On the asymptotics of eigenfunctions of Sturm-Liouville operators with distribution potentials

We consider a Sturm-Liouville operator in the space L ₂[0, π] and derive asymptotic formulas for the eigenvalues and eigenfunctions of this operator for the case of Dirichlet-Neumann boundary conditions. The leading and second terms of the asymptotics are found in closed form.     

Theorems on asymptotics for singular Sturm-Liouville operators with various boundary conditions

We consider the Sturm-Liouville operator L(y) = ?d ² y/dx ² + q(x)y in the space L ₂[0, π], where the potential q(x) is a complex-valued distribution of the first order of singularity; i.e., q(x) = u′(x), u ∈ L ²[0, π]. (Here the derivative is understood in the sense of distributions.) We obtain asymptotic formulas for the eigenvalues and eigenfunctions of the operator in the case of the Neumann-Dirichlet conditions [y ^[1](0) = 0, y(π) = 0] and Neumann conditions [y ^[1](0) = 0, y ^[1](π) = 0] and refine similar formulas for all types of boundary conditions. The leading and second terms of asymptotics are found in closed form.     

Asymptotic analysis of non-self-adjoint Hill operators

We obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L _t (q) with a potential q ∈ L ₁[0,1] and t-periodic boundary conditions, t ∈ (?π, π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L ₂(?∞,∞) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator provided that the potential q satisfies sufficient conditions.     

Uniform asymptotics of the eigenvalues and eigenfunctions of the Dirac system with an integrable potential

We consider the Dirac operator on a finite interval with a potential belonging to some set X completely bounded in the space L₁[0, π] and with strongly regular boundary conditions. We derive asymptotic formulas for the eigenvalues and eigenfunctions of the operator; moreover, the constants occurring in the estimates for the remainders depend on the boundary conditions and the set X alone.     

Asymptotics of the spectrum of the Sturm-Liouville operator with regular boundary conditions

On the interval (0, π), we consider the spectral problem generated by the Sturm-Liouville operator with regular but not strongly regular boundary conditions. For an arbitrary potential q(x) ∈ L ₁ (0, π) [q(x) ∈ L ₂(0, π)], we establish exact asymptotic formulas for the eigenvalues of this problem.     

A mapping method in inverse Sturm-Liouville problems with singular potentials

In the space L ₂[0, π], the Sturm-Liouville operator L _D(y) = ?y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L ₂[0, π], i.e., q ∈ W ₂ ^?1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L _D is solved in the subspace of odd real functions σ(π/2 ? x) = ?σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.     

Equiconvergence of the trigonometric Fourier series and the expansion in eigenfunctions of the Sturm-Liouville operator with a distribution potential

We consider the Sturm-Liouville operator L = ?d ²/dx ² + q(x) with the Dirichlet boundary conditions in the space L ₂[0, π] under the assumption that the potential q(x) belongs to W ₂ ^?1 [0, π]. We study the problem of uniform equiconvergence on the interval [0, π] of the expansion of a function f(x) in the system of eigenfunctions and associated functions of the operator L and its Fourier sine series expansion. We obtain sufficient conditions on the potential under which this equiconvergence holds for any function f(x) of class L ₁. We also consider the case of potentials belonging to the scale of Sobolev spaces W ₂ ^?θ [0, π] with ½ < θ ≤ 1. We show that if the antiderivative u(x) of the potential belongs to some space W ₂ ^θ [0, π] with 0 < θ < 1/2, then, for any function in the space L ₂[0, π], the rate of equiconvergence can be estimated uniformly in a ball lying in the corresponding space and containing u(x). We also give an explicit estimate for the rate of equiconvergence.     

Inverse spectral problem for integro-differential operators

In this paper, we study the inverse spectral problem on a finite interval for the integro-differential operator ? which is the perturbation of the Sturm-Liouville operator by the Volterra integral operator. The potential q belongs to L ₂[0, π] and the kernel of the integral perturbation is integrable in its domain of definition. We obtain a local solution of the inverse reconstruction problem for the potential q, given the kernel of the integral perturbation, and prove the stability of this solution. For the spectral data we take the spectra of two operators given by the expression for ? and by two pairs of boundary conditions coinciding at one of the finite points.     

On the equiconvergence rate of trigonometric series expansions and eigenfunction expansions for the Sturm-Liouville operator with a distributional potential

We study the Sturm-Liouville operator L = ?d ²/dx ² + q(x) in the space L ₂[0, π] with the Dirichlet boundary conditions. We assume that the potential has the form q(x) = u′(x), u ∈ W ₂ ^θ [0, π], 0 < θ < 1/2. We consider the problem on the uniform (on the entire interval [0, π]) equiconvergence of the expansion of a function f(x) in a series in the system of root functions of the operator L with its Fourier expansion in the system of sines. We show that if the antiderivative u(x) of the potential belongs to any of the spaces W ₂ ^θ [0, π], 0 < θ < 1/2, then the equiconvergence rate can be estimated uniformly over the ball u(x) ∈ B _R = {v(x) ∈ W ₂ ^θ [0, π] | ∥v∥_{W 2} ^θ ≤ R} for any function f(x) ∈ L ₂[0, π].     

Sampling and recovery of multidimensional bandlimited functions via frames

In this paper, we investigate frames for L₂d[−π,π] consisting of exponential functions in connection to oversampling and nonuniform sampling of bandlimited functions. We derive a multidimensional nonuniform oversampling formula for bandlimited functions with a fairly general frequency domain. The stability of said formula under various perturbations in the sampled data is investigated, and a computationally manageable simplification of the main oversampling theorem is given. Also, a generalization of Kadec's 1/4 theorem to higher dimensions is considered. Finally, the developed techniques are used to approximate biorthogonal functions of particular exponential Riesz bases for L₂[−π,π], and a well-known theorem of Levinson is recovered as a corollary.     

On a new class of boundary value problems for the Sturm-Liouville operator

We consider the spectral problem generated by the Sturm-Liouville operator with complex-valued potential q(x) ∈ L ₂(0, π) and degenerate boundary conditions. We show that the set of potentials q(x) for which there exist associated functions of arbitrarily high order in the system of root functions is everywhere dense in L ₁(0, π).     

Equiconvergence of spectral decompositions for the Dirac system with potential in Lebesgue spaces

The problem of equiconvergence of spectral decompositions corresponding to the systems of root functions of two one-dimensional Dirac operators ?_P,U and ?_0,U with potential P summable on a finite interval and Birkhoff-regular boundary conditions is analyzed. It is proved that in the case of P ∈ L_?[0, π], ? ∈ (1,∞], equiconvergence holds for every function f ∈ L_μ[0, π], μ ∈ [1,∞], in the norm of the space L_ν[0, π], ν ∈ [1,∞], if the indices ?, μ, and ν satisfy the inequality 1/? + 1/μ ? 1/ν ≤ 1 (except for the case when ? = ν = ∞ and μ = 1). In particular, in the case of a square summable potential, the uniform equiconvergence on the interval [0, π] is proved for an arbitrary function f ∈ L₂[0, π].     

Generalization of Berg-Dimovski convolution in spaces of analytic functions

In the space H(G) of functions analytic in a ρ-convex region G equipped with the topology of compact convergence, we construct a convolution for the operator J _π+L where J _ρ is the generalized Gel’fond-Leont’ev integration operator and L is a linear continuous functional on H(G). This convolution is a generalization of the well-known Berg-Dimovski convolution. We describe the commutant of the operator J _π+L in ?(G) and obtain the representation of the coefficient multipliers of expansions of analytic functions in the system of Mittag-Leffler functions.     

Dirac system with potential lying in Besov spaces

We study the spectral properties of the Dirac operator L_P,U generated in the space (L₂[0, π])² by the differential expression By′ + P(x)y and by Birkhoff regular boundary conditions U, where y = (y₁, y₂)^t, \(B = \left( {\begin{array}{*{20}{c}} { - i}&0 \\ 0&i \end{array}} \right)\), and the entries of the matrix P are complexvalued Lebesgue measurable functions on [0, π]. We also study the asymptotic properties of the eigenvalues {λ_n}_n∈Z of the operator L_P,U as n → ∞ depending on the “smoothness” degree of the potential P; i.e., we consider the scale of Besov spaces B_1,∞^θ, θ ∈ (0, 1). In the case of strongly regular boundary conditions, we study the asymptotic behavior of the system of normalized eigenfunctions of the operator L_P,U, and in the case of regular but not strongly regular boundary conditions, we find the asymptotics of two-dimensional spectral projections.     

On the equivalence of the centered Gaussian measure in L 2 with the correlation operator (−d 2/dx 2)−1 and the conditional Wiener measure

Let w and µ be respectively the conditional Wiener measure in C ₀([0, 1]) and the centered Gaussian measure in L ₂[0, 1] with the correlation operator (?d ²/dx ²)^?1. We prove the equivalence of these two measures in the following sense: for any Borel set A ? L ₂[0, 1] the set A ∩ C ₀([0, 1]) is a Borel subset of C ₀([0, 1]) and µ(A) = w(A∩C ₀([0, 1])).     

On the completeness of the system of root functions of the Sturm-Liouville operator with degenerate boundary conditions

We consider the spectral problem generated by the Sturm-Liouville equation with arbitrary complex-valued potential q(x) ∈ L ₁(0, π) and with degenerate boundary conditions. We obtain sufficient conditions for the completeness of the system of eigenfunctions and associated functions of this operator.     

Basis Properties of the System of Root Functions of the Sturm–Liouville Operator with Degenerate Boundary Conditions: I

The spectral problem for the Sturm–Liouville operator with an arbitrary complex-valued potential q(x) of the class L¹(0, π) and degenerate boundary conditions is considered. We prove that the system of root functions of this operator is not a basis in the space L²(0, π).     

Widths of classes of periodic differentiable functions in the space L 2[0, 2π]

We obtain exact values of different n-widths for classes of differentiable periodic functions in the space L ₂[0, 2π] satisfying the constraint $$ \left( {\int_0^h {\omega _m^p \left( {f^{\left( r \right)} ;t} \right)dt} } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \leqslant \Phi \left( h \right) $$ , where 0 < h < ∞, 1/r < p ≤ 2, r ∈ ?, and ω _m (f ^(r); t) is the modulus of continuity of mth order of the derivative f ^(r)(x) ∈ L ₂[0, 2π].     

Estimates of Riesz Constants for the Dirac System with an Integrable Potential

We consider the Dirac operator on the interval [0, π] with an integrable potential P = (p_ij (x)) _i,j=1 ² and strongly regular boundary conditions U. It is well known that for any integrable potential P the system {y_n}_n∈Z of root functions of the strongly regular operator L_P,U is a Riesz basis in the space H = L₂[0, π] × L₂[0, π]. We obtain estimates, uniform on every compact set of potentials, of the Riesz constants ||T||||T^?1||, where T is the operator of transition to an orthonormal basis.