in one variable) of the Macdonald–Mehta integral is obtained, for which also two alternative, direct proofs are presented.     

Determining radii of meromorphy via orthogonal polynomials on the unit circle

Using a convergence theorem for Fourier–Padé approximants constructed from orthogonal polynomials on the unit circle, we prove an analogue of Hadamard's theorem for determining the radius of m-meromorphy of a function analytic on the unit disk and apply this to the location of poles of the reciprocal of Szeg functions.