Bilal Bilalov

Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan · Non-harmonic analysis

We consider a non-local boundary value problem for the Laplace equation in unbounded studding the weak and strong solvability of that problem in the framework of the weighted Sobolev space $W^{1,p}_\nu$, with a Muckenhoupt weight. We proved that if any weak solution belongs to the space $W_{\nu}^{2,p}$, then it is also a strong solution and satisfi...

This paper considers the method of Riemann boundary value problems of the theory of analytic functions to study basis properties of perturbed systems of exponents in rearrangement invariant Banach function spaces. This method is demonstrated by the example of a system of exponents with a linear phase, depending on a complex parameter. The study of...

We consider $m$-th order linear, uniformly elliptic equation $\mathcal{L}u = f$ with non-smooth coefficients in Banach-Sobolev space $W_{X_{w} }^{m} (\Omega )$ generated by weighted general Banach Function Space (BFS) $X_{w} (\Omega)$ on a bounded domain $\Omega \subset \mathbb{R}^n.$ Supposing boundedness of the Hardy-Littlewood Maximal and Calder...

In this paper a Banach function space (b.f.s. in short) X(Ω) on a n-dimensional bounded domain Ω with Lebesgue measure is considered. The Banach function space X(S) on the (n−1)-dimensional surface S⊂Ω¯ generated by the norm of the space X(Ω) is defined. The Sobolev function space WXm(Ω) is defined using the norm of the space X(Ω), as well as the c...

In this chapter an m-th order elliptic equation is considered in Sobolev spaces generated by the norm of a grand Lebesgue space. Subspaces are determined in which the shift operator is continuous, and local solvability (in the strong sense) is established in these subspaces. It is established an interior and up-to boundary Schauder-type estimates w...

Non-standard grand-Lebesgue and Morrey spaces are considered in this work, together with grand-Sobolev and Morrey–Sobolev spaces generated by them. Dirichlet problems for Laplace equation in different versions are considered in these spaces in a bounded domain of n-dimensional space with sufficiently smooth boundary. These spaces are non-separable,...

We consider a higher order elliptic equation with nonsmooth coefficients with respect to rearrangement invariant spaces on the domain \( \Omega\subset{𝕉}^{n} \). Separable subspaces of these spaces are distinguished in which infinitely differentiable and compactly supported functions are dense; Sobolev spaces generated by these subspaces are determ...

In this work, it is considered an elliptic operator L of mth order with nonsmooth coefficients in a non-standard grand Sobolev space Wq)m(Ω) on a bounded domain Ω⊂Rn generated by the norm of the grand Lebesgue space Lq)(Ω). Under weaker restrictions on the coefficients of the operator, we prove the solvability (in the strong sense) in the small in...

We consider a one-dimensional mixed problem for one class of third-order partial differential equations with nonlinear right-hand side. The concept of generalized solution of this problem is introduced. By the Fourier method, the problem of existence and uniqueness of generalized solutions for this problem is reduced to the problem of solvability o...

In this paper a second order elliptic equation with nonsmooth coefficients is considered in grand-Sobolev classes Wq)2Ω on a bounded n-dimensional domain Ω⊂Rn with a sufficiently smooth boundary ∂Ω, generated by the norm of the grand-Lebesgue space Lq)Ω. These spaces are non-separable and therefore the definition of a reasonable solution in them fa...

Weighted Smirnov classes in bounded and unbounded domains are defined in this work. Nonhomogeneous Riemann problems with a measurable coefficient whose argument is a piecewise continuous function are considered in these classes. A Muckenhoupt type condition is imposed on the weight function and the orthogonality condition is found for the solvabili...

UDC 517.9 One-dimensional mixed problem for one class of third order partial differential equation with nonlinear right-hand side is considered. The concept of generalized solution for this problem is introduced. By the Fourier method, the problem of existence and uniqueness of generalized solution for this problem is reduced to the problem of solv...

This work deals with the rearrangement invariant Banach function space X and Banach Hardy classes generated by this space, which consist of analytic functions inside and outside the unit circle. In these Hardy classes we consider homogeneous and nonhomogeneous Riemann problems with piecewise continuous coefficient. We define new characteristic of t...

An exponential system with piecewise linear phase depending on some parameters is considered in this work. Basis properties of this system (such as completeness, minimality and basicity) are studied in a subspace of Morrey space where continuous functions are dense. A sufficient condition for the completeness (minimality or basicity) of this system...

This work deals with the mth order elliptic equation with non-smooth coefficients in grand-Sobolev space generated by the norm of the grand-Lebesgue space L q ) ( Ω ) , 1 < q < + ∞ . These spaces are non-separable, and therefore, to use classical methods for treating solvability problems in these spaces, you need to modify these methods. To this ai...

We study the basis properties of eigenfunctions in Lebesgue spaces for a spectral problem for a discontinuous second-order differential operator with spectral parameter in the discontinuity (transmission) conditions. This problem arises when solving the problem on the vibrations of a loaded spring with fixed endpoints. An abstract theorem on the st...

In this paper, the basis properties (completeness, minimality and basicity) of the system of exponents are investigated in weighted Morrey spaces, where the weight function is defined as a product of power functions. Although the same properties of the system of exponents, as well as their perturbations, are well studied in weighted Lebesgue spaces...

for the perturbed system of exponentials exp(i(n − β sign n)t), for n ∈ Z, where β is a complex parameter, we find a necessary and sufficient condition on β under which this system constitutes a basis for the Morrey space on (−π, π). The system is of particular interest in the theory of nonharmonic Fourier series; the study of its basis property in...

In the paper it is considered the generalized Faber polynomials defined inside and outside a regular curve on the complex plane. The weighted Smirnov spaces corresponding to bounded and unbounded regions are defined. It is proved that the generalized Faber polynomials forms a basis in weighted Smirnov spaces, if the weight function satisfies the Mu...

The concepts of Bessel families and frames in non-separable Hilbert spaces are introduced in this work. Besselianness criterion for a family is found. Similar to the usual case, analysis, synthesis and frame operators are defined, their properties are studied. Many results related to usual frames are extended to new case. Examples are given.

We consider a spectral problem for a second order discontinuous differential operator with spectral parameter in the boundary condition. We present a method for establishing the basicity of eigenfunctions for such problem. We also consider a direct expansion of a Banach space with respect to subspaces and we propose a method for constructing a basi...

In this paper, we discuss few existence result for solution of an infnite system of fractional differential equations of order α (1 < α < 2), with three point boundary value problem in the interval [0, T]. The problem is studied in the classical Banach sequence spaces c0 and ℓp (1 ≤ p < ∞), using Hausdorff measure of noncompactness and Darbo type f...

Riemann boundary value problem of analytic function theory in weighted Hardy classes with variable summability index is considered in this work. The Fredholmness of this problem is investigated under certain conditions on coefficients and a weight. The general solution for homogeneous problem is obtained in weighted Hardy classes with variable summ...

In the present work it is considered the system of functions a (t) eint − b (t) e−int, n ∈ N, with complex-valued coefficients a (·); b (·): [0, π] → C. Sufficient conditions on the coefficients of the system are found in order for the system to form a basis in Lebesgue spaces with variable exponent. © 2017, Institute of Mathematics and Mechanics,...

The concept of ℱst-fundamentality is introduced in uniform spaces, generated by some filter ℱ. Its equivalence to the concept of ℱ-convergence in uniform spaces is proved. This convergence generalizes many kinds of convergence, including the well-known statistical convergence.

A method for constructing a basis for a Banach space from bases for its subspaces is proposed. The case of isomorphic subspace bases and the case when no corresponding isomorphisms are required are considered separately. The completeness, minimality, uniform minimality, and basis property with parentheses of the corresponding systems are studied. T...

A part of an exponential system with degenerate coefficients is considered. The frame properties (completeness, minimality, basicity, atomic decomposition) of this system in Hardy classes are studied in the case where the coefficients may not satisfy the Muckenhoupt condition.

В монографию включены некоторые аспекты теории аппроксимации и базисов. Приведены основные положения, понятия и факты теории аппроксимации так конечномерного, так и бесконечномерного случая пространств. По некоторым позициям изложения отличны от традиционных. Даны некоторые обобщения базисов в направлении тензорных произведений, билинейных отображе...

This work considers the Riemann boundary value problem with the piecewise continuous coefficient in Morrey-Hardy classes. Under some conditions on the coefficient, the Fredholmness of this problem is studied and the general solution of homogeneous and nonhomogeneous problems in Morrey-Hardy classes is constructed.

Statistical convergence in Lebesgue spaces is considered in this paper. A criterion for statistical convergence is given. It is shown that the known Tauberian theorems for scalar case are valid in this case, too.

The concept of μ-statistical convergence at a point for measurable functions in measurable space with a measure is introduced in this work. This concept is a generalization of a similar idea about the sequence of numbers. We also introduce the concept of μ-statistical fundamentality at a point, and the equivalence of these two concepts is proved. T...

The concept of μ-statistical convergence at a point for measurable functions in measurable space with a measure is introduced in this work. This concept is a generalization of a similar idea about the sequence of numbers. We also introduce the concept of μ-statistical fundamentality at a point, and the equivalence of these two concepts is proved. T...

The statistical convergence in metric spaces is considered. Its equivalence to the statistical fundamentality in complete metric spaces is proved. Introduced the concept of $p$-strong convergence, and proved its equivalence to the statistical convergence. Tauberian theorems concerning statistical convergence in metric spaces are given.

Frames in Hilbert and Banach spaces are considered and their properties in the context of Noetherian mapping are studied in this paper. Atomic decompositions in Banach spaces are also considered. The concept of K-closeness is introduced. The stability of frame properties and atomic decompositions with respect to K-closeness is proved.

The concept of t -frame associated with the tensor product of Hilbert spaces is introduced. All the properties of ordinary frames are extended to this case. Noetherian perturbation of t -frames is considered. The stability of t -frameness with respect to quadratic closeness is proved.

In this paper, we consider Banach spaces of functions of Morrey-Hardy, Morrey-Smirnov and Morrey-Lebesgue type. Basicity of exponential system and its parts in these spaces is proved.

Fuzzy metric space is considered. The concepts of fuzzy completeness, fuzzy minimality, fuzzy biorthogonality, fuzzy basicity and fuzzy space of coefficients are introduced. Strong completeness of fuzzy space of coefficients with regard to fuzzy metric and strong basicity of canonical system in this space are proved. Strong basicity criterion in fu...

A double system of generalized Faber polynomials with complex-valued coefficients is considered. Under certain conditions on the coefficients, the basicity of this system is proved in Lebesgue spaces L p (Γ), 1<p<+∞, where Γ is a Lyapunov or Radon curve on the complex plane. Besides, basicity of systems of generalized Faber polynomials is proved in...

We consider a double power system with degenerate coefficients. Under certain conditions we obtain a completeness criterion for this system in the Lebesgue function space.

Frames in Hilbert and Banach spaces are considered and their properties in the context of Noetherian mapping are studied in this paper. Atomic decompositions in Banach spaces are also considered. The concept of K-closeness is introduced. The stability of frame properties and atomic decompositions with respect to K-closeness is proved.

It is proved that the arbitrary nondegenerate system in a linear complete topological space has a correspondence complete topological space of coefficients with canonical basis. Basicity criterion for systems in such spaces is given in terms of coefficient operator.

Systems of sines with degenerate coefficients are considered in this paper. Frame properties of these systems in Lebesgue spaces are studied.

A system of exponents with a piecewise linear phase is considered in the paper. The criteria of basicity, completeness and minimality of this system in Lebesgue space of functions with variable summability exponent are established.

Under study is the basis property in L 2 of a system of Kostyuchenko type. In particular, some criterion is established for the basis property of the Kostyuchenko system under the natural constraints on a parameter in this system.

Intuitionistic fuzzy normed space is defined using concepts of $t$-norm and $t$-conorm. The concepts of fuzzy completeness, fuzzy minimality, fuzzy biorthogonality, fuzzy basicity, and fuzzy space of coefficients are introduced. Strong completeness of fuzzy space of coefficients with regard to fuzzy norm and strong basicity of canonical system in t...

Exponential systems in Lebesgue function spaces with variable exponent are considered. The basis properties of these exponential systems are studied in the spaces under certain conditions on the function. The space endowed with the norm is Banach and the set of compactly supported infinitely differentiable functions is found to be everywhere dense....

Some problems of the theory of bases are considered. The known notions for bases as a space of coefficients, natural isomorphism and etc. are taken to the case of systems possessing definite properties. Some theorems on basicity of closed system are also given.

A problem on optimal control of processes described by totality of a parabolic equation and ordinary differential equation with moving sources is investigated in the paper. For some classes of linear and nonlinear boundary value problems with impulse functions, the existence and uniqueness problems of generalized solution are studied. In particular...

We consider the systems of exponents {expi(n-αsignn)t} n∈ℤ , 1∪{expi(n-αsignn)t} n≠0 , cosines{cos(n-α)t} n≥0 (1∪{cos(n-α)t} n≥1 ) and sines{sin(n-α)t} n≥1 . The basis properties of these systems are completely studied in the spaces L p t with variable exponent p(t).

A method for establishing the basicity of the perturbed system from the basis in Banach spaces is cited. The obtained results are applied to the system of exponentials.

The classical concepts of Bessel and Hilbert systems and Riesz bases in Hilbert spaces were introduced in Bari's fundamental work where different criteria were stated for a given system to be a Bessel or Hilbert one or a Riesz basis. These concepts were extended to Banach spaces in different directions. A number of theorems were proposed that playe...

We introduce a system of exponential functions with shift, study its basis properties in L 2, and examine its connections with the Kostyuchenko problem.

The system of powers with generated coefficients is considered. Under defined conditions, the equivalence of completeness of this system in L p to the completeness of system of exponents with generation is proved.

The relations between the Paley-Wiener and N. K. Bari theorems on close bases in general situations are shown in the paper.

We establish relatlons between basis properties of dual and unitary systems of powers in the Lebesgue spaces. Moreover, we suggest a method to establish the basicity of the system of cosines in Sobolev spaces.

We consider basis properties of the classical system of exponents in ordinary and weight Lebesgue spaces of functions with variable suitability exponent.

Generalizations of the Stone-Weierstrass and Bishop approximation theorems are presented. Given an algebra, a subspace in a continuous function space coinciding with the closure of this algebra is constructed. Analogs of these results are obtained in the case where the set of functions under consideration is not an algebra, but its closure is relat...

We present several generalizations of the classical Bari theorem on the Riesz basis property of close systems in Hilbert spaces to Banach spaces. We introduce the corresponding definitions and formulate theorems on the basis property of close systems in Banach spaces.

Without Abstract

We consider power systems with complex-valued coeficients. We establish a necessary and suficient condition for completeness and minimality and also a necessary condition for the basis property of these systems in Lebesgue spaces.

In the paper, some methods of establishment of basicity of systems in Banach spaces considered as basic or subspaces with respect to others, are given.

We consider some systems of exponential functions, cosines, and sines with complex-valued coefficients and establish a necessary and sufficient condition for completeness and minimality of these systems in Lebesgue spaces.

In the present paper, we consider the basis property of systems of exponentials, cosines, and sines which are eigenfunctions of an ordinary dierential equation with boundary conditions containing a small parameter. Such dierential operators arise in the solution of mixed type equations. Moiseev [1{4] successfully used the basis property of sine and...

The system of functions of the form f n (t)≡∑ k=1 l a k (t)(ω k (t)),t∈(a,b) is considered. Relations between basis properties of systems {x n (t)} considered in L p (c,d) and systems {f n (t)} considered in L p (a,b), p≥1 are also studied.

Considered is the system of exponents: E(λ)=exp(iλnt), n=0,±1,..., where λn=n+δn, {δn} - some sequence of complex numbers. At δn=βsign(n), where β - the complex parameter, in other works were obtained necessary and sufficient conditions of the basicity for the system E(λ) in Lp(-π,π), 1<p<+∞. In the present work the analogue of these results is obt...

The system of exponents E(λ k )≡e iλ k t ,k=0,±1,⋯, is considered, where {λ k } is a sequence of complex numbers with the asymptotics λ k =n±β ± +O(|n| -α ± ),n→±∞, and β ± ∈ℂ are complex parameters. A necessary and sufficient condition for this system to be a base in L p ,1<p<+∞, is established.