On Some Discrete Equations in a Half-Space

On Some Discrete Equations in a
Half-Space
Alexander V. Vasilyev ∗ , Vladimir B. Vasilyev ∗∗
∗ Belgorod State University, Belgorod 308015, Russia (e-mail:
alexvassel@gmail.com).
∗∗ Lipetsk State Technical University, Lipetsk 398600, Russia (e-mail:
vbv57@inbox.ru)
Abstract: We consider discrete multi-dimensional singular integral equation with Calderon-
Zygmund kernel in a discrete half-space. Taking into account the Fourier transform properties
and the corresponding properties of Calderon-Zygmund operators we study the solvability for
such equations.
Keywords: Calderon-Zygmund operator, discrete convolution, solvability
1. INTRODUCTION
We consider discrete operator generated by Calderon-
Ztgmund kernel K(x), with is defined for functions of
discrete variable uh(˜x), ˜x ∈ Z m h , where Zm h is integer
lattice (modulo h) in Rm , and corresponding equation
auh(˜x) +
K(˜x − ˜y)uh(˜y)h m = vh(˜x), ˜x ∈ Z m h,+,(1)
˜y∈Z m
h,+
in the discrete half-space Zm h,+ = {˜x ∈ Zm h
uh, vh ∈ L2(Zm h,+ ) ≡ l2 h .
: ˜ xm > 0} ,
By definition we let K(0) = 0, and the symbol of operator
uh(˜x) ↦→ au(˜x) +
K(˜x − ˜y)uh(˜y)h m , ˜x ∈ Z m h ,
˜y∈Z m
h
is periodic function
σh(ξ) = a +
˜x∈Z m
h
e −iξ˜x K(˜x)h m
with basic cube period [−πh −1 ; πh −1 ] m .
The sum in (2) is defined as a limit of partial sums for
cubes QN
lim
e −iξ˜x K(˜x)h m ,
N→∞
˜x∈QN
QN = ˜x ∈ Z m h : |˜x| ≤ N, |˜x| = max
1≤k≤m |˜xk|
.
Remind the symbol of classical Calderon-Zygmund operator
Mikhlin et al. (1986) is defined by Fourier transform
of the kernel K(x) in principal value sense
σ(ξ) = lim K(x)e
N→∞
iξx dx.
ε→0
ε

(1992) one can assert that (4) holds at least for continuous
symbol σ(ξ) on sphere S m−1 and transmission condition
σ(0; +1) = σ(0; −1).
3. DISCRETE CONVOLUTIONS ON A HALF-AXIS
A convolution of two functions f, g on a straight line is
defined by the integral
+∞
(f ⋆ g)(x) = f(x − y)g(y)dy,
−∞
which exists always if f, g ∈ L2(R). It is a continual
convolution. A discrete convolution is defined analogously.
If f, g are functions of discrete variable, i.e. sequences, then
(f ⋆ g)(n) ≡
f(n − k)g(k) ≡
fn−kgk, (5)
k∈Z
k∈Z
fk ≡ fk, g(k) ≡ gk, k ∈ Z,
which exists for f, g ∈ l2.
The Fourier transform of discrete function is defined by
the formula
(F f)(ξ) ≡ ˜ f(ξ) =
fke −ikξ , ξ ∈ [−π, π].
k∈Z
Application of Fourier transform to (5) leads to the standard
formula
F (f ⋆ g) = ˜ f · ˜g,
which implies immediately a solvability criteria for discrete
convolution equation
au(n) +
M(n − k)u(k) = v(n), (6)
k∈Z
a is constant, M, v are given discrete functions, u is sought.
The function a + ˜ M(ξ), ξ ∈ [−π, π], is called symbol of
the equation (6).
So, the equation (6) is uniquely solvable iff its symbol never
vanishes, M, v ∈ l2.
A situation is essentially complicated, if one considers
the equation (6) not on whole lattice Z, but on Z+ =
{0, 1, 2, ...},i.e.
au(n) +
M(n − k)u(k) = v(n), n ∈ Z+; (7)
k∈Z+
here the discrete function M is defined on whole Z, but
given function v (and sought function u) on Z+ only.
Let’s introduce two projectors
u(n), n ≥ 0
0, n ≥ 0
(P+u)(n) =
(P−u)(n) =
0, n < 0,
u(n), n < 0,
and discrete convolution operator M : u(n) ↦−→ au(n) +
M(n − k)u(k), then the equation (7) can be written
k∈Z+
in the form
P+Mu+ = f+, (8)
where the functions u+(sought), f+ (given) are defined on
Z+. It’s looked easily, that the equation (8) is equivalent
from solvability viewpoint to so-called paired equation
(M1P+ + M2P−)U = F (9)
on whole lattice Z, M2 = I (identity operator).
Formal using the discrete Fourier transform leads to summation
of divergence series
e −ikξ , (10)
k∈Z+
and to destroy this divergence one adds the factor e is and
then tends to limit under s → 0. This activity gives the
form for the operator P+ as Fourier image.
Write
k∈Z+
e −ikξ e iks =
k∈Z+
e −ik(ξ+is) =
k∈Z+
e −ikζ , ζ = ξ + is.
The corrected series is absolutely converges, and its sum
is
e −ikζ = 1/2 − i/2 cot ζ/2.
Thus,
k∈Z+
(F P+u) = 1/2ũ(ξ) − i/2 lim
π
s→0+
−π
cot
ζ − τ
ũ(τ)dτ.
2
Let’s note a similar integral (as a principal value) arises
under summation of the series (10) by standard method
(the Dirichlet kernel and transfer to limit in partial sums
Edwards (1979)), and leads to the periodic variant of the
Hilbert transform
(Hu)(x) = v.p.
π
−π
cot
x − t
2 u(t)dt.
If we deal with projector P−, then the sum (9) takes the
form
= i/2 + i/2 cot(ζ/2),
and we have the formula
(F P−u) = −1/2ũ(ξ) + i/2 lim
π
s→0+
−π
cot
ζ − τ
ũ(τ)dτ.
2
4. PERIODIC RIEMANN BOUNDARY PROBLEM
Let’s introduce the function
Φ(ζ) = 1
π
cot
4πi
−π
ζ − t
2 φ(t)dt,
and assume that φ(t) satisfies the Hölder condition on
[−π, π],
|φ(t1) − φ(t2)| ≤ c|t1 − t2| α ,
∀t1, t2 ∈ [−π, π], 0 < α ≤ 1, φ(−π) = φ(π).
The limit values (s → ±0) can be calculated by transfer
from [−π, π] to unit circle and applying classical Plemelj-
Sokhotskii formulas, and then we have
Theorem 1. The formulas
Φ ± (ξ) = ± φ(t)
2
+ 1
2πi v.p.
π
−π
cot
ξ − t
φ(t)dt, (11)
2
hold, Φ ± (ξ) denotes limit values Φ ± (ζ) under s → ±0.

These formulas lead to the following statement for periodic
Riemann boundary problem: finding two functions Φ ± (z),
which are holomorphic in the half-strips
Π± = {z ∈ C : z = t + is, t ∈ [−π, π], ± s > 0},
and their boundary values under s → 0± satisfy linear
relation
Φ + (t) = G(t)Φ − (t) + g(t), t ∈ [−π, π],
where G(t), g(t) are given functions on [−π, π].
If we suppose, that G(t) ∈ C[−π, π], G(−π) = G(π), then
the index of function G on the segment [−π, π] is called
variation of the arg G(t) divided 2π, under t varies from
−π to π . It is entire number denoted by κ.
Theorem 2. If G(t) satisfies the Hölder condition, κ=0,
then periodic Riemann boundary problem has unique
solution Φ ± (t) ∈ L2[−π, π], which is constructed with the
help of fuction Φ(ζ).
5. EQUATIONS IN CONTINUAL CASE AND THE
CLASSICAL RIEMANN BOUNDARY PROBLEM
5.1 A half-axis case
The reduction of (9) to so-called characteristic singular
integral equation is constructed with the help of special
Hilbert transform Gakhov (1990), Gohberg et al.
(1992), King (2009)
(Hu)(x) ≡ 1
πi v.p.
+∞
u(s)
s − x ds
≡ 1
πi
lim
N→+∞
ε→0+
(
x−ε
−N
+
−∞
N
x+ε
) u(s)
s − x ds.
The properties of this operator are studied, and particularly
the operator H : L2(R) → L2(R) is linear bounded
operator, its spectra is ±1, and H 2 = I.
Besides, the following two operators
P = 1/2(I + H), Q = 1/2(I − H)
are projectors on subspace A(R) ⊂ L2(R) of functions
admitting analytic continuation into upper complex halfplane
C+, and B(R) ⊂ L2(R) of functions admitting
analytic continuation into lower complex half-plane C−
so that
A(R) ⊕ B(R) = L2(R).
The following identities hold
P 2 = P, Q = I − P, Q 2 = Q, P Q = QP = 0.
If we will denote as above P+, P− the restriction operators
on positive and negative half-axis, then it’s easily verified,
that Gohberg et al. (1992)
F P+ = QF, F P− = P F. (12)
Further, applying the Fourier transform to one-dimensional
equation (9) we obtain the following
1
2 σM1(ξ)(I
1
− H) Ũ(ξ) +
2 σM2(ξ)(I + H) Ũ(ξ) = ˜ F (ξ),
where σM1 , σM2 are symbols of operators M1, M2. After
summands collection one can write the full equation as
following
σM1 (ξ) + σM2 (ξ)
Ũ(ξ)+
+ σM1(ξ) + σM2(ξ)
v.p.
2πi
2
+∞
−∞
Ũ(η)
η − ξ dη = ˜ F (ξ). (13)
The equation (13) is well-known in the theory of singular
integral equations Gakhov (1990), and is called
characteristic singular integral equation. Its solution is
related closely with classical boundary Riemann problem
for upper and lower half-plane C±. It is formulated by the
following way: finding two functions Φ ± (t) defined on R,
which admit an analytic continuation into C± and satisfy
on straight line R the linear relation
Φ + (t) = G(t)Φ−(t) + g(t), (14)
where G(t), g(t) are given on R functions. If we will denote
σM1 (t) + σM2 (t) σM1 (t) − σM2 (t)
a(t) = , b(t) = ,
2
2
then we will see that equation (13) in the spase L2(R) and
the problem (14) for Φ ± ∈ L2(R) are equivalent Gakhov
(1990), so, the coefficient G(t) and right hand side g(t) are
easily defined by a and b:
a(t) + b(t)
˜F (t)
G(t) = , g(t) =
a(t) − b(t) a(t) − b(t) ,
and vice versa, given problem (14) corresponds to related
characteristic singular integral equation (13). It was shown
[3] the solvability condition for the equation (13) is determined
by certain topological invariant, whis is called index.
Let’s note in our case
G(t) = σM1 (t)σ−1 (t). (15)
M2
We suppose the following with respect to the function
(15). Let’s denote R one-point compactification of R and
assume that G(t) is continuous on R and never vanishing.
The variation of argument of G(t) divided 2π, when t
varies from −∞ to +∞, is called index æ of this function.
If æ = 0, then the solution of the equation (13) is unique,
and it can be written by exact formula with the help of
the Hilbert transform Gakhov (1990).
5.2 A half-space case
Let’s go back to the equation (9) assuming M1, M2 are
Calderon-Zygmund operators (like in equation (3)), and
P+, P− we mean the restriction operators on half-space
R m + = {x = (x1, ..., xm), ± xm > 0}.
Obviously, the previous constructions will be valid with
some little supplement. If we as earlier denote the Fourier
tranmsform by F , then we have the following relations
F P+ = Qξ ′F, F P− = Pξ ′F,
P = 1/2(I + Hξ ′), Q = 1/2(I − Hξ ′),
where Hξ ′ is Hilbert transform on variable ξm, ξ ′ =
(ξ1, ..., ξm−1) is fixed:
(Hξ ′u)(ξ′ , ξm) ≡ 1
πi v.p.
+∞
−∞
u(ξ ′ , τ)
dτ.
τ − ξm

The equation (13) for this case will take the form of the
following equation with the parameter ξ ′ :
σM1 (ξ′ , ξm) + σM2 (ξ′ , ξm)
Ũ(ξ)+
+ σM1 (ξ′ , ξm) + σM2 (ξ′ , ξm)
v.p.
2πi
2
+∞
−∞
Ũ(ξ ′ , η)
dη =
η − ξm
˜ F (ξ).(16)
It corresponds to Riemann boundary problem (with parameter
ξ ′ also) with coefficient
G(ξ ′ , ξm) = σM1(ξ ′ , ξm)σ −1
M2 (ξ′ , ξm). (17)
For unique solvability of the equation (16) we need the
index of G(ξ ′ , ξm) on variable ξm is equal to 0.
The symbol of the Calderon-Zygmund operator is very
specific, it is a function homogeneous of order 0, i.e.
indeed it is defined on unit sphere S m−1 . Let m ≥ 3. Fix
ξ ′ ∈ S m−2 and suppose G(0, −1) = G(0, +1).Varying ξm
from −∞ to +∞ the function G(ξ) will vary along the
arc of big half-circle across south pole (0, −1) and north
pole (0, +1). At the same time the symbol’s values will go
along closed curve in a complex plane. These all curves
for different ξ ′ will be homotopic, and therefore they have
the same entire-valued index æ with respect to 0. The
condition æ = 0 implies the unique solvability for the
equation (16).
6. BACK TO A DISCRETE CASE
Here we return again to the equations in a discrete context
assuming that in equation (9) P± are restriction operators
on Z m h,± , M1, M2 are discrete Calderon-Zygmund
operators generated by kernels M1(x), M2(x), which are
bounded in the space L2(Z m h ).
For functions of discrete variable defined on the lattice Z m h
its discrete Fourier transform is given by the formula
u(˜x) ↦−→ 1
(2π) m
u(˜x)e −i˜x·ξ h m ≡ ũ(ξ), ξ ∈ [−π, π] m .
˜x∈Z m
h
Such Fourier transform has the same properties as usual
Fourier transform Sobolev et al. (1997).
According to theorem 1 and Sec.5 we introduce periodic
analogue of the Hilbert transform on variable ξm (ξ ∈
[−π, π] m , ξ ′ is fixed) by the formula
(H per
ξ ′ u)(ξm) = 1
2πi
πh −1
−πh −1
u(t) cot
and periodic analogues of projectors (12)
P per
ξ ′
= 1/2(I + H per
ξ ′ ), Q per
ξ ′
h(t − ξm)
dt (18)
2
= 1/2(I − H per
ξ ′ ).
Insted of the equation (16) we obtain its periodical analogue
σ1,h(ξ ′ , ξm) + σ2,h(ξ ′ , ξm)
Ũ(ξ)+
2
+ σ1,h(ξ ′ , ξm) + σ2,h(ξ ′ , ξm)
×
4πi
×v.p.
πh −1
−πh −1
Ũ(ξ ′ , η) cot
h(η − ξm)
dη =
2
˜ F (ξ). (19)
where σ1,h, σ2,h are symbols (2) of discrete operators
M1, M2. Naturally, the equation (19) will be related to
corresponding periodic Riemann bounary problem, for
which its unique solvability conditions are in theorem 2.
In our case it is
Ind σ1,h(·, ξm)σ −1
2,h (·, ξm) = 0.
7. LIMIT TRANSFER FROM A DISCRETE CASE TO
CONTINUOUS ONES
First here we have remember that images of symbols σ and
σh are the same [9]. Moreover, the index is entire-valued
characteristic both in continual case (if the transmission
condition σ(0, −1) = σ(0, +1) holds)and in discrete (periodic)
ones. Looking for the variation σh(·, ξm) along arcs
of big half-circles on S m−1 and taking into account that
lim
h→0 σh(ξ) = σ(ξ), ∀ξ ∈ S m−1 ,
we conclude, that for transmission condition we have
Theorem 3. The equations (1) and (3) are solvable or
unsolvable at the same time.
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