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This article discusses the existence conditions of dual frames and their uniqueness for the Γ -translation generated frames systems in L 2 (G) , where G be a locally compact group, and Γ is a closed abelian subgroup of G. Necessary and sufficient conditions of such duals associated with the infimum cosine angles are obtained. The problems are first addressed in the context of multiplication-generated frame systems in L 2 (X ; H) , where X is a σ -finite measure space, then obtained the results for locally compact group setups. [ABSTRACT FROM AUTHOR] more...

Given a σ-compact and metrizable locally compact abelian (LCA) group G, we give new perturbation results for Riesz bases for the matrix-valued function space L2(G, ℂs×r), s and r ∈ ℕ. [ABSTRACT FROM AUTHOR] more...

The paper considers boundary value problems generated by an ordinary differential expression of the nth order and arbitrary boundary conditions with linear dependence on the spectral parameter both in the equation and the boundary conditions. Classes of problems are defined, which are called regular and strongly regular. Linear operators in the space are assigned to these problems, and the corresponding adjoint operators are constructed in explicit form. In the general form, we solve the problem of selecting "superfluous" eigenfunctions, which was previously studied only for the special cases of second- and fourth-order equations. Namely, a criterion is found for selecting m eigen- or associated (root) functions of a regular problem so that the remaining system of root functions forms a Riesz basis or a Riesz basis with parenthesis in the original space . [ABSTRACT FROM AUTHOR] more...

This paper addresses the dilation problem on (dual) frames for Krein spaces. We characterize Riesz bases for Krein spaces and equivalence ((J1,J2)-unitary equivalence) between frames for Krein spaces; prove that every frame (dual frame pair) for a Krein space can be dilated to a Riesz basis (dual Riesz basis pair) for a larger Krein space, and that the corresponding J-orthogonal complementary frame (J-joint complementary frame) is unique up to equivalence ((J1,J2)-joint equivalence). Also we illustrate that two equivalent Parseval frames for Krein spaces need not be (J1,J2)-unitarily equivalent and that not every Parseval frame can be dilated to a J-orthonormal basis for a larger Krein space, and derive a result on matrices of finite size as application. [ABSTRACT FROM AUTHOR] more...

This paper is devoted to introduce a new concept of bases for the range of the operator $K$. Actually, we consider controlled $K$-orthonormal and controlled $K$-Riesz bases which are a generalization of ordinary bases in Hilbert spaces. In the sequel, we give some characterizations of these sequences. Next, we construct some new controlled bases with operator theory tools. Finally, some dual characterizations are given. more...

A Riesz basis for a separable Hilbert space H is the image of an orthonormal basis under a bounded, linear and bijective operator acting on H. Equivalently, it is an exact frame that shares the properties of a basis for H. Let G be a σ -compact and metrizable locally compact abelian group, and s and r be positive integers. It is illustrated that the image of a matrix-valued orthonormal basis under a bijective, bounded and linear operator acting on the matrix-valued signal space L 2 (G , ℂ s × r) may not be a frame, hence not a basis of the space L 2 (G , ℂ s × r). We introduce a notion of matrix-valued Riesz basis in the space L 2 (G , ℂ s × r) , where the adjointability of a bounded linear operator in the definition of Riesz basis with respect to the matrix-valued inner product plays a crucial role. We establish the existence of matrix-valued Riesz bases of the space L 2 (G , ℂ s × r). Extending results for standard Riesz bases of separable Hilbert spaces, we give necessary and sufficient conditions, and a characterization of matrix-valued Riesz bases of the space L 2 (G , ℂ s × r). [ABSTRACT FROM AUTHOR] more...

In this paper, we study an analogue of Tricomi problem for a parabolichyperbolic equation of the first type. To solve the problem for the heat equation, we use properties of the spectral problem with a spectral parameter in the boundary conditions. We show the basicity of the spectral problem with the spectral parameter in the boundary conditions. The conditions for the correctness of the formulated Tricomi problem are found. [ABSTRACT FROM AUTHOR] more...

By the method of separation of variables, we construct the solution of the investigated problem with an unknown coefficient in the differential equation. We study the properties of the spectral problem for the second-order differential equation with involution. The dependence of the spectrum of the operator of the problem, its multiplicity, and the structure of the system of root functions and particular solutions of the problem on the involutive part of the equation is analyzed. We establish the conditions for the existence and uniqueness of the solution to the inverse problem. To determine the unknown coefficient, we find and solve the Volterra integral equation of the second kind. [ABSTRACT FROM AUTHOR] more...

We study positive operator-valued measures generated by projections on one-dimensional subspaces. A special attention is paid to the case in which subspaces are spanned by vectors forming a Riesz basis. It is shown that the measurement fulfilled by such measure is informationally complete for quantum states being a convex hull of projections on subspaces spanned by the system of biorthogonal vectors. We also discuss the properties of different quantum channels associated with a discrete measurement. Finally, we show that our measurement allows to introduce a quantum instrument taking values in the set of two points. [ABSTRACT FROM AUTHOR] more...

In this paper, a nonlocal problem of the Bitsadze–Samarskii type for a parabolic-hyperbolic equation with the Gerasimov–Caputo operator is studied. The spectral method is used to solve the problem. Using this method, the problem under consideration is reduced to the study of a spectral boundary value problem for a second-order ordinary differential equation with respect to the spatial variable. The spectral properties of the obtained, as well as the adjoint problem, are investigated. Eigenvalues are found, as are the corresponding root functions, and their basis property is proved. Further, under certain conditions on given functions, uniqueness and existence theorems for the solution of the problem are proved. At the proof of uniqueness of the solution, the completeness of the system of eigenfunctions is used, and the solution of the problem is constructed in the form of an absolutely and uniformly convergent biorthogonal series. [ABSTRACT FROM AUTHOR] more...

This paper characterize important concepts such as the Riesz bases and pair of dual frames for tensor product of Hilbert spaces and introduce some special properties of pair of dual frames in these spaces. We show that Riesz bases and pair of dual frames in H ⊗ K are preserved under invertible and unitary operators on H . Finally, We present a representation for Hilbert–Schmidt operators by a pair of dual frames. [ABSTRACT FROM AUTHOR] more...

This article primarily aims at introducing a novel operational calculus of Mikusiński's type for the Dzherbashian‐Nersesian operator. Using this calculus, we are able to derive exact solutions for the forward and backward source problems of a differential equation that features Dzherbashian‐Nersesian operator in time and intertwined with nonlocal boundary conditions. The initial condition is expressed in terms of Riemann‐Liouville integral. Solutions are presented using Mittag‐Leffler (ML) type functions. The outcomes related to the existence and uniqueness subject to certain conditions of regularity on the input data are developed. [ABSTRACT FROM AUTHOR] more...

Although Naimark dilation theorem was originally stated in 1940, it still finds many important applications in various areas. The objective of this paper is to introduce a method for explicitly constructing the vectors of complementary frames in the Naimark dilation theorem, specifically in cases where the initial Parseval frame contains a Riesz basis as a subset. These findings serve as a foundation for the construction of dual frames. [ABSTRACT FROM AUTHOR] more...

A diffusion equation involving integral convolution in time variable with arbitrary kernel and nonlocal boundary conditions is considered. The existence and uniqueness results for two inverse problems of determining source terms (space- and time-dependent sources) along with diffusion concentration from appropriate over-specified conditions are presented. A bi-orthogonal system of functions is used to have series representation of the solutions of the inverse problems. Several special cases such as standard diffusion, multi-term diffusion, and tempered diffusion equations are discussed, and some examples are provided. [ABSTRACT FROM AUTHOR] more...

In this article, we study the output regulation of two coupled heat equations with recycle, where the disturbances are distributed in all channels, the control is implemented at the left boundary |$x=0$| and the point temperature at |$x=2$| is the only noncollocated measurement. The output control is divided into two parts: one is to stabilize the unstable heat system by the noncollocated static feedback, and the other is to achieve output regulation by solving the regulator equation and estimating the disturbances, where an observer is constructed to estimate the state of heat system and exosystem synchronously. The closed-loop system is proved to be well-posed and uniformly bounded using the Riesz basis approach, and the integral output tracks successfully the reference signal exponentially. The numerical simulations are carried out to demonstrate the proposed controller is effective and the tracking error converges to zero. [ABSTRACT FROM AUTHOR] more...

When a signal measured by woven frames is transmitted, the order of some of these coefficients may change, resulting in undesirable signal reconstruction. In this paper, we investigate the permutations that if two frames are woven, one of the frames will be woven with a permutation of the other frame. We also examine the existence of permutations that a Riesz basis and reordered Riesz basis are woven. [ABSTRACT FROM AUTHOR] more...

We introduce the Zak transform on L 2 (R 2 n) associated with the Weyl transform. By making use of this transform, we define a bracket map and prove that the system of twisted translates { T (k , l) t ϕ : k , l ∈ Z n } is a frame sequence iff 0 < A ≤ ϕ , ϕ (ξ , ξ ′) ≤ B < ∞ , for a.e (ξ , ξ ′) ∈ Ω ϕ , where Ω ϕ = { (ξ , ξ ′) ∈ T n × T n : ϕ , ϕ (ξ , ξ ′) ≠ 0 } . We also prove a similar result for the system { T (k , l) t ϕ : k , l ∈ Z n } to be a Riesz sequence. For a given function belonging to the principal twisted shift-invariant space V t (ϕ) , we find a necessary and sufficient condition for the existence of a canonical biorthogonal function. Further, we obtain a characterization for the system { T (k , l) t ϕ : k , l ∈ Z } to be a Schauder basis for V t (ϕ) in terms of a Muckenhoupt A 2 weight function. [ABSTRACT FROM AUTHOR] more...

We consider the indefinite Sturm–Liouville differential expression a (f) : = - 1 w 1 r f ′ ′ , where a is defined on a finite or infinite open interval I with 0 ∈ I and the coefficients r and w are locally summable and such that r(x) and (sgn x) w (x) are positive a.e. on I. With the differential expression a we associate a nonnegative self-adjoint operator A in the Krein space L w 2 (I) which is viewed as a coupling of symmetric operators in Hilbert spaces related to the intersections of I with the positive and the negative semi-axis. For the operator A we derive conditions in terms of the coefficients w and r for the existence of a Riesz basis consisting of generalized eigenfunctions of A and for the similarity of A to a self-adjoint operator in a Hilbert space L | w | 2 (I) . These results are obtained as consequences of abstract results about the regularity of critical points of nonnegative self-adjoint operators in Krein spaces which are couplings of two symmetric operators acting in Hilbert spaces. [ABSTRACT FROM AUTHOR] more...

In this paper, we consider the concept of R-duality with respect to Riesz bases. In particular, we study the concept of R-duality with respect to Riesz bases, constructed by anti-linear operators, for Bessel sequences. Using these anti-linear operators, we give some characterizations for frames and Riesz bases. [ABSTRACT FROM AUTHOR] more...

In this paper, the inverse problem of the Bitsadze–Samarsky type is studied for a fractional order equation with a Hadamard–Caputo fractional differentiation operator. The problem is solved using the spectral method. The spectral aspects of the obtained problem are investigated, root functions are found, and their basis property is proved. The conjugate problem is investigated. The uniqueness and existence theorems for a regular solution to this problem are proved. more...

In this paper, the inverse problem for a fourth-order parabolic equation with a variable complex-valued coefficient is studied by the method of separation of variables. The properties of the eigenvalues of the Dirichlet and Neumann boundary value problems for a non-self-conjugate fourth-order ordinary differential equation with a complex-valued coefficient are established. Known results on the Riesz basis property of eigenfunctions of boundary value problems for ordinary differential equations with strongly regular boundary conditions in the space L2(−1,1) are used. On the basis of the Riesz basis property of eigenfunctions, formal solutions of the problems under study are constructed and theorems on the existence and uniqueness of solutions are proved. When proving theorems on the existence and uniqueness of solutions, the Bessel inequality for the Fourier coefficients of expansions of functions from space L2(−1,1) into a Fourier series in the Riesz basis is widely used. The representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are derived. The convergence of the obtained solutions is discussed. more...

We study boundary output tracking for a flexible beam with tip payload and boundary nonlinear disturbance. By constructing an infinite-dimensional extended state observer (ESO), the total disturbance is estimated on line. A servomechanism corresponding to the reference signal and servomechanism-based output feedback control law are designed. Under such control law, we prove that the performance boundary output tracks exponentially the reference signal. By virtue of the Riesz basis approach, the Riesz basis generation and exponential stability of an coupled system are verified, and thereby the boundedness of closed-loop system is obtained. Some numerical simulations are presented to illustrate the effectiveness. [ABSTRACT FROM AUTHOR] more...

We consider the linear degenerate wave equation, on the interval (0, 1) wtt - (x\α wx)x = p(t)m&$955;(x)w, with bilinear control p and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference tra jectory, the "ground state." Under some classical and generic assumptions on \mu, we prove that there exists a threshold value for time, T\α = 2 1 \α, such that the reachable set is a neighborhood of the ground state. If T < T\α it is contained in a C1-submanifold of infinite codimension. Finally, it T = T\α and \α \in [0,1) it is a C1-submanifold of codimension 1 and if \α\in (1,2) the reachable set is a neighborhood of the ground state. The case \α = 1 remains open. This extends to the degenerate case the work [K. Beauchard, J. Differential Equations, 250 (2011), pp. 2064--2098] and adapts to bilinear controls the work [F. Alabau-Boussouira, P. Cannarsa, and G. Leugering, SIAM J. Control Optim., 55 (2017), pp. 2052--2087]. Our proofs are based on a careful analysis of the spectral problem and on Ingham type results, which are extensions of Kadec's 1/4 theorem. [ABSTRACT FROM AUTHOR] more...

We show that there exists a bounded subset of R such that no system of exponentials can be a Riesz basis for the corresponding Hilbert space. An additional result gives a lower bound for the Riesz constant of any putative Riesz basis of the two-dimensional disk. [ABSTRACT FROM AUTHOR] more...

The solution of the investigated problem with an unknown coefficient in the equation was constructed by using the method of separation of variables. The properties of the induced spectral problem for the second-order differential equation with involution are studied. The dependence on the equation involutive part of the spectrum and its multiplicity as well as the structure of the system of root functions and partial solutions of the problem were investigated. The conditions for the existence and uniqueness of the solution of the inverse problem have been established. To determine the required coefficient, Volterra's integral equation of the second kind was found and solved. more...

We consider a general twisted shift-invariant system, $V^{t}(\mathcal {A})$ , consisting of twisted translates of countably many generators and study the problem of obtaining a characterization for the system $V^{t}(\mathcal {A})$ to form a frame sequence or a Riesz sequence. We illustrate our theory with some examples. In addition to these results, we study a dual twisted shift-invariant system and also obtain an orthonormal sequence of twisted translates from a given Riesz sequence of twisted translates. [ABSTRACT FROM AUTHOR] more...

In this paper, we study the exact boundary controllability of the linear fourth-order Schrödinger equation, with variable physical parameters and clamped boundary conditions on a bounded interval. The control acts on the first spatial derivative at the right endpoint. We prove that this control system is exactly controllable at any time T > 0 . The proofs are based on a detailed spectral analysis and the use of nonharmonic Fourier series. [ABSTRACT FROM AUTHOR] more...

In this paper, we study an analogue of Tricomi problem for a parabolic–hyperbolic equation. One boundary condition in the parabolic part of the domain contains the third derivative of the space variable. A spectral problem with a spectral parameter in the boundary condition arises, when solving this problem. The system of eigenfunctions of this problem (after removing any one eigenfunction) forms a Riesz basis. The conditions for the correctness of the formulated Tricomi problem are found. [ABSTRACT FROM AUTHOR] more...

Recently D.T. Stoeva proved that if two Bessel sequences in a separable Hilbert space H are biorthogonal and one of them is complete in H, then both sequences are Riesz bases for H. This improves a well known result where completeness is assumed on both sequences. In this note we present an alternative proof of Stoeva’s result which is quite short and elementary, based on the notion of Riesz-Fischer sequences. [ABSTRACT FROM AUTHOR] more...

Abstract The paper is devoted to finding a solution and restoring the right-hand side of the heat equation with reflection of the argument in the second derivative, with a complex-valued variable coefficient. We prove a theorem on the Riesz basis property for eigenfunctions of the second-order differential operator with involution in the second derivative. We establish the existence and uniqueness of the solution of the studied problems by the method of separation of variables more...

An inverse source problem (ISP) of recovering space‐dependent source term along with diffusion concentration for a two‐dimensional diffusion equation involving integral convolution of arbitrary memory kernel in time is considered. The unique existence of the solution is proved using a bi‐orthogonal system of functions obtained from the associated non‐self‐adjoint spectral problem and its adjoint problem which form Riesz basis in L2[(0,1)×(0,1)]$$ {L}&#x0005E;2\left[\left(0,1\right)\times \left(0,1\right)\right] $$. In addition, some particular cases of ISP are described as special cases of our results. [ABSTRACT FROM AUTHOR] more...

In this paper, we study the controllability of a coupled first-order hyperbolic-elliptic system in the interval $ (0, 1) $ by a Dirichlet boundary control acting at the left endpoint of the hyperbolic component only. Using the multiplier approach and compactness-uniqueness argument, we establish the exact controllability for the hyperbolic component of the model, at any time $ T>1 $. We explore the method of moments to conclude the exact controllability at the critical time $ T = 1 $. For the case of small time, that is for $ T<1 $, we show that the system is not null controllable. Further, using a Gramian-based approach introduced by Urquiza, we prove the exponential stabilization of the corresponding closed-loop system with arbitrary prescribed decay rate by means of boundary feedback control law. [ABSTRACT FROM AUTHOR] more...

Reproducing systems in L 2 (R) such as wavelet and Gabor dual frames have been extensively studied, but reducing systems in L 2 (R +) with R + = (0 , ∞) have not. In practice, L 2 (R +) models the causal space since the time variable cannot be negative. Due to R + not being a group under addition, L 2 (R +) admits no nontrivial shift invariant system and thus admits no traditional wavelet or Gabor analysis. However, L 2 (R +) admits nontrivial dilation systems due to R + being a group under multiplication. This paper addresses the frame theory of a class of dilation-and-modulation ( M D ) systems generated by a finite family in L 2 (R +) . We obtain a parametric expression of M D -frames, and a density theorem for such M D -systems which is parallel to that of traditional Gabor systems in L 2 (R) . It is well known that an arbitrary Gabor frame must admit dual frames with the same structure. Interestingly, it is not the case for M D -frames. We prove that an M D -frame admits M D -dual frames if and only if log b a is an integer, where a and b are dilation and modulation parameters, respectively. And in this case, we characterize and express all M D -dual generators for an arbitrarily given M D -frame. Some examples are also provided. [ABSTRACT FROM AUTHOR] more...

For a weight function r ∈ L 1 [ - 1 , 1 ] with sign change at 0 we consider two indefinite regular Sturm–Liouville problems: - f ′ ′ = λ r f with Neumann boundary conditions and - (u ′ r) ′ = λ u with Dirichlet boundary conditions. It is known that the eigenfunctions may (or may not) form a Riesz basis in the naturally associated weighted Hilbert spaces only simultaneously. Here, we study the situation where this property fails to hold. Nevertheless, we obtain eigenfunction expansions on smaller domains or in spaces with stronger or weaker topology. As the main result, we identify explicit functions in the "original" spaces which do not allow an eigenfunction expansion. Furthermore, we find a sequence of functions which violates a certain HELP inequality and indicate limits of other related results in Kreĭn spaces. [ABSTRACT FROM AUTHOR] more...

We study spectral properties of a nonself-adjoint problem for a (2n) th order operator of differentiation with nonlocal conditions representing multipoint perturbations of the Birkhoff strongly regular self-adjoint conditions. We establish sufficient conditions under which the system of eigenfunctions is complete and, under certain additional assumptions, forms a Riesz basis. We construct a set of transformation operators, each element of which maps the system of eigenfunctions of the unperturbed problem into the system of eigenfunctions of a certain isospectral problem. The cases of problems with Birkhoff regular and irregular two-point perturbations are analyzed. Finally, we establish conditions for the existence and uniqueness of the solution of the studied problem. [ABSTRACT FROM AUTHOR] more...

In the present work, two-dimensional mixed problems with the Caputo fractional order differential operator are studied using the Fourier method of separation of variables. The equation contains a linear transformation of involution in the second derivative. The considered problem generalizes some previous problems formulated for some fourth-order parabolic-type equations. The basic properties of the eigenfunctions of the corresponding spectral problems, when they are defined as the products of two systems of eigenfunctions, are studied. The existence and uniqueness of the solution to the formulated problem is proved. [ABSTRACT FROM AUTHOR] more...

This paper is concerned with performance output tracking for an Euler–Bernoulli beam equation with moment boundary control and shear boundary disturbance. An infinite‐dimensional disturbance estimator is designed to estimate the total disturbance. By compensating the total disturbance, a servomechanism corresponding to the reference signal and servomechanism‐based output feedback control law are designed. It is proved that under such control law, the performance output tracks exponentially the reference signal and the involved states of closed‐loop system are bounded. The most important contribution is to deal with the shear boundary term stemmed from the error system between the disturbance estimator and the original system. The admissibility does not hold for such shear boundary term, while the corresponding boundary terms in the existing literature was proved to be admissible. Two key steps are presented to cope with such problem: First, the semigroup generation and exponential stability for a coupled beam system are verified by Riesz basis approach; second, the admissibility of a control operator for semigroup governed by such coupled beam system is proved. Moreover, Sobolev embedding theorem is introduced to simplify the proof of the boundedness of the closed‐loop systems with respect to the available literature. Some numerical simulations are presented to illustrate the effectiveness. [ABSTRACT FROM AUTHOR] more...

In this paper, we are interested in a boundary damped wave problem with a singular potential. Using a careful spectral analysis, asymptotic expressions of the eigenvalues and eigenvectors of the system operator are derived in terms of the dissipative coefficient and the potential. The Riesz basis property of eigenfunctions and generalized eigenfunctions is also studied. As a consequence, we obtained the exponential stability. [ABSTRACT FROM AUTHOR] more...

In this paper, we consider the feedback boundary stabilization of a flexible beam clamped at one end to a rigid disk, and attached at the other end to a tip mass. The rigid disk is assumed to rotate freely with a non-constant angular velocity. To stabilize the system, we propose a feedback control which consists of a boundary control force and a boundary control moment applied at the tip mass, and a torque control applied on the disk. The well-posedness of the system is proved using the semigroup theory. In the case where the angular velocity of the disk is constant, the system is a Riesz spectral system by the method of spectral analysis. Consequently, the spectrum-determined growth condition and the exponential stability are held. Moreover, for the homogeneous beam, if the control force is the only control applied, the lack of exponential stability is proved and the polynomial stability is established using Borichev-Tomilov's Theorem. The numerical simulations of the solution, the distribution of the eigenvalues, and the energy for the linear subsystem are presented to support the theoretical results. [ABSTRACT FROM AUTHOR] more...

We study the energy decay of a damped Euler-Bernoulli beam which is subject to a point-wise feedback force representing a Shape Memory Alloy (SMA) cable. The problem we consider is that of the paper by [Liu et al. Int. J. Struct. Stab. Dyn.17 (2017) 1750076] but, for simplicity, our mod-elization does not take into account the additional stiffness term they considered. An explicit expression is given for the resolvent of the underlying operator as well as its eigenvalues and eigenfunctions. We show the exponential decay of the energy. The fastest decay rate is given by the supremum of the real part of the spectrum of the infinitesimal generator of the underlying semigroup since we prove the existence of a Riesz basis. To the question "Do Shape Memory Alloy cables restrain the vibrations of girder bridges?", the experiments in [Liu et al. Int. J. Struct. Stab. Dyn.17 (2017) 1750076] answer positively. Our study does not allow to give a definite answer yet. The only presence of these cables may not be enough. Some physical parameters have to be chosen carefully. [ABSTRACT FROM AUTHOR] more...

We have explored the concept of Riesz multiresolution analysis (Riesz MRA) on a locally compact Abelian group G, and have done a detailed study of the methods of construction of a Riesz wavelet from the given Riesz MRA. For simplicity, we have assumed the order of dilations to be two, i.e. we have worked with dyadic dilations. We have proved that precisely one function is required to construct a Riesz wavelet basis for the space L²(G). [ABSTRACT FROM AUTHOR] more...

The decomposition of an image can be done in the following way: The image is split into a low-resolution part, which can be described by a smaller number of samples than the original image, and a signal difference, which describes the difference between the low-resolution image and the real coded image. Therefore, this low-resolution image is also decomposed into a low-resolution image and a difference image, making more efficient coding possible. This decomposition is repeated several times, so that a hierarchical image decomposition is created. Thus, the low-resolution image is only half the size of the original image. This reduced image is enlarged to the size of the original image. The result is a detailed image that is the same size as the original image. Our problem is: "Can we build algorithms allowing the decomposition and reconstruction of a signal in a fuzzy environment? We will answer in the affirmative. This construction is first made possible by one-dimensional fuzzy multiresolution analysis, which will later be extended to two dimensions. In the first part, we recall some definitions and main results obtained on the existence of fuzzy multiresolution analysis in the one-dimensional case, for the decomposition of a fuzzy image. The second part, based on this multi-resolution analysis, presents a fast construction algorithm for the analysis and synthesis of a fuzzy signal. Finally, the third part is nothing else than an extension of this algorithm to 2 dimensions, by the fuzzy multiresolution two-dimensional analysis. more...