Entire Analytic Functions Research Articles

Distribution of eigenvalues of a perturbed differentiation operator on the interval

In this paper, we construct a characteristic determinant of the spectral problem of a first-order differential equation on an interval with an integral perturbation in the boundary value condition, which is an entire analytic function of the spectral parameter. Based on the formula for the characteristic determinant, conclusions are drawn about the asymptotic behavior of the spectrum of the perturbed spectral problem depending on the modulus of continuity of the subinteral function.

Countably Generated Algebras of Analytic Functions on Banach Spaces

In the paper, we study various countably generated algebras of entire analytic functions on complex Banach spaces and their homomorphisms. Countably generated algebras often appear as algebras of symmetric analytic functions on Banach spaces with respect to a group of symmetries, and are interesting for their possible applications. Some conditions of the existence of topological isomorphisms between such algebras are obtained. We construct a class of countably generated algebras, where all normalized algebraic bases are equivalent. On the other hand, we find non-isomorphic classes of such algebras. In addition, we establish the conditions of the hypercyclicity of derivations in countably generated algebras of entire analytic functions of the bounded type. We use methods from the theory of analytic functions of several variables, the theory of commutative Fréchet algebras, and the theory of linear dynamical systems.

Topological transitivity of translation operators in a non-separable Hilbert space

We consider a Hilbert space of entire analytic functions on a non-separable Hilbert space, associated with a non-separable Fock space. We show that under some conditions operators, like the differentiation operators and translation operators, are topologically transitive in this space.

Partial Inverse Sturm-Liouville Problems

This paper presents a review of both classical and modern results pertaining to partial inverse spectral problems for differential operators. Such problems consist in the recovery of differential expression coefficients in some part of the domain (a finite interval or a geometric graph) from spectral characteristics, while the coefficients in the remaining part of the domain are known a priori. Usually, partial inverse problems require less spectral data than complete inverse problems. In this review, we pay considerable attention to partial inverse problems on graphs and to the unified approach based on the reduction of partial inverse problems to Sturm-Liouville problems with entire analytic functions in a boundary condition. We not only describe the results of selected studies but also compare them with each other and establish interconnections.

Zeros of an entire function connected by a loaded first-order differential operator on a segment

In the paper, we consider the problem on eigenvalues of a loaded differential operator of the first order with a periodic boundary condition on the interval [–1; 1], that is, equation contains a load at the point (–1) and the function of bounded variation (t), with the condition Φ(−1) = Φ(1) = 1 . A characteristic determinant of spectral problem is constructed for the considered loaded differentiation operator, which is an entire analytical function on the spectral parameter. On the basis of the characteristic determinant formula, conclusions are proved about the asymptotic behavior of the spectrum and eigenfunctions of the loaded spectral problem for the differentiation operator, the characteristic determinant of which is an entire analytic function of the spectral parameter l. A theorem on the location of eigenvalues on the complex plane l is formulated, where the regular growth of an entire analytic function is indicated. A theorem is proved on the asymptotics of the zeros of an entire function, that is, the eigenvalues of the original considered spectral problem for a loaded differential operator of differentiation, and the asymptotic properties of an entire function with distribution of roots are studied.

Inverse Sturm–Liouville Problem with Spectral Parameter in the Boundary Conditions

In this paper, for the first time, we study the inverse Sturm–Liouville problem with polynomials of the spectral parameter in the first boundary condition and with entire analytic functions in the second one. For the investigation of this new inverse problem, we develop an approach based on the construction of a special vector functional sequence in a suitable Hilbert space. The uniqueness of recovering the potential and the polynomials of the boundary condition from a part of the spectrum is proved. Furthermore, our main results are applied to the Hochstadt–Lieberman-type problems with polynomial dependence on the spectral parameter not only in the boundary conditions but also in discontinuity (transmission) conditions inside the interval. We prove novel uniqueness theorems, which generalize and improve the previous results in this direction. Note that all the spectral problems in this paper are investigated in the general non-self-adjoint form, and our method does not require the simplicity of the spectrum. Moreover, our method is constructive and can be developed in the future for numerical solution and for the study of solvability and stability of inverse spectral problems.

CONVERGENCE OF URYSOHN-TYPE NONLINEAR INTEGRAL OPERATORS AT p–LEBESGUE POINT

In this paper, a general form of the family of Urysohn-type nonlinear integral operators with kernel $K_{\lambda }\left( x,t,g\right) $ is discussed and theorems about the point convergence of this family at ${p}$--Lebesgue points of a function in $L_{p}$ are given. Here, $\lambda$ is the accumulation point and is a positive parameter that changes in the real numbers. Kernel function $K_{\lambda }\left( x,t,g\left( t\right) \right)$ is an entire analytic function with respect to its third variable.

Algebras of entire functions containing functions of unbounded type on a Banach space

In this paper, we consider algebras of entire analytic functions which are bounded on a prescribed family of balls in a Banach space. We investigate the structures of such algebras and describe their spectra in terms of spectra of algebras of uniformly continuous analytic functions. Some partial examples are considered. In particular, we have complete descriptions of the spectra for the case of Tsirelson space and for $c_0$.

Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

On nonlocal perturbation of the problem on eigenvalues of differentiation operator on a segment

This work is devoted to the construction of a characteristic polynomial of the spectral problem of a first-order differential equation on an interval with a spectral parameter in a boundary value condition with integral perturbation which is an entire analytic function of the spectral parameter. Based on the characteristic polynomial formula, conclusions about the asymptotics of the spectrum of the perturbed spectral problem are established.

A partial inverse Sturm‐Liouville problem on an arbitrary graph

The Sturm‐Liouville operator with singular potentials of class on a graph of arbitrary geometrical structure is considered. We study the partial inverse problem, which consists in the recovery of the potential on a boundary edge of the graph from a subspectrum under the assumption that the potentials on the other edges are known a priori. We obtain (i) the uniqueness theorem, (ii) a reconstruction algorithm, (iii) global solvability, and (iv) local solvability and stability for this inverse problem. Our method is based on reduction of the partial inverse problem on a graph to the Sturm‐Liouville problem on a finite interval with entire analytic functions in the boundary condition.

Classes of Entire Analytic Functions of Unbounded Type on Banach Spaces

In this paper we investigate analytic functions of unbounded type on a complex infinite dimensional Banach space X. The main question is: under which conditions is there an analytic function of unbounded type on X such that its Taylor polynomials are in prescribed subspaces of polynomials? We obtain some sufficient conditions for a function f to be of unbounded type and show that there are various subalgebras of polynomials that support analytic functions of unbounded type. In particular, some examples of symmetric analytic functions of unbounded type are constructed.

Spectral approximations of strongly degenerate elliptic differential operators

We establish analytical estimates of spectral approximations errors for strongly degenerate elliptic differential operators in the Lebesgue space $L_q(\Omega)$ on a bounded domain $\Omega$. Elliptic operators have coefficients with strong degeneration near boundary. Their spectrum consists of isolated eigenvalues of finite multiplicity and the linear span of the associated eigenvectors is dense in $L_q(\Omega)$. The received results are based on an appropriate generalization of Bernstein-Jackson inequalities with explicitly calculated constants for quasi-normalized Besov-type approximation spaces which are associated with the given elliptic operator. The approximation spaces are determined by the functional $E\left(t,u\right)$, which characterizes the shortest distance from an arbitrary function ${u\in L_q(\Omega)}$ to the closed linear span of spectral subspaces of the given operator, corresponding to the eigenvalues such that not larger than fixed ${t>0}$. Such linear span of spectral subspaces coincides with the subspace of entire analytic functions of exponential type not larger than ${t>0}$. The approximation functional $E\left(t,u\right)$ in our cases plays a similar role as the modulus of smoothness in the functions theory.

On pointwise convergence of the family of Urysohn‐type integral operators

In this paper, we consider a family of nonlinear integral operators of Urysohn‐type and study the pointwise convergence of the family at characteristic points of L1−function. The kernel Kλ(x,t,u(t)) depends on the positive parameter λ changing on the set of numbers with the accumulation point at infinity and Kλ(x,t,u(t)) is an entire analytic function of variable u, which is a bounded function belonging to L1(R).

Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces

We show that the dual Bp·locΩ′ of the variable exponent Hörmander space Bp(·)loc(Ω) is isomorphic to the Hörmander space B∞c(Ω) (when the exponent p(·) satisfies the conditions 0<p-≤p+≤1, the Hardy-Littlewood maximal operator M is bounded on Lp(·)/p0 for some 0<p0<p- and Ω is an open set in Rn) and that the Fréchet envelope of Bp(·)loc(Ω) is the space B1loc(Ω). Our proofs rely heavily on the properties of the Banach envelopes of the p0-Banach local spaces of Bp(·)loc(Ω) and on the inequalities established in the extrapolation theorems in variable Lebesgue spaces of entire analytic functions obtained in a previous article. Other results for p(·)≡p, 0<p<1, are also given (e.g., all quasi-Banach subspace of Bploc(Ω) is isomorphic to a subspace of lp, or l∞ is not isomorphic to a complemented subspace of the Shapiro space hp-). Finally, some questions are proposed.

Duals of variable exponent Hörmander spaces ( $$0< p^- \le p^+ \le 1$$ 0 < p - ≤ p + ≤ 1 ) and some applications

In this paper we characterize the dual $$\bigl (\mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )'$$ of the variable exponent Hormander space $$\mathcal {B}^c_{p(\cdot )} (\Omega )$$ when the exponent $$p(\cdot )$$ satisfies the conditions $$0 < p^- \le p^+ \le 1$$ , the Hardy-Littlewood maximal operator $$M$$ is bounded on $$L_{p(\cdot )/p_0}$$ for some $$0 < p_0 < p^-$$ and $$\Omega $$ is an open set in $$\mathbb {R}^n$$ . It is shown that the dual $$\bigl (\mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )'$$ is isomorphic to the Hormander space $$\mathcal {B}^{\mathrm {loc}}_\infty (\Omega )$$ (this is the $$p^+ \le 1$$ counterpart of the isomorphism $$\bigl (\mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )' \simeq \mathcal {B}^{\mathrm {loc}}_{\widetilde{p'(\cdot )}} (\Omega )$$ , $$1 < p^- \le p^+ < \infty $$ , recently proved by the authors) and hence the representation theorem $$\bigl ( \mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )' \simeq l^{\mathbb {N}}_\infty $$ is obtained. Our proof relies heavily on the properties of the Banach envelopes of the steps of $$\mathcal {B}^c_{p(\cdot )} (\Omega )$$ and on the extrapolation theorems in the variable Lebesgue spaces of entire analytic functions obtained in a precedent paper. Other results for $$p(\cdot ) \equiv p$$ , $$0 < p < 1$$ , are also given (e.g. $$\mathcal {B}^c_p (\Omega )$$ does not contain any infinite-dimensional $$q$$ -Banach subspace with $$p < q \le 1$$ or the quasi-Banach space $$\mathcal {B}_p \cap \mathcal {E}'(Q)$$ contains a copy of $$l_p$$ when $$Q$$ is a cube in $$\mathbb {R}^n$$ ). Finally, a question on complex interpolation (in the sense of Kalton) of variable exponent Hormander spaces is proposed.

Haydock’s recursive solution of self-adjoint problems. Discrete spectrum

Haydock’s recursive solution is shown to underline a number of different concepts such as (i) quasi-exactly solvable models, (ii) exactly solvable models, (iii) three-term recurrence solutions based on Schweber’s quantization criterion in Hilbert spaces of entire analytic functions, and (iv) a discrete quantum mechanics of Odake and Sasaki. A recurrent theme of Haydock’s recursive solution is that the spectral properties of any self-adjoint problem can be mapped onto a corresponding sequence of polynomials {pn(E)} in energy variable E. The polynomials {pn(E)} are orthonormal with respect to the density of states n0(E) and energy eigenstate |E〉 is the generating function of {pn(E)}. The generality of Haydock’s recursive solution enables one to see the different concepts from a unified perspective and mutually benefiting from each other. Some results obtained within the particular framework of any of (i) to (iv) may have much broader significance.

Fractals as macroscopic manifestation of squeezed coherent states and brain dynamics

Recent results on the relation between self-similarity and squeezed coherent states are presented. I consider fractals which are generated iteratively according to a prescribed recipe, the so-called deterministic fractals. Fractal properties are incorporated in the framework of the theory of the entire analytical functions and deformed coherent states. Conversely, fractal properties of squeezed coherent states are recognized. This sheds some light on the understanding of the dynamical origin of fractals and their global nature emerging from local deformation processes. The self-similarity in brain background activity suggested by laboratory observations of power-law distributions of power spectral densities of electrocorticograms is also discussed and accounted in the frame of the dissipative many-body model of brain.

Fractals, coherent states and self-similarity induced noncommutative geometry

The self-similarity properties of fractals are studied in the framework of the theory of entire analytical functions and the q-deformed algebra of coherent states. Self-similar structures are related to dissipation and to noncommutative geometry in the plane. The examples of the Koch curve and logarithmic spiral are considered in detail. It is suggested that the dynamical formation of fractals originates from the coherent boson condensation induced by the generators of the squeezed coherent states, whose (fractal) geometrical properties thus become manifest. The macroscopic nature of fractals appears to emerge from microscopic coherent local deformation processes.

On variable exponent Lebesgue spaces of entire analytic functions

In this article we introduce the variable Lebesgue spaces of entire analytic functions Lp(⋅)K. A maximal inequality of Jawerth is generalized to our context and inequalities of Plancherel–Polya–Nikolʼskij type are obtained. We calculate the dual of the space Lp(⋅)K when the function χK is an Lp(⋅)-Fourier multiplier and a number of consequences of this result (on sequence space representations) is given. Finally, a Fourier multiplier theorem by Triebel is extended to the setting of the variable Lebesgue spaces.