The most actively developing modern area of scientific knowledge is the theory of correctly and incorrectly posed problems, most of which have practical value and require decision making in uncertain or contradictory conditions. The development and justification of methods for solving such a complex class of problems as ill-posed ones is an urgent problem of the present time. The theory of ill-posed problems is an apparatus of scientific research for many scientific ar-eas, such as differentiation of approximately given functions, solving inverse boundary value problems, solving problems of linear programming and control systems, solving degenerate or ill-conditioned systems of linear equations, etc. The concept of a well-posed problem is due to J. Hadamard (1923), who took the point of view that every mathematical problem corresponding to some physical or technological problem must be well-posed. In fact, what physical interpre-tation can a solution have if an arbitrary small change in the data can lead to large changes in the solution? Moreover, it would be difficult to apply approximation methods to such problems. This put the expediency of studying ill-posed problems in doubt [34]. For ill-posed problems of the question arises: What is meant by an approximate solution? Clearly, it should be so defined that it is stable under small changes of the original information. A second question is: What algorithms are there for the construction of such solutions? Answers to these basic questions were given by A.N. Tikhonov (see [2]).

The concept of conditional correctness first appeared in the work of Tikhonov [2], and then in the studies of Lavrent’ev [37, 38]. In a theoretical study of the conditional correctness (cor-rectness according to Tikhonov) of an ill-posed problem of the existence of a solution and its belonging to the correctness set, it is postulated in the very formulation of the problem. The study of uniqueness issues in a conditionally well-posed formulation does not essentially differ from the study in a classically well-posed formulation, and the stability of the solution from the data of the problem is required only from those variations of the data that do not deduce solu-tions from the well-posedness set. After establishing the uniqueness and stability theorems in the study of the conditional correctness of ill-posed problems, the question arises of construct-ing effective solution methods, i.e. construction of regularizing operators. It is known that the Cauchy problem for elliptic equations and for systems of elliptic equations belongs to the class of ill-posed problems (see, for example, [1, 2, 29, 34], [36]-[38], [46]-[48]). Boundary value problems, as well as numerical solutions of some problems, are considered in [3]-[6], [30]-[33], [35, 39, 44], [49]-[51].

Based on the results of previous works [37, 38], [40]-[42], [46, 47] we have constructed the Carleman matrix and based on it the approximate solution of the Cauchy problem for the ma-trix factorization of the Helmholtz equation. In this article, we find an explicit formula for an approximate solution of the Cauchy problem for matrix factorizations of the Helmholtz equa-tion in a multidimensional bounded domain of an odd-dimensional space R . The case of an even-dimensional space will be considered in other scientific studies of the authors. Our ap-proximate solution formula also includes the construction of a family of fundamental solutions of the Helmholtz operator in space. This family is parametrized by some entire function K(z), the choice of which depends on the dimension of the space. In this work, relying on the results of previous works [7]-[16], we similarly obtain better results with approximate estimates due to a special selection of the function K(z). This helped to get good results when finding an approximate solution based on the Carleman matrix. The Carleman matrix or Carleman func-tion for some elliptic equations and systems was considered in the following studies [7]-[16], [28, 37, 38], [40]-[43], [46, 47].

When solving correct problems, sometimes it is not possible to find the value of the vector function on the entire boundary. Finding the value of a vector function on the entire boundary for systems of elliptic type with constant coefficients (see, for example, [7]-[27]) is one of the topical problems in the theory of differential equations.

For the last decades, interest in classical ill-posed problems of mathematical physics has remained. This direction in the study of the properties of solutions of the Cauchy problem for the Laplace equation was started in [37, 38, 46, 47] and subsequently developed in [7]-[16], [28], [40]-[43].

Let R , (m = 2k + 1, k ≥ 1) be the m− dimensional real Euclidean space,

ζ = (ζ1, . . . , ζm) ∈ R , η = (η1, . . . , ηm) ∈ R ,

′ m−1 ′ m−1

ζ = (ζ1, . . . , ζm−1) ∈ R , η = (η1, . . . , ηm−1) ∈ R .

Next, we use the following notation:

′ ′ 2 2

r = |η − ζ| , α = |η − ζ | , z = i a + α + ηm, a ≥ 0,

T T T   ∂ζ = (∂ζ , . . . , ∂ζ ) , ∂ζ = χ , χ =  . . .  − transposed vector χ, χm

T 0 n m

W(ζ) = (W1(ζ), . . . , Wn(ζ)) , v = (1, . . . , 1) ∈ R , n = 2 , m ≥ 3,

w1 0 · · · 0 0 w2 · · · 0 E(w) = − diagonal matrix, w = (w1, . . . , wn) ∈ R .

· · · · · · . · · · 0 0 0 wn

We also consider a bounded simply-connected domain Ω ⊂ R , having a piecewise smooth boundary ∂Ω = Σ D, where Σ is a smooth surface lying in the half-space ηm > 0 and D is the plane ηm = 0.

P(χ ) is an (n × n)−dimensional matrix satisfying:

∗ T T 2 2 0

P (χ )P(χ ) = E((|χ| + λ )v ),

∗ T T

where P (χ ) is the Hermitian conjugate matrix of P(χ ), λ ∈ R, the elements of the matrix P(χ ) consist of a set of linear functions with constant coefficients from the complex plane C.

Let us consider the following first order systems of linear partial differential equations with constant coefficients P (∂ζ) W(ζ) = 0, (2.1) in the domain Ω, where P (∂ζ) is the matrix differential operator of the first-order.

Also consider the set S (Ω) = W : Ω −→ R ,

here W is continuous on Ω = Ω ∪ ∂Ω and W satisfies the system (2.1).

The Cauchy problem for system (2.1) is formulated as follows:

Let f : Σ −→ R be a continuous given function on Σ. Suppose W(η) ∈ S(Ω) and

W(η)| = f(η), η ∈ Σ. (2.2)

Our purpose is to determine the function W(η) in the domain Ω when its values are known on Σ.

If W(η) ∈ S(Ω), then the following integral representation holds

W(ζ) = L(η, ζ; λ)W(η)dsη, ζ ∈ Ω, (2.3)

where

0 ∗ T

L(η, ζ; λ) = E Γm(λr)v P (∂ζ) P(t ).

Here t = (t1, . . . , tm)−is the unit exterior normal, drawn at a point η, the surface ∂Ω, Γm(λr)− is the fundamental solution of the Helmholtz equation in R , (m = 2k + 1, k ≥ 1), where Γm(λr) defined by the following formula (see [45]):

H (λr) (m−2) 2 (m−2)/2 Γ (λr) = B λ / , (m−2)/2

Bm = , m = 2k + 1, k ≥ 1.

(m−2)/2 2i(2π)

Let K(z) be an entire function taking real values for real z, (z = a + ib; a, b ∈ R) such that

p (p)

K(a) ̸= 0, sup b K (z) = N(a, p) < ∞,

−∞ < a < ∞, p = 0, 1, . . . , m.

We define the function Ψ(η, ζ; λ) at η ̸= ζ by the following equality

1 ∂ K(z) cos(λa) Ψ(η, ζ; λ) = Im √ da,

k−1 2 2

cmK(ζm) ∂s z − ζm a + α 0 (2.6)

m = 2k + 1, k ≥ 1,

k −k m

where cm = (−1) 2 (2k − 1)!(m − 2)πωm, k ≥ 1; ωm is the area of a unit sphere in R .

In equality (2.6), choosing the function K(z) in the form

K(z) = exp(σz ), K(ζm) = exp(σζm), σ > 0, (2.7)

we get

−σζm k−1 2

e ∂ exp(σz ) cos(λa) Ψσ(η, ζ; λ) = Im √ da. (2.8)

k−1 2 2

cm ∂s z − ζm a + α

The integral representation (2.3) will be true if instead Γm(λr) we substitute the function

Ψσ(η, ζ; λ) = Γm(λr) + Gσ(η, ζ; λ), (2.9)

where Gσ(η, ζ; λ)− is the regular solution of the Helmholtz equation with respect to the variable η, including the point η = ζ.

Then the integral representation (2.3) can be written as:

W(ζ) = Lσ(η, ζ; λ)W(η)dsη, ζ ∈ Ω, (2.10)

where

0 ∗ T

Lσ(η, ζ; λ) = E Ψσ(η, ζ; λ)v P (∂ζ) P(t ).

Theorem 3.1. Suppose W(η) ∈ S(Ω) satisfies the following boundary condition

|W(η)| ≤ M, η ∈ D. (3.1)

If Z Wσ(ζ) = Lσ(η, ζ; λ)W(η)dsη, ζ ∈ Ω, (3.2)

k+1 −σζm

|W(ζ) − Wσ(ζ)| ≤ M C(ζ)σ e , σ > 1, ζ ∈ Ω. (3.3)

Here and next the functions bounded on compact subsets of the domain Ω, we denote by C(ζ).

Proof. Here, using the integral representation (2.10) and the equality (3.2), we get the following equality Z Z W(ζ) = Lσ(η, ζ; λ)W(η)dsη + Lσ(η, ζ; λ)W(η)dsη =

= Wσ(ζ) + Lσ(η, ζ; λ)W(η)dsη, ζ ∈ Ω.

Using boundary condition (3.1), we estimate the following

|W(ζ) − Wσ(ζ)| ≤ Lσ(η, ζ; λ)W(η)dsη ≤

≤ |Lσ(η, ζ; λ)| |W(η)| dsη ≤ M |Lσ(η, ζ; λ)| dsη, ζ ∈ Ω.

Next, we estimate the integrals |Ψσ(η, ζ; λ)| dsη, dsη, j = 1, m − 1

and dsη on the part D of the plane ηm = 0.

To do this, first, we separate the imaginary part of the equality (2.8), and as a result we get the following equality

Z ∞ √

2 2 2 2

σ(ηm−ζm) k−1 −σ(a +α ) 2 2

e ∂ e cos σ a + α Ψ (η, ζ; λ) =  − cos(λa)da+

σ k−1 2 2

cm ∂s a + r

Z ∞ √

k−1 −σ(a +α ) 2 2

∂ e (ηm − ζm) sin σ a + α cos(λa) + √ da , ζ > 0.

k−1 2 2 m

∂s a + r a2 + α2 Taking into account equality (3.5), we have

k+1 −σζm

|Ψσ(η, ζ; λ)| dsη ≤ C(ζ)σ e , σ > 1, ζ ∈ Ω, (3.6)

Next, we use the following equality to estimate the second integral

∂Ψσ(η, ζ; λ) ∂Ψσ(η, ζ; λ) ∂s ∂Ψσ(η, ζ; λ) = = 2(yη − ζj) , ∂ηj ∂s ∂ηj ∂s

s = α , j = 1, 2, . . . , m − 1.

Given equality (3.5) and equality (3.7), we obtain

∂Ψσ(η, ζ; λ) 2

k+1 −σζm

dsη ≤ C(ζ)σ e , σ > 1, ζ ∈ Ω. (3.8)

Now, we estimate the integral dsη.

Taking into account equality (3.5), we obtain

∂Ψσ(η, ζ; λ) 2

k+1 −σζm

dsη ≤ C(ζ)σ e , σ > 1, ζ ∈ Ω, (3.9)

From inequalities (3.6), (3.8), (3.9) and (3.4), we obtain an estimate (3.3).

Corollary 3.2. The limiting equality

lim Wσ(ζ) = W(ζ),

holds uniformly on each compact set from the domain Ω.

Theorem 4.1. Let W(η) ∈ S(Ω) satisfy condition (3.1), and on a smooth surface Σ the inequality

|W(η)| ≤ δ, 0 < δ < M, (4.1)

ζm ζm

k+1 η ¯2 η ¯2

|W(ζ)| ≤ C(ζ)σ M m δ m , σ > 1, ζ ∈ Ω. (4.2)

Proof. Using the integral formula (2.10), we have

W(ζ) = Lσ(η, ζ; λ)W(η)dsη + Lσ(η, ζ; λ)W(η)dsη, ζ ∈ Ω.

Then we estimate the following

|W(ζ)| ≤ Lσ(η, ζ; λ)W(η)dsη + Lσ(η, ζ; λ)W(η)dsη , ζ ∈ Ω. (4.3)

Given inequality (4.1), we estimate the first integral of inequality (4.3).

Lσ(η, ζ; λ)W(η)dsη ≤ |Lσ(η, ζ; λ)| |W(η)| dsη ≤

≤ δ |Lσ(η, ζ; λ)| dsη, ζ ∈ Ω.

To do this, we estimate the integrals |Ψσ(η, ζ; λ)| dsη, dsη, j =

1, m − 1 and dsη on a smooth surface Σ.

Taking into account equality (3.5), we have

k+1 σ(ηm−ζm)

|Ψσ(η, ζ; λ)| dsη ≤ C(ζ)σ e , σ > 1, ζ ∈ Ω. (4.5)

To estimate the second integral, on using the equalities (3.5) and (3.7), we obtain

∂Ψσ(η, ζ; λ) 2 2

k+1 σ(ηm−ζm)

dsη ≤ C(ζ)σ e , σ > 1, ζ ∈ Ω, (4.6)

To estimate the integral dsη, on using the equality (3.5), we obtain

∂Ψσ(η, ζ; λ) 2 2

k+1 σ(ηm−ζm)

dsη ≤ C(ζ)σ e , σ > 1, ζ ∈ Ω. (4.7)

From (4.5) – (4.7) and (4.4), we obtain

k+1 σ(ηm−ζm)

Lσ(η, ζ; λ)W(η)dsη ≤ C(ζ)σ δ e , σ > 1, ζ ∈ Ω. (4.8)

The following is known

k+1 −σxm

Nσ(y, x; λ)U(y)dsy ≤ M C(x)σ e , σ > 1, x ∈ G. (4.9)

Now taking into account (4.8) – (4.9) and (4.3), we have

C(ζ)σ 2 2

ση ¯m −σζm

|W(ζ)| ≤ (δ e + M)e , σ > 1, ζ ∈ Ω. (4.10)

Choosing σ from the equality σ = ln , (4.11) η ¯2 δ we obtain an estimate (4.2).

Let W(η) ∈ S(Ω) and instead W(η) on Σ with its approximation fδ(η) are given, respec-tively, with an error 0 < δ < M,

max |W(η) − fδ(η)| ≤ δ. (4.12)

We put Z Wσ(δ)(ζ) = Lσ(η, ζ; λ)fδ(η)dsη, ζ ∈ Ω. (4.13)

The following is true

Theorem 4.2. Let W(η) ∈ S(Ω) on the part of the plane ηm = 0 satisfy condition (3.1).

ζm ζm

k+1 η ¯2 η ¯2

W(ζ) − W (ζ) ≤ C(ζ)σ M m δ m , ζ ∈ Ω. (4.14)

σ(δ) Proof. From the integral formulas (2.10) and (4.13), we have

W(ζ) − Wσ(δ)(ζ) = Lσ(η, ζ; λ)W(η)dsη − Lσ(η, ζ; λ)fδ(η)dsη =

∂Ω Σ

Z Z Z = Lσ(η, ζ; λ)W(η)dsη + Lσ(η, ζ; λ)W(η)dsη − Lσ(η, ζ; λ)fδ(η)dsη =

Σ D Σ

= Lσ(η, ζ; λ) {W(η) − fδ(η)} dsη + Lσ(η, ζ; λ)W(η)dsη.

Using conditions (3.1) and (4.12), we estimate the following:

W(ζ) − Wσ(δ)(ζ) = Lσ(η, ζ; λ) {W(η) − fδ(η)} dsη +

+ Lσ(η, ζ; λ)W(η)dsη ≤ |Lσ(η, ζ; λ)| |{W(η) − fδ(η)}| dsy+

Z Z Z + |Lσ(η, ζ; λ)| |W(η)| dsη ≤ δ |Lσ(η, ζ; λ)| dsη + M |Lσ(η, ζ; λ)| dsη.

D Σ D

Now, repeating the proof of Theorems 3.1 and 4.1, we obtain

C(λ, ζ)σ 2 2

ση ¯m −σζm

W(ζ) − Wσ(δ)(ζ) ≤ (δ e + M)e .

Next, choosing σ from equality (4.11), we obtain an estimate (4.14).

Corollary 4.3. The limit equality

lim Wσ(δ)(ζ) = W(ζ),

holds uniformly on every compact set from the domain G.

In this paper, we have explicitly found a regularized solution to the ill-posed Cauchy problem for matrix factorizations of the Helmholtz equation in a multidimensional bounded domain. It is assumed that a solution to the problem exists and is continuously differentiable in a closed domain with exactly given Cauchy data. For this case, an explicit formula for the continuation of the solution is established, as well as a regularization formula for the case when, under the indicated conditions, instead of the Cauchy data, their continuous approximations with a given error in the uniform metric are given. We have obtained a stability estimate for the solution of the Cauchy problem in the classical sense.

Thus, functional Wσ(δ)(ζ) determines the regularization of the solution for problems (2.1) and (2.2).

The author expresses his sincere gratitude to the reviewers and editors for their useful comments.