A Parseval-type theorem applied to certain integral transforms on generalized functions

In this paper the Parseval theorem for Laplace and Stieltjestransforms which was proved by Yurekli (1989, IMA J. Appl. Math.,42, 241–249) for conventional functions is proved forgeneralized functions. The theorem yields some corollaries similarto the corollaries in Yurekli (1989).     

A class of Nevanlinna functions related to singular Sturm-Liouville problems

The class of Nevanlinna functions consists of functions which are holomorphic off the real axis, which are symmetric with respect to the real axis, and whose imaginary part is nonnegative in the upper halfplane. The Kac subclass of Nevanlinna functions is defined by an integrability condition on the imaginary part. In this note a further subclass of these Kac functions is introduced. It involves an integrability condition on the modulus of the Nevanlinna functions (instead of the imaginary part). The characteristic properties of this class are investigated. The definition of the new class is motivated by the fact that the Titchmarsh-Weyl coefficients of various classes of Sturm-Liouville problems (under mild conditions on the coefficients) actually belong to this class.

     

Eigenvalues of periodic Sturm-Liouville problems by the Shannon-Whittaker sampling theorem

We are concerned with the computation of eigenvalues of a periodic Sturm-Liouville problem using interpolation techniques in Paley-Wiener spaces. We shall approximate the Hill discriminant by sampling a few of its values and then find its zeroes which are the square roots of the eigenvalues. Computable error estimates are provided together with eigenvalue enclosures.

     

On sampling theory associated with the resolvents of singular Sturm-Liouville problems

This paper is concerned with the sampling theory associated with resolvents of eigenvalue problems. We introduce sampling representations for integral transforms whose kernels are Green's functions of singular Sturm-Liouville problems provided that the singular points are in the limit-circle situation, extending the results obtained in the regular problems.

     

Global inversion theorem to unique solvability of Dirichlet problems

Banach-Mazur-Caccioppoli global inversion theorem is applied to obtain a generalization of a result due to Ambrosetti and Prodi concerning unique solvability of a Dirichlet problem for a second-order differential equation.     

Paley–Wiener and Boas theorems for singular Sturm–Liouville integral transforms

This paper deals with a class of integral transforms arising from a singular Sturm–Liouville problem y″−q(x)y=−λy, x(a,b), in the limit-point case at one end or both ends of the interval (a,b). The paper completely solves the problem of characterization of the image of a function that has compact support (Paley–Wiener theorem) and also of a function that vanishes on some interval (Boas problem) under this class of transforms. The characterizations are obtained with no restriction on q(x) other than being locally integrable.     

An integral transform associated to the Poisson integral and inversion of Flett potentials

We introduce a new weighted wavelet-like transform, generated by the Poisson integral and a “wavelet measure.” By making use of the relevant Calderón-type reproducing formula, we obtain an explicit inversion formula for the Flett potentials which are interpreted as negative fractional powers of the operator (E+Λ), where Λ=(−Δ)^1/2, Δ is the Laplacian and E is the identity operator.     

EXISTENCE OF SOLUTIONS OF STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS FOR NONLINEAR SECOND ORDER IMPULSIVE DIFFERENTIAL EQUATIONS IN BANACH SPACES

In this paper, the fixed point theorem is applied to investigate the existence of solutions of Sturm-Liouville boundary value problems for nonlinear second order impulsive differential equations in Banaeh spaces.     

Existence of positive solutions for Sturm-Liouville boundary value problems on the half-line

In this paper, we establish sufficient conditions to guarantee the existence of at least one positive solution, a unique positive solution, and multiple positive solutions for the Sturm-Liouville boundary value problem on the half-line. By using an effective operator, the fixed point theorems in cone, especially Krasnoselskii fixed point theorem, can be applied to such systems and then existence criteria are established. The interesting point of the results is that the nonlinear term f can be sign-changing.     

Criteria of the three cases for non-self-adjoint singular Sturm-Liouville difference equations

This paper is concerned with the criteria of the three cases for non-self-adjoint singular Sturm-Liouville difference equations. It has been known that this class of equations was classified into cases I, II and III by R.H. Wilson (Proc. R. Soc. A 401 (2005)). In this paper, several criteria of the cases I, II and III are established, respectively.     

New expressions of the modified generalized integral transform via the translation theorem with applications

In this paper, we obtain very natural basic formulas for the modified generalized integral transform (MGIT) on function space. In order to do this, we first introduce an MGIT of functionals on function space. We next establish some basic formulas with respect to the MGIT and the first variation. Finally, we obtain a new version of the Cameron–Storvick theorem via the translation theorem. Some applications are demonstrated as examples which are used in classification of nanoparticles.     

On extremal problems related to integral transforms of a class of analytic functions

For γ?0 and β<1 given, let Pγ(β) denote the class of all analytic functions f in the unit disk with the normalization f(0)=f^′(0)−1=0 and satisfying the condition

     

A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Dunkl transform

A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Dunkl transform on the real line is proved. To this end we state, in terms of Bessel functions, an orthonormal system which is complete in . This orthonormal system is a generalization of the classical exponential system defining Fourier series.

     

Estimates for an oscillatory integral operator related to restriction to space curves

We consider the oscillatory integral operator defined by

where 1$">, and is a real-valued function in . This operator may be thought of as a variable-curve version of the adjoint of the Fourier restriction operator for space curves. Under a certain nondegeneracy condition on , we obtain estimates for with a suitable bound for the operator norm . This generalizes a result of Hörmander for the plane to higher dimensions.

     

A parallel version of a fast algorithm for singular integral transforms

The mathematical foundation of an algorithm for fast and accurate evaluation of singular integral transforms was given by Daripa [9,10,12]. By construction, the algorithm offers good parallelization opportunities and a lower computational complexity when compared with methods based on quadrature rules. In this paper we develop a parallel version of the fast algorithm by redefining the inherently sequential recurrences present in the original sequential formulation. The parallel version only utilizes a linear neighbor-to-neighbor communication path, which makes the algorithm very suitable for any distributed memory architecture. Numerical results and theoretical estimates show good parallel scalability of the algorithm. This revised version was published online in June 2006 with corrections to the Cover Date.     

Existence results for non-linear singular integral equations with hilbert kernel in banach spaces

A class of non-linear singular integral equations with Hilbert kernel and a related class of quasi-linear singular integro-differential equations are investigated by applying Schauder's fixed point theorem in Banach spaces.