The exploration of regularizing operators is crucial in the context of the Cauchy problem for the Helmholtz equation, as it facilitates the stabilization of solutions that are otherwise sensi-tive to perturbations in data. Regularization techniques, by incorporating prior knowledge or additional constraints, can effectively mitigate the inherent instability often present in ill-posed problems. Notably, the Carleman estimate serves as a powerful tool in deriving bounds for solu-tions and establishing the existence of regularizing operators. These estimates not only enhance our understanding of solution behavior but also lead to effective numerical schemes for solv-ing complex inverse problems. In recent years, advancements in computational algorithms have significantly transformed our approach to these challenges. The iterative methods rooted in the Kozlov-Maz’ya-Fomin framework have shown remarkable success in practical applications, pro-viding robust strategies for regularization that improve convergence rates. Leveraging such algo-rithms enables practitioners to obtain approximate solutions that adhere closely to the underlying physical phenomena, thereby enhancing predictive accuracy. Furthermore, interdisciplinary col-laborations draw upon insights from applied mathematics and computational physics, promoting innovative methodologies tailored to various scientific domains. As the frontiers of research in matrix factorizations continue to expand, understanding the interplay between theoretical con-structs and practical applications remains essential for tackling the complexities associated with the Helmholtz equation and its myriad implications across disciplines. The complexity of the Cauchy problem for elliptic equations arises from the interplay between the uniqueness of so-lutions and the characteristics of the data defined on an incomplete boundary. In contrast to Fredholm equations, where one can leverage well-established methods from functional analy-sis, the elliptic setting necessitates a more nuanced approach owing to the non-closed nature of the dense set of solvable data. This results in significant challenges for proving existence and regularity, leading to the necessity for innovative techniques and advanced tools that can oper-ate within this less structured framework. Furthermore, the stability of solutions relies heavily on the geometric and analytical properties of the underlying domain and the operators involved. The works of Aizenberg, Kytmanov, and Tarkhanov provided critical advancements, establishing foundational results that explore stability conditions and perturbation theories specific to elliptic equations. Their contributions laid the groundwork for subsequent investigations into the ro-bustness of solutions amid varying data configurations, enhancing the theoretical understanding of inverse problems. In particular, the incorporation of compactness into the solution space, as suggested by Tikhonov, aids in addressing the instability that arises from non-closed data sets. This restriction allows researchers to apply compactness arguments to derive well-posedness in particular scenarios, illuminating pathways for resolving the broader challenges posed by the elliptic Cauchy problem. The interplay of uniqueness, stability, and data density continues to be an area ripe for exploration, promising rich avenues for future research (see, for instance [1, 18]).

The regularization techniques employed in the context of hyperbolic equations typically in-volve the introduction of additional parameters or constraints that allow for more tractable so-lutions. These techniques can be particularly significant in problems where classical solutions may become ill-defined or exhibit discontinuities. For instance, using smoothing functions or convolution operators can lead to well-defined approximations that effectively capture the un-derlying dynamics of the system, allowing one to analyze the behavior of solutions even in challenging scenarios. Moreover, the adaptability of regularization formulas is a vital aspect of their utility; they can be calibrated according to the specifics of the initial conditions or other in-fluencing factors. This flexibility enhances their application across various domains, from fluid dynamics to wave propagation, encouraging researchers to devise customized approaches tai-lored to the peculiarities of their particular problems. As a result, the theoretical groundwork laid in exploring Cauchy problems becomes invaluable for practitioners seeking reliable com-putational techniques. In essence, the exploration of regularization formulas transcends mere theoretical investigation, fostering synergies between mathematical theory and practical appli-cation. With ongoing advancements in computational methods and numerical algorithms, the prospect of harnessing these regularization strategies in real-world scenarios remains promising, offering a pathway to overcoming challenges inherent in the absence of classical solutions.

This issue pertains to ill-posed problems, indicating a degree of instability. It is acknowl-edged that the Cauchy problem for elliptic equations exhibits instability in response to minor alterations in the data, which signifies its incorrectness (refer to Hadamard, for example, see [3], p. 39). There exists a substantial body of literature on this topic (see, for instance, [2]-[8]). In his work, N.N. Tarkhanov [15] introduced a criterion for determining the solvability of a broader category of boundary value problems concerning elliptic systems. In scenarios involving unsta-ble problems, the operator’s image is not closed. Consequently, the conditions for solvability cannot be expressed solely in terms of continuous linear functionals. Thus, within the Cauchy problem for elliptic equations that utilize data from a portion of the boundary, solutions typically exist uniquely. This problem is valid for a set of data that is dense everywhere, though this set is not closed. As a result, the theoretical framework for the solvability of these problems is far more complex and profound than that related to Fredholm equations. Initial findings in this area emerged in the mid-1980s through the contributions of L.A. Aizenberg, A.M. Kytmanov, and N.N. Tarkhanov (see, for example, [16]). The distinctiveness of the solution is derived from Holmgren’s general theorem (refer to [7]), while the problem’s conditional stability is estab-lished through the research of A.N. Tikhonov (see [1]), provided we limit the set of potential solutions to a compact space.

In pursuit of understanding these complexities, researchers have employed diverse mathemat-ical tools and theoretical frameworks. The exploration of operator theory, particularly within the context of unbounded operators, plays a pivotal role. By identifying appropriate domains, one can establish the continuity and boundedness critical to yielding solvable mathematical models. Moreover, advancements in functional analysis have fostered significant developments, particu-larly in the realm of spectral theory, which offers insights into the behavior of solutions under various conditions and constraints. The interplay between regularization techniques and nu-merical methods has also garnered attention. Techniques such as Tikhonov regularization and Landweber iteration provide robust avenues for approximating solutions to ill-posed problems. By refining these approximations, researchers are now capable of tackling problems that previ-ously appeared intractable. The Cauchy problem related to the Helmholtz equation, for example, exemplifies how these approaches can offer potential solutions that adhere to physical principles while addressing mathematical shortcomings. Furthermore, boundary value problems present a fertile ground for investigation. The complexities associated with non-standard boundary con-ditions often necessitate innovative computational strategies. Recent studies have highlighted the importance of adaptive meshes and iterative solvers, demonstrating their effectiveness in en-hancing the accuracy of solutions across diverse geometries and conditions. As the frontier of mathematical physics continues to expand, collaboration across disciplines remains crucial in deciphering the myriad challenges that lie ahead.

The challenge of reconstructing solutions in this context lies in the inherent limitations im-posed by the boundary conditions. When dealing with first-order elliptic systems, the boundary data often provides insufficient information for a unique solution across the entire domain. As a result, the inverse problem becomes crucial, where one seeks to determine the underlying behav-ior of the solution based on partial or indirect information. This scenario highlights the signifi-cance of various mathematical methods, such as operator theory and functional analysis, which can be employed to address the gaps in data and derive approximations of the desired solutions. Further complicating the situation is the nature of the Helmholtz operator itself. Its factorization involves understanding the interplay between differential operators and the associated bound-ary value problems. In many cases, the factorization leads to a simplification of the underlying equations, but it also raises new questions about the well-posedness and stability of the resulting solution. Researchers have explored various techniques to ensure that the solutions remain robust despite the restricted boundary conditions, often involving regularization methods and stability analysis. Moreover, recent advancements in the field of numerical methods have provided new avenues for tackling these reconstructive challenges. Computational approaches, particularly those based on iterative methods and state-of-the-art algorithms, have shown promise in recov-ering solutions from incomplete data. These methodologies not only enhance our understanding of elliptic systems but also expand their applicability across various scientific domains, from en-gineering to physics, where such systems are prevalent. Consequently, the ongoing investigation into these problems stands at the forefront of modern mathematical research, reflecting both their theoretical significance and practical implications.

The Cauchy problem associated with many elliptic equations has a singular solution, indicat-ing that this problem can be addressed for a data set that is dense throughout the space, although this data set is not regarded as closed. Consequently, the study of the solvability of such prob-lems is quite intricate. References [10]-[21] provide an in-depth exploration of the characteristics and attributes of these issues. In works [22]-[32], approximate solutions to the ill-posed Cauchy problem are analyzed through various factorizations of the Helmholtz operator. Building upon these findings, explicit regularized solutions for the Cauchy problem have been derived for dif-ferent factorizations associated with the Helmholtz operator, as discussed in [33]-[49].

Let R be a m−dimensional real Euclidean space,

x = (x1, . . . , xm) ∈ R , y = (y1, . . . , ym) ∈ R ,

′ m−1 ′ m−1

x = (x1, . . . , xm−1) ∈ R , y = (y1, . . . , ym−1) ∈ R ,

We introduce the following notation:

′ ′ 2 2 2

r = |y − x| , α = |y − x | , w = i u + α + ym, w0 = iα + ym, u ≥ 0, s = α ,

w = iτ u2 + α2 + β, w = iτ α + β, β = τ y , τ = tg , ρ > 1,

Gρ = {y : |y | < τ ym, ym > 0}, ∂Gρ = {y : |y | = τ ym, ym > 0},

T ξ1 ∂ ∂ ∂ ∂ T T   = , . . . , , → ξ , ξ =  ...  −transposed vector ξ, ∂x ∂x1 ∂xm ∂x ξm

T 0 n m

U(x) = (U1(x), . . . , Un(x)) , u = (1, . . . , 1) ∈ R , n = 2 , m ≥ 2, z1 0 ... 0 0 z2 ... 0 E(z) = −diagonal matrix, z = (z1, . . . , zn) ∈ R .

... ... . ...

0 0 ... zn Let D(ξ ) be a (n × n)−dimensional matrix with elements consisting of a set of linear functions with constant coefficients of the complex plane for which the following condition is satisfied:

∗ T T 2 2 0

D (ξ )D(ξ ) = E((|ξ| + λ )u ),

∗ T T

where D (ξ ) is the Hermitian conjugate matrix D(ξ ), λ is a real number.

We will consider the following system of linear first order partial differential equations

D U(x) = 0, (1.1)

where D is the matrix of first-order differential operators.

Gρ ⊂ R be a bounded simply-connected domain, the boundary of which consists of the surface of the cone ∂Gρ, and a smooth piece of the surface S, lying in the cone Gρ, i.e., ∂Gρ = S T, T = ∂Gρ\S.

Let (0, . . . , xm) ∈ Gρ, xm > 0.

We denote by A(Gρ) the class of vector functions in the domain Gρ continuous on Gρ = Gρ ∂Gρ and satisfying system (1.1).

Problem 1. Suppose U(y) ∈ A(Gρ) and

U(y)| = f(y), y ∈ S. (1.2)

Here, f(y) a given continuous vector-function on S. It is required to restore the vector func-tion U(y) in the domainGρ, based on it’s values f(y) on S.

If U(y) ∈ A(Gρ), then the following integral formula of Cauchy type is valid

U(x) = N(y, x; λ)U(y)dsy, x ∈ Gρ. (1.3)

∂Gρ

N(y, x; λ) = E(φm(λr)u )D D(t ).

Here t = (t1, . . . , tm)−is the unit exterior normal, drawn at a point y, the surface ∂Gρ, φm(λr) is the fundamental solution of the Helmholtz equation in R , where φm(λr) defined by the following formula:

H (λr) (m−2) 2 (m−2)/2 φ (λr) = P λ / , (m−2)/2 r (1.4) Pm = .

(m−2)/2 2i(2π)

Here H (λr) is the Hankel function of the first kind of (m − 2) 2−th order (see, for (m−2)/2 instance [9]).

We denote by K(w) is an entire function taking real values for real w,(w = u + iv, u, v−real numbers) and satisfying the following conditions:

p (p)

K(u) ̸= 0, sup v K (w) = B(u, p) < ∞, v≥1 (1.5) −∞ < u < ∞, p = 0, 1, ..., m.

We define the function Φ(y, x; λ) at y ̸= x by the following equalities 1 ∂ K(w) uI0(λu) Φ(y, x; λ) = Im √ du,

k−1 2 2

cmK(xm) ∂s w − xm u + α (1.6)

m = 2k, k ≥ 1, cm = (−1) (k − 1)!(m − 2)ωm,

1 ∂ K(w) cos(λu) Φ(y, x; λ) = Im √ du,

k−1 2 2

cmK(xm) ∂s w − xm u + α (1.7)

k −k

m = 2k + 1, k ≥ 1, cm = (−1) 2 (2k − 1)!(m − 2)πωm.

Where I0(λu) = J0(iλu)−is the Bessel function of the first kind of zero order, ωm−area of a unit sphere in space R .

In the formula (1.6) and (1.7), choosing

1/ρ 1/ρ K(w) = Eρ(σ w), K(xm) = Eρ(σ γ), γ = τ xm, σ > 0. (1.8)

we get −1 1/ρ k−1 1/ρ Eρ (σ γ) ∂ Eρ(σ w) uI0(λu) Φσ(y, x; λ) = Im √ du,

k−1 2 2

cm ∂s w − xm u + α (1.9)

m = 2k, k ≥ 1,

−1 1/ρ k−1 1/ρ Eρ (σ γ) ∂ Eρ(σ w) cos(λu) Φσ(y, x; λ) = Im √ du,

k−1 2 2

cm ∂s w − xm u + α (1.10)

m = 2k + 1, k ≥ 1.

Here Eρ(σ w) is the entire Mittag-Leffler function (see [7]).

The formula (1.3) is true if instead φm(λr) of substituting the function

Φσ(y, x; λ) = φm(λr) + gσ(y, x; λ). (1.11)

Then the integral formula (1.3) has the form:

U(x) = Nσ(y, x; λ)U(y)dsy, x ∈ Gρ. (1.12)

∂Gρ

Nσ(y, x; λ) = E(Φσ(y, x; λ)u )D D(t )

equation in a multidimensional bounded domain

The Cauchy problem for matrix factorizations of the Helmholtz equation presents a complex challenge in applied mathematics and computational physics, particularly within the context of multidimensional bounded domains. This problem involves determining the unknown parameter fields of the equation from incomplete or noisy measurements, a situation often encountered in fields such as geophysics and biomedical imaging. The Helmholtz equation itself, characterized by the wave nature of solutions, necessitates careful consideration of boundary conditions and matrix structures that arise in its factorization approaches. Exploring the interplay between lin-ear algebra and numerical analysis is essential for understanding the intricate behavior of the solutions to the Helmholtz equation. Recent advancements in computational power and algo-rithms have further enhanced our ability to simulate and solve these types of problems effec-tively, paving the way for innovative applications across various scientific domains.

Theorem 2.1. Let U(y) ∈ A(Gρ) it satisfy the inequality

|U(y)| ≤ M, y ∈ T = ∂Gρ\S, x ∈ Gρ. (2.1)

If

Uσ(x) = Nσ(y, x; λ)U(y)dsy, x ∈ Gρ, (2.2)

then the following estimates are true:

|U(x) − Uσ(x)| ≤ Cρ(λ, x)M σ exp(−σγ ), σ > 1, x ∈ Gρ, (2.3)

∂U(x) ∂Uσ(x) k ρ − ≤ Cρ(λ, x)M σ exp(−σγ ), σ > 1, x ∈ Gρ, j = 1, m, (2.4)

k+1 ρ

|U(x) − Uσ(x)| ≤ Cρ(x)M σ exp(−σγ ), σ > 1, x ∈ Gρ, (2.5)

∂U(x) ∂Uσ(x) k+1 ρ − ≤ Cρ(x)M σ exp(−σγ ), σ > 1, x ∈ Gρ, j = 1, m. (2.6)

Here and below functions bounded on compact subsets of the domain Gρ, we denote by Cρ(λ, x) and Cρ(x).

Corollary 2.2. For each x ∈ Gρ the equalities are true

∂Uσ(x) ∂U(x) lim Uσ(x) = U(x), lim = , j = 1, m.

σ→∞ σ→∞ ∂xj ∂xj

We denote by Gε the set

Gε = {(x1, . . . , xm) ∈ Gρ, a > xm ≥ ε, a = max ψ(x ), 0 < ε < a}.

Here, at m = 2, ψ(x1)−is a surface, and at m > 2, ψ(x )−is a surface. It is easy to see that the set Gε ⊂ Gρ is compact.

∂Uσ(x) Corollary 2.3. If x ∈ Gε , then the families of functions {Uσ(x)} and converge

uniformly for σ → ∞, i.e.:

∂Uσ(x) ∂U(x) Uσ(x)⇒U(x), ⇒ , j = 1, m.

Suppose that the curve S is given by the equation

′ ′ m−1

ym = ψ(y ), y ∈ R

where ψ(y ) is a single-valued function satisfying the Lyapunov conditions.

We put

′ ′2 ′

a = max ψ(y ), b = max 1 + ψ (y ).

Theorem 2.4. Let U(y) ∈ A(Gρ) satisfy condition (1.9), and on a smooth curve S the inequality

|U(y)| ≤ δ, 0 < δ < M, y ∈ S. (2.7)

Then the following estimates is true

γ ρ γ ρ

k 1−( ) ( ) |U(x)| ≤ Cρ(λ, x)σ M δ , σ > 1, x ∈ Gρ, (2.8)

γ ρ γ ρ

∂U(x) k 1− ( a ) ( a ) ≤ Cρ(λ, x)σ M δ , σ > 1, x ∈ Gρ, j = 1, m. (2.9)

γ ρ γ ρ

k+1 1−( ) ( ) |U(x)| ≤ Cρ(x)σ M δ , σ > 1, x ∈ Gρ, (2.10)

γ ρ γ ρ

∂U(x) k+1 1− ( a ) ( a ) ≤ Cρ(x)σ M δ , σ > 1, x ∈ Gρ, j = 1, m. (2.11)

Here is a = max Re w .

Let U(y) ∈ A(Gρ) and instead of functions U(y) on S with its approximations fδ(y)respectively, with an error 0 < δ < M,

max |U(y) − fδ(y)| ≤ δ. (2.12)

We put Z Uσ(δ)(x) = Nσ(y, x; λ)fσ(y)dsy, x ∈ Gρ. (2.13)

Theorem 2.5. Let U(y) ∈ A(Gρ) on the part of the plane ym = 0 satisfy condition (1.9) Then the following estimates is true

γ ρ γ ρ

k 1−( ) ( ) U(x) − Uσ(δ)(x) ≤ Cρ(λ, x)σ M δ , σ > 1, x ∈ Gρ, (2.14)

∂U (x) γ ρ γ ρ ∂U(x) σ(δ) k 1− ( a ) ( a ) − ≤ Cρ(λ, x)σ M δ , σ > 1, x ∈ Gρ, j = 1, m. (2.15)

γ ρ γ ρ

k+1 1−( ) ( ) U(x) − Uσ(δ)(x) ≤ Cρ(x)σ M δ , σ > 1, x ∈ Gρ, (2.16)

∂U (x) γ ρ γ ρ ∂U(x) σ(δ) k+1 1− ( a ) ( a ) − ≤ Cρ(x)σ M δ , σ > 1, x ∈ Gρ, j = 1, m. (2.17)

Corollary 2.6. For each x ∈ Gρ, the equalities are true

∂Uσ(δ)(x) ∂U(x) lim Uσ(δ)(x) = U(x), lim = , j = 1, m.

δ→0 δ→0 ∂xj ∂xj

∂Uσ(δ)(x) Corollary 2.7. If x ∈ Gε, then the families of functions {Uσ(δ)(x)} and converge

uniformly for δ → 0, i.e.:

∂Uσ(δ)(x) ∂U(x) Uσ(δ)(x)⇒U(x), ⇒ , j = 1, m.

In this paper, we have found an approximate solution to the problem based on the properties of the Carleman matrix. If the Carleman matrix is known, then it is no longer difficult to find a regularized solution in explicit form. In this case, we have that the solution to the problem exists and is continuously differentiable in a closed region with exactly specified Cauchy data.

We note that for solving applicable problems, the approximate values of U(x) and ∂U(x) , x ∈ Gρ, j=1,m should be found.

As a result, we constructed a family of vector functions U(x, fδ) =Uσ(δ)(x) and ∂U(x, fδ) ∂Uσ(δ)(x) = , (j=1,m), which depend on the parameter σ. It is additionally proved

that under specific conditions and a special choice of the parameter σ=σ(δ), at δ → 0, the family ∂Uσ(δ)(x) ∂U(x) Uσ(δ)(x) and are convergent to a solution U(x) and its derivative , x ∈ Gρ at

∂Uσ(δ)(x) point x ∈ Gρ. Here we will call Uσ(δ)(x) and the regularized solution of the problems

(1.1) and (1.2).

equation in a bounded domain R

In addressing this, matrix factorization techniques become instrumental. These methods leverage properties of linear algebra to decompose the matrix representation of the differential operator associated with the Helmholtz equation. By transforming the problem into a manageable format, one can employ regularization strategies to mitigate the sensitivity to noise and improve numeri-cal stability. The use of singular value decomposition (SVD) and other factorization approaches enables the reconstruction of the wave field from insufficient or noisy boundary data. Further-more, applying appropriate boundary conditions and leveraging variational principles leads to results that are both theoretically robust and practically relevant. The interplay between analyt-ical techniques and numerical simulations is crucial in understanding the behavior of solutions in various geometries and for different boundary conditions. Thus, the Cauchy problem for ma-trix factorizations of the Helmholtz equation remains an active area of research with profound implications across scientific disciplines.

G ⊂ R is a bounded simply-connected domain with piecewise smooth boundary consisting of the plane T: ym= 0 and a smooth surface S lying in the half-space ym> 0, i.e., ∂G=S∪T, T = ∂G\S.

We denote by A(G) the class of vector functions in the domain G continuous on G = G ∂G and satisfying system (1.1).

Problem 2. Suppose U(y) ∈ A(G) and

U(y)| = f(y), y ∈ S. (3.1)

Here, f(y) a given continuous vector-function on S. It is required to restore the vector func-tion U(y) in the domainG, based on it’s values f(y) on S.

If U(y) ∈ A(G), then the following integral formula of Cauchy type is valid

U(x) = N(y, x; λ)U(y)dsy, x ∈ G. (3.2)

N(y, x; λ) = E(φm(λr)u )D D(t ).

Here t = (t1, . . . , tm)−is the unit exterior normal, drawn at a point y, the surface ∂G, φm(λr) is the fundamental solution of the Helmholtz equation in R , where φm(λr) defined by the formula (1.4).

We define the function Φ(y, x; λ) at y ̸= x by the equalities (1.6) and (1.7).

In the formula (1.6) and (1.7), choosing K(w) = exp(σw), K(xm) = exp(σxm), σ > 0. (3.3) we get

−σxm k−1

e ∂ exp(σw) uI0(λu) Φσ(y, x; λ) = Im √ du,

k−1 2 2

cm ∂s w − xm u + α (3.4)

m = 2k, k ≥ 1,

−σxm k−1

e ∂ exp(σw) cos(λu) Φσ(y, x; λ) = Im √ du,

k−1 2 2

cm ∂s w − xm u + α (3.5)

m = 2k + 1, k ≥ 1.

The formula (3.2) is true if instead φm(λr) of substituting the function

Φσ(y, x; λ) = φm(λr) + gσ(y, x; λ). (3.6) Then the integral formula (3.2) has the form:

U(x) = Nσ(y, x; λ)U(y)dsy, x ∈ G. (3.7)

Nσ(y, x; λ) = E(Φσ(y, x; λ)u )D D(t )

Theorem 3.1. Let U(y) ∈ A(G) it satisfy the inequality

|U(y)| ≤ M, y ∈ T = ∂G\S, x ∈ G. (3.8) If Z Uσ(x) = Nσ(y, x; λ)U(y)dsy, x ∈ G, (3.9)

then the following estimates are true

k −σxm

|U(x) − Uσ(x)| ≤ C(λ, x)M σ e , σ > 1, x ∈ G, (3.10)

∂U(x) ∂Uσ(x) k −σx − ≤ C(λ, x)M σ e , σ > 1, x ∈ G, j = 1, m, (3.11)

k+1 −σxm

|U(x) − Uσ(x)| ≤ C(x)M σ e , σ > 1, x ∈ G, (3.12)

∂U(x) ∂Uσ(x) k+1 −σx − ≤ C(x)M σ e , σ > 1, x ∈ G, j = 1, m. (3.13)

Here and below functions bounded on compact subsets of the domain G, we denote by C(λ, x) and C(x).

Corollary 3.2. For each x ∈ G, the equalities are true

∂Uσ(x) ∂U(x) lim Uσ(x) = U(x), lim = , j = 1, m.

σ→∞ σ→∞ ∂xj ∂xj

We denote by Gε the set

Gε = {(x1, . . . , xm) ∈ G, a > xm ≥ ε, a = max ψ(x ), 0 < ε < a}.

Here, at m = 2, ψ(x1)−is a curve, and at m > 2, ψ(x )−is a surface. It is easy to see that the set Gε ⊂ G is compact.

∂Uσ(x) Corollary 3.3. If x ∈ Gε, then the families of functions {Uσ(x)} and converge

uniformly for σ → ∞, i.e.:

∂Uσ(x) ∂U(x) Uσ(x)⇒U(x), ⇒ , j = 1, m.

Suppose that the curve S is given by the equation

′ ′ m−1

ym = ψ(y ), y ∈ R

where ψ(y ) is a single-valued function satisfying the Lyapunov conditions.

We put

′ ′2 ′

a = max ψ(y ), b = max 1 + ψ (y ).

Theorem 3.4. Let U(y) ∈ A(G) satisfy condition (1.12), and on a smooth surface S the inequal-ity

|U(y)| ≤ δ, 0 < δ < M, y ∈ S. (3.14)

Then the following estimates is true

xm xm

k 1−

|U(x)| ≤ C(λ, x)σ M a δ a , σ > 1, x ∈ G, (3.15)

∂U(x) xm xm

k 1−

≤ C(λ, x)σ M a δ a , σ > 1, x ∈ G, j = 1, m. (3.16)

xm xm

k+1 1−

|U(x)| ≤ C(x)σ M a δ a , σ > 1, x ∈ G, (3.17)

∂U(x) xm xm

k+1 1−

≤ C(x)σ M a δ a , σ > 1, x ∈ G, j = 1, m. (3.18)

Let U(y) ∈ A(G) and instead of functions U(y) on S with its approximations fδ(y)respectively, with an error 0 < δ < M,

max |U(y) − fδ(y)| ≤ δ (3.19)

We put Z

Uσ(δ)(x) = Nσ(y, x; λ)fσ(y)dsy, x ∈ G. (3.20)

Theorem 3.5. Let U(y) ∈ A(G) on the part of the plane ym = 0 satisfy condition (3.8).

Then the following estimates is true

xm xm

k 1−

U(x) − U (x) ≤ C(λ, x)σ M a δ a , σ > 1, x ∈ G, (3.21)

σ(δ)

∂U(x) ∂U (x) xm xm

σ(δ) k 1−

− ≤ C(λ, x)σ M a δ a , σ > 1, x ∈ G, j = 1, m. (3.22)

xm xm

k+1 1−

U(x) − U (x) ≤ C(x)σ M a δ a , σ > 1, x ∈ G, (3.23)

σ(δ)

∂U(x) ∂U (x) xm xm

σ(δ) k+1 1−

− ≤ C(x)σ M a δ a , σ > 1, x ∈ G, j = 1, m. (3.24) Corollary 3.6. For each x ∈ G , the equalities are true

∂Uσ(δ)(x) ∂U(x) lim Uσ(δ)(x) = U(x), lim = , j = 1, m.

δ→0 δ→0 ∂xj ∂xj

∂Uσ(δ)(x) Corollary 3.7. If x ∈ Gε, then the families of functions {Uσ(δ)(x)} and converge

uniformly for δ → 0, i.e.:

∂Uσ(δ)(x) ∂U(x) Uσ(δ)(x)⇒U(x), ⇒ , j = 1, m.

In this paper, we have found an approximate solution to the problem based on the properties of the Carleman matrix. If the Carleman matrix is known, then it is no longer difficult to find a regularized solution in explicit form. In this case, we have that the solution to the problem exists and is continuously differentiable in a closed region with exactly specified Cauchy data.

We note that for solving applicable problems, the approximate values of U(x) and ∂U(x) , x ∈ G, j=1,m should be found.

As a result, we constructed a family of vector functions U(x, fδ) =Uσ(δ)(x) and ∂U(x, fδ) ∂Uσ(δ)(x) = , (j=1,m), which depend on the parameter σ. It is additionally proved

that under specific conditions and a special choice of the parameter σ=σ(δ), at δ → 0, the family ∂Uσ(δ)(x) ∂U(x) Uσ(δ)(x) and are convergent to a solution U(x) and its derivative , x ∈ G at

∂Uσ(δ)(x) point x ∈ G. Here we will call Uσ(δ)(x) and the regularized solution of the problems

(1.3) and (3.1).

equation in an unbounded multidimensional domain

To tackle the Cauchy problem effectively, one must consider the appropriate boundary condi-tions and how they relate to the unbounded domain. The utilization of matrix decomposition techniques, such as singular value decomposition or the QR factorization, provides unique ad-vantages in analyzing the stability and convergence of the solutions. These factorizations en-able a clearer understanding of the underlying structure of the solution space, facilitating nu-merical methods that can efficiently approximate solutions even in complex scenarios. More-over, exploring the interplay between the Helmholtz equation and various numerical method-ologies—including finite element methods and boundary integral equations—underscores the versatility of matrix factorization approaches. Establishing a robust theoretical framework is essential for deriving convergence results and ensuring that numerical solutions closely adhere to the analytical counterparts derived from the original Cauchy problem. Ultimately, advancing our understanding of matrix factorizations within the context of the Helmholtz equation not only enhances solution techniques but also broadens the applicability of these methods across diverse fields such as acoustics, electromagnetism, and geophysics. The ongoing research in this area promises to unlock new insights and improve computational efficiency, thereby addressing some of the pressing challenges in applied mathematics and theoretical physics.

G ⊂ R is an unbounded simply-connected domain with piecewise smooth boundary con-sisting of the plane T: ym= 0 and a smooth surface S lying in the half-space ym> 0, i.e., ∂G=S ∪ T, T = ∂G\S.

We denote by A(G) the class of vector functions in the domain G continuous on G = G ∂G and satisfying system (1.1).

Problem 3. Suppose U(y) ∈ A(G) and

U(y)| = f(y), y ∈ S. (4.1) Here, f(y) a given continuous vector-function on S. It is required to restore the vector func-tion U(y) in the domain G, based on it’s values f(y) on S.

If U(y) ∈ A(G), then the following integral formula of Cauchy type is valid

U(x) = N(y, x; λ)U(y)dsy, x ∈ G. (4.2)

N(y, x; λ) = E(φm(λr)u )D D(t ).

Here t = (t1, . . . , tm)−is the unit exterior normal, drawn at a point y, the surface ∂G, φm(λr) is the fundamental solution of the Helmholtz equation in R , where φm(λr) defined by the formula (1.4).

Let G ⊂ R be an unbounded domain, with a piecewise smooth boundary ∂G (∂G−extends to infinity).

We denote by GR the part G lying inside the circle of radius R with center at zero:

GR = {y : y ∈ G, |y| < R}, GR = G\GR, R > 0.

Theorem 4.1. Let U(y) ∈ A(G), G be a finitely connected unbounded domain in R , with piecewise-smooth boundary ∂G. If for each fixed x ∈ G we have the equality

lim N(y, x; λ)U(y)dsy = 0, (4.3)

then the formulas (1.6) and (1.7) is true.

We denote by Aρ(G) is the class of vector-valued functions from A(G), satisfying the fol-lowing growth condition:

Aρ(G) = {U(y) ∈ A(G), |U(y)| ≤ exp[0(exp ρ |y |)], y → ∞, y ∈ G}.

We define the function Φ(y, x; λ) at y ̸= x by the equalities (1.6) and (1.7).

In the formula (1.6) and (1.7), choosing

K(w) = exp(σw), k ≥ 1, σ > 0, (w − xm + 2h) (4.4) K(xm) = exp(σxm), 0 < xm < h, h = .

(2h) ρ we get

−σxm k−1

e ∂ exp(σw) uI0(λu) Φσ(y, x; λ) = Im √ du,

−k k−1 k 2 2

cm(2h) ∂s (w − xm + 2h) (w − xm) u + α (4.5)

m = 2k, k ≥ 1,

−σxm k−1

e ∂ exp(σw) cos(λu) Φσ(y, x; λ) = Im √ du,

−k k−1 k 2 2

cm(2h) ∂s (w − xm + 2h) (w − xm) u + α (4.6)

m = 2k + 1, k ≥ 1.

The formula (4.2) is true if instead φm(λr) of substituting the function

Φσ(y, x; λ) = φm(λr) + gσ(y, x; λ). (4.7) Then the integral formula (4.2) has the form:

U(x) = Nσ(y, x; λ)U(y)dsy, x ∈ G. (4.8) Nσ(y, x; λ) = E(Φσ(y, x; λ)u )D D(t )

Here is a = max Re w .

Theorem 4.2. Let U(y) ∈ Aρ(G) it satisfy the inequality

|U(y)| ≤ M, y ∈ T = ∂G\S, x ∈ G. (4.9)

If Z Uσ(x) = Nσ(y, x; λ)U(y)dsy, x ∈ G, (4.10)

then the following estimates are true

k −σxm

|U(x) − Uσ(x)| ≤ Cρ(λ, x)M σ e , σ > 1, x ∈ G, (4.11)

∂U(x) ∂Uσ(x) k −σx − ≤ Cρ(λ, x)M σ e , σ > 1, x ∈ G, j = 1, m, (4.12)

k+1 −σxm

|U(x) − Uσ(x)| ≤ Cρ(x)M σ e , σ > 1, x ∈ G, (4.13)

∂U(x) ∂Uσ(x) k+1 −σx − ≤ Cρ(x)M σ e , σ > 1, x ∈ G, j = 1, m. (4.14)

Here and below functions bounded on compact subsets of the domain G, we denote by Cρ(λ, x) and Cρ(x).

Corollary 4.3. For each x ∈ G, the equalities are true

∂Uσ(x) ∂U(x) lim Uσ(x) = U(x), lim = , j = 1, m.

σ→∞ σ→∞ ∂xj ∂xj

We denote by Gε the set

Gε = {(x1, . . . , xm) ∈ G, a > xm ≥ ε, a = maxψ(x ), 0 < ε < a}.

Here, at m = 2, ψ(x1)−is a curve, and at m > 2, ψ(x )−is a surface. It is easy to see that the set Gε ⊂ G is compact.

∂Uσ(x) Corollary 4.4. If x ∈ Gε then the families of functions {Uσ(x)} and converge uni-

formly for σ → ∞, i.e.:

∂Uσ(x) ∂U(x) Uσ(x)⇒U(x), ⇒ , j = 1, m.

Suppose that the curve S is given by the equation

′ ′ m−1

ym = ψ(y ), y ∈ R ,

where ψ(y ) is a single-valued function satisfying the Lyapunov conditions.

We put

′ ′2 ′

a = maxψ(y ), b = max 1 + ψ (y ).

Theorem 4.5. Let U(y) ∈ Aρ(G) satisfy condition (1.12), and on a smooth curve S the inequality

|U(y)| ≤ δ, 0 < δ < M, y ∈ S. (4.15)

Then the following estimates is true

xm xm

k 1−

|U(x)| ≤ C (λ, x)σ M a δ a , σ > 1, x ∈ G, (4.16)

∂U(x) xm xm

k 1−

≤ C (λ, x)σ M a δ a , σ > 1, x ∈ G, j = 1, m. (4.17)

at m = 2k + 1, , k ≥ 1:

xm xm

k+1 1−

|U(x)| ≤ C (x)σ M a δ a , σ > 1, x ∈ G, (4.18)

∂U(x) xm xm

k+1 1−

≤ C (x)σ M a δ a , σ > 1, x ∈ G, j = 1, m. (4.19)

Let U(y) ∈ Aρ(G) and instead of functions U(y) on S with its approximations fδ(y) respec-tively, with an error 0 < δ < 1,

max |U(y) − fδ(y)| ≤ δ. (4.20)

We put Z Uσ(δ)(x) = Nσ(y, x; λ)fσ(y)dsy, x ∈ G. (4.21)

Theorem 4.6. Let U(y) ∈ Aρ(G) on the part of the plane ym = 0 satisfy condition (4.9).

Then the following estimates is true

xm xm

k 1−

U(x) − U (x) ≤ C (λ, x)σ M a δ a , σ > 1, x ∈ G, (4.22)

σ(δ) ρ

∂U(x) ∂U (x) xm xm

σ(δ) k 1−

− ≤ C (λ, x)σ M a δ a , σ > 1, x ∈ G, j = 1, m. (4.23)

xm xm

k+1 1−

U(x) − U (x) ≤ C (x)σ M a δ a , σ > 1, x ∈ G, (4.24)

σ(δ) ρ

∂U(x) ∂U (x) xm xm

σ(δ) k+1 1−

− ≤ C (x)σ M a δ a , σ > 1, x ∈ G, j = 1, m. (4.25)

Corollary 4.7. For each x ∈ G, the equalities are true

∂Uσ(δ)(x) ∂U(x) limUσ(δ)(x) = U(x), lim = , j = 1, m.

δ→0 δ→0 ∂xj ∂xj

∂Uσ(δ)(x) Corollary 4.8. If x ∈ Gε, then the families of functions {Uσ(δ)(x)} and converge

uniformly for δ → 0, i.e.:

∂Uσ(δ)(x) ∂U(x) Uσ(δ)(x)⇒U(x), ⇒ , j = 1, m.

In this paper, we have found an approximate solution to the problem based on the properties of the Carleman matrix. If the Carleman matrix is known, then it is no longer difficult to find a regularized solution in explicit form. In this case, we have that the solution to the problem exists and is continuously differentiable in a closed region with exactly specified Cauchy data.

We note that for solving applicable problems, the approximate values of U(x) and ∂U(x) , x ∈ Gρ, j=1,m should be found.

As a result, we constructed a family of vector functions U(x, fδ) =Uσ(δ)(x) and ∂U(x, fδ) ∂Uσ(δ)(x) = , (j=1,m), which depend on the parameter σ. It is additionally proved

that under specific conditions and a special choice of the parameter σ=σ(δ), at δ → 0, the family ∂Uσ(δ)(x) ∂U(x) Uσ(δ)(x) and are convergent to a solution U(x) and its derivative , x ∈ Gρ at

∂Uσ(δ)(x) point x ∈ Gρ. Here we will call Uσ(δ)(x) and the regularized solution of the problems

(1.1) and (4.1).

To establish the groundwork for our analysis, we first consider the mathematical structure of the Helmholtz equation, emphasizing the significance of its matrix factorizations. These factoriza-tions play a pivotal role in addressing the ill-posed nature of the Cauchy problem, particularly within the framework of both bounded and unbounded domains. We investigate the interplay be-tween the analytical properties of the solution and the characteristics of the Cauchy data, reveal-ing the conditions under which stable extensions can be derived. Subsequent to the derivation of explicit formulae, we apply a regularization strategy that employs both spectral and non-spectral approaches. This duality not only facilitates the stabilization of solutions but also ensures that the convergence behavior remains consistent even in the presence of perturbations. We demon-strate that these regularized solutions retain the essential features of the original problem while mitigating the adverse effects of noise in the data. The stability estimation we present hinges on classical functional analysis principles, allowing for a comprehensive evaluation of the solu-tion’s robustness. By performing a thorough sensitivity analysis, we identify critical thresholds that delineate the boundaries of stability, thus equipping practitioners with essential insights for application in various fields, including acoustics, electromagnetism, and geophysical imaging. Our investigation further elucidates the significant role of matrix factorizations in the stability of solutions to the Helmholtz equation under diverse conditions. By formulating the problem within the context of operator theory, we establish connections between the eigenvalue spectra of the governing operators and the stability of numerical solutions. In particular, we derive bounds on the condition numbers associated with various factorization schemes, which serve as indicators of stability in the presence of noisy Cauchy data. This analytical framework not only enriches our understanding of the equation’s eigenfunctions but also provides a pathway for enhancing computational algorithms. Moreover, the application of regularization techniques, particularly in ill-posed problems, becomes paramount in ensuring that our solutions are not only mathemat-ically coherent but also practically viable. By drawing from a variety of regularization method-ologies, including Tikhonov and Landweber approaches, we showcase how integrating multiple regularization pathways can yield superior outcomes. This multidimensional strategy allows for a more robust framework capable of contending with real-world data imperfections, enhancing the adaptability of the Helmholtz solution across applications in engineering and physics. Lastly, our sensitivity analysis reveals that the interplay between the Cauchy data and the Helmholtz equation’s inherent properties delineates critical operational thresholds. These insights empower practitioners to make informed decisions that optimize performance while safeguarding against instability, underscoring the importance of this research within both theoretical and applied con-texts.

Through this research, we contribute to a deeper mathematical understanding while providing practical methodologies for real-world applications. By translating our mathematical findings into actionable methodologies, we pave the way for improved practices in fields reliant on precise modeling of wave phenomena.

In conclusion, the authors expresses his sincere gratitude to the professor of Anand Interna-tional College of Engineering, the Department of Mathematics, Professor, Praveen Agarwal and reviewer for useful advice.