On the boundary value problem for the Sturm–Liouville equation with the discontinuous coefficient

We consider a boundary value problem for the Sturm–Liouville equation with piecewise‐constant leading coefficient. We prove that some integral representations for the solutions of the considered equation can be obtained by using classical transformation operators for the Sturm–Liouville operator at the end points of a finite interval. We also investigate the spectral characteristics of the boundary value problem, prove the completeness and expansion theorem. Copyright © 2012 John Wiley & Sons, Ltd.     

Resolvent Operator and Self-Adjointness of Sturm–Liouville Operators with a Finite Number of Transmission Conditions

In this paper, we consider a Sturm–Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at a finite number of interior points. We introduce a Hilbert space formulation such that the problem under consideration can be interpreted as an eigenvalue problem for a suitable self-adjoint linear operator. We construct Green function of the problem and resolvent operator. We establish the self-adjointness of the discontinuous Sturm–Liouville operator.     

On the convexity coefficient of Musielak–Orlicz function spaces equipped with the Orlicz norm

In Hudzik and Landes, the convexity coefficient of Musielak–Orlicz function spaces over a non-atomic measure space equipped with the Luxemburg norm is computed whenever the Musielak–Orlicz functions are strictly convex see [6]. In this paper, we extend this result to the case of Musielak–Orlicz spaces equipped with the Orlicz norm. Also, a characterization of uniformly convex Musielak–Orlicz function spaces as well as k-uniformly convex Musielak–Orlicz spaces equipped with the Orlicz norm is given.     

Inverse problems for Sturm–Liouville operators with interior discontinuities and boundary conditions dependent on the spectral parameter

In this paper, we discuss the inverse problem for Sturm–Liouville operators with arbitrary number of interior discontinuities and boundary conditions having fractional linear function of spectral parameter on the finite interval [0,1]. Using Weyl function techniques, we establish some uniqueness theorems for the Sturm–Liouville operator. Copyright © 2012 John Wiley & Sons, Ltd.     

Inverse problems for the matrix Sturm–Liouville equation with a Bessel-type singularity

The matrix Sturm–Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. We consider inverse problems by the Weyl matrix and by the spectral data for this equation. Constructive solutions, based on the method of spectral mappings, are obtained for these inverse problems.     

Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition

The paper deals with the Sturm–Liouville eigenvalue problem with the Dirichlet boundary condition at one end of the interval and with the boundary condition containing entire functions of the spectral parameter at the other end. We study the inverse problem, which consists in recovering the potential from a part of the spectrum. This inverse problem generalizes partial inverse problems on finite intervals and on graphs and also the inverse transmission eigenvalue problem. We obtain sufficient conditions for global solvability of the studied inverse problem, which prove its local solvability and stability. In addition, application of our main results to the partial inverse Sturm–Liouville problem on the star-shaped graph is provided.     

On the Reconstruction of the Sturm-Liouville Problems with Spectral Parameter in the Discontinuity Conditions

In this paper, we are concerned with the problem of recovering the Sturm–Liouville problem under the circumstance of the discontinuity conditions involved spectral parameter at finite interior points of a finite interval. We provide procedures for constructing their potentials and boundary conditions either from the Weyl function, or from spectral data, or from two spectra in terms of the method of spectral mappings.     

Approximation in <Emphasis Type="Italic">L</Emphasis> <Subscript>2</Subscript> by partial integrals of the Fourier transform over the eigenfunctions of the Sturm–Liouville operator

For approximations in the space L₂(?₊) by partial integrals of the Fourier transform over the eigenfunctions of the Sturm–Liouville operator, we prove Jackson’s inequality with exact constant and optimal argument in the modulus of continuity. The optimality of the argument in the modulus of continuity is established using the Gauss quadrature formula on the half-line over the zeros of the eigenfunction of the Sturm–Liouville operator.     

Sturm–Liouville operators in the half axis with shifted potentials

We consider Sturm–Liouville operators in the half axis generated by shifts of the potential and prove that Lebesgue measure is equivalent to a measure defined as an average of spectral measures which correspond to these operators. This is then used to obtain results on stability of spectral types under change of parameters such as boundary conditions, local perturbations, and shifts. In particular if for a set of shifts of positive measure the corresponding operators have α-singular, singular continuous and (or) point spectrum in a fixed interval, then this set of shifts has to be unbounded. Moreover, there are large sets of boundary conditions and local perturbations for which the corresponding operators enjoy the same property.     

The logarithmic Sobolev inequality for the Wasserstein diffusion

We prove that the Dirichlet form associated with the Wasserstein diffusion on the set of all probability measures on the unit interval, introduced in von Renesse and Sturm (Entropic measure and Wasserstein diffusion. Ann Probab, 2008) satisfies a logarithmic Sobolev inequality. This implies hypercontractivity of the associated transition semigroup. We also study functional inequalities for related diffusion processes.     

Approximation in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> by Partial Integrals of the Multidimensional Fourier Transform over the Eigenfunctions of the Sturm–Liouville Operator

For approximations in the space L₂(?₊ ^d ) by partial integrals of the multidimensional Fourier transform over the eigenfunctions of the Sturm–Liouville operator, we prove the Jackson inequality with sharp constant and optimal argument in the modulus of continuity. The multidimensional weight that defines the Sturm–Liouville operator is the product of onedimensional weights. The one-dimensional weights can be, in particular, power and hyperbolic weights with various parameters. The optimality of the argument in the modulus of continuity is established by means of the multidimensional Gauss quadrature formula over zeros of an eigenfunction of the Sturm–Liouville operator. The obtained results are complete; they generalize a number of known results.     

Sturm–Liouville operators on time scales

We establish the connection between Sturm–Liouville equations on time scales and Sturm–Liouville equations with measure-valued coefficients. Based on this connection, we generalize several results for Sturm–Liouville equations on time scales, which have been obtained by various authors in the past.     

Equiconvergence of Spectral Decompositions for Sturm–Liouville Operators with a Distributional Potential in Scales of Spaces

Doklady Mathematics - We study the equiconvergence of spectral decompositions for two Sturm–Liouville operators on the interval [0, π] generated by the differential expressions...     

A Criterion of Convergence of Lagrange–Sturm–Liouville Processes in Terms of One-Sided Module of Variation

We obtain a criterion of uniform convergence inside the interval (0, π) of interpolation processes determined by eigenfunctions of the regular Sturm–Liouville problem with a continuous potential of bounded variation. The criterion is formulated in terms of one-sided modulus of variation.     

Pointwise behaviour of Orlicz–Sobolev functions

We study the regularity of Orlicz–Sobolev functions on metric measure spaces equipped with a doubling measure. We show that each Orlicz–Sobolev function is quasicontinuous and has Lebesgue points outside a set of capacity zero and that the discrete maximal operator is bounded in the Orlicz–Sobolev space. We also show that if the Hardy–Littlewood maximal operator is bounded in the Orlicz space $L^{\Psi}(X)We study the regularity of Orlicz–Sobolev functions on metric measure spaces equipped with a doubling measure. We show that each Orlicz–Sobolev function is quasicontinuous and has Lebesgue points outside a set of capacity zero and that the discrete maximal operator is bounded in the Orlicz–Sobolev space. We also show that if the Hardy–Littlewood maximal operator is bounded in the Orlicz space , then each Orlicz–Sobolev function can be approximated by a H?lder continuous function both in the Lusin sense and in norm. The research is supported by the Centre of Excellence Geometric Analysis and Mathematical Physics of the Academy of Finland.     

Self‐adjoint Sturm–Liouville problems with an infinite number of boundary conditions

We study Sturm–Liouville (SL) problems on an infinite number of intervals, adjacent endpoints are linked by means of boundary conditions, and characterize the conditions which determine self‐adjoint operators in a Hilbert space which is the direct sum of the spaces for each interval. These conditions can be regular or singular, separated or coupled. Furthermore, the inner products of the summand spaces may be multiples of the usual inner products with different spaces having different multiples. We also extend the GKN Theorem to cover the infinite number of intervals theory with modified inner products and discuss the connection between our characterization and the classical one with the usual inner products. Our results include the finite number of intervals case.     

Eigenvalue problems for Sturm Liouville equations with transmission conditions

In this study, we consider a Sturm Liouville type boundary-value problem with eigenparameter-dependent boundary conditions and with two supplementary transmission conditions at one inner point of a finite interval under consideration. We modify some techniques of classical Sturm Liouville theory and suggest a new approach for the investigation of eigenvalues and eigenfunctions of this type of boundary-value problem.     

Guaranteed Error Bounds for Eigenvalues of Singular Sturm‐Liouville Problems

We develop a simple oscillation theory for singular Sturm‐Liouville problems and combine it with recent asymptotic results, and with the AWA interval‐arithmetic code for integration of initial value problems with guaranteed error bounds, to obtain eigenvalue approximations with guaranteed error bounds for a class of singular Sturm‐Liouville problems. We believe that this is the first time that this has been achieved for singular eigenvalue problems.     

Linear Sturm–Liouville problems with general homogeneous linear multi‐point boundary conditions

In this paper, we study second order linear Sturm–Liouville problems involving one or two homogeneous linear multi‐point boundary conditions in the most general form. We obtain conditions for the existence of a sequence of positive eigenvalues with consecutive zero counts of the eigenfunctions. Furthermore, we reveal the interlacing relations between the eigenvalues of such Sturm–Liouville problems and those of Sturm–Liouville problems with certain two‐point separated boundary conditions.     

Using Time Scales to Study Multi-Interval Sturm–Liouville Problems with Interface Conditions

We consider a Sturm–Liouville problem defined on multiple intervals with interface conditions. The existence of a sequence of eigenvalues is established and the zero counts of associated eigenfunctions are determined. Moreover, we reveal the continuous and discontinuous nature of the eigenvalues on the boundary condition. The approach in this paper is different from those in the literature: We transfer the Sturm–Liouville problem with interface conditions to a Sturm–Liouville problem on a time scale without interface conditions and then apply the Sturm–Liouville theory for equations on time scales. In this way, we are able to investigate the problem in a global view. Consequently, our results cover the cases when the potential function in the equation is not strictly greater than zero and when the domain consists of an infinite number of intervals.