CUBO A Mathematical Journal

Vol.11, No¯ 01, (145–162). March 2009

On a new notion of holomorphy and its applications

Wolfgang Sproessig

Freiberg University of Mining and Technology,

Faculty of Mathematics and Informatics,

Agricola-Strasse 1,

09596 Freiberg, Germany.

email: sproessig@math.tu-freiberg.de

and

Le Thu Hoai

Hanoi University of Technology,

Faculty for Applied Mathematics and Informatics,

Dai Co Viet Street 1,

10000 Hanoi, Vietnam.

ABSTRACT

This paper devotes a new general notion of holomorphy which works in the continous

and discrete cases. With the help of methods of a general operator theory the so

called L-holomorphy is introduced. Realizations of this calculus follow. New versions

of Taylor- and Taylor–Gontcharov formulae are deduced. The results are applied for

the solution of higher order systems of differential equations.

RESUMEN

Este artículo es dedicado a una nueva noción de holomorfía la cual funciona en los

casos continuo y discreto. Con la ayuda de métodos de la teoría general de operadores

la llamada L-holomorfia es presentada. Realizaciones de este cálculo siguen. Nuevas

versiones de fórmulas de Taylor-y Taylor-Gontcharov son deducidas. Los resultados son

aplicados para la solución de sistemas de orden superior de ecuaciones diferenciales.

Key words and phrases: Generalized holomorphic functions, Taylor-Gontcharov formulae,

Plemelj projections, higher order boundary value problems.



146 Wolfgang Sproessig and Le Thu Hoai CUBO
11, 1 (2009)

1 Introduction.

The aim of this article is to introduce a very general notion of holomorphy by the help of three

general operators in Banach spaces which have to satisfy some conditions. This introduction is

oriented at the theory of right invertible operators. We refer to the well-known book of V.S.

Ryabenskij [11] (1987), W. Schempp and F.J. Delvos [2] (1990) and the article by M. Tasche [16]

(1981). The advantage of our approach is the fact that holomorphy can be considered in the

continuous and discrete case within one calculus. We continue the line of action we have followed

in books [6],[7],[5]. In the second part we present a large number of realisations. Here we use above

all results of the common research with K. Gürlebeck confer again in [6], [7] and [4]. Finally, some

classes of boundary value problems of higher order will be considered. In that connection new

formulae of Taylor- and Taylor-Gontcharov type are obtained. All our considerations take place in

the scale of Sobolev and Besov spaces as well as its discrete analogue.

2 A general holomorphy

Let X,Y,Z be Banach spaces. We introduce the bounded linear operators T,Tr and P with the

following properties

(i) T : X → im T ⊂ Y is injective.

(ii) Tr : Y → Z is a generalized trace operator .

(iii) The operator P : imTr ∩ Y → Y satisfies the property PTrPu = Pu.

Furthermore, we assume

(i) im TrT ⊂ kerP ,

(ii) im T ∩ ker Tr = {0}.

Remark 1. We also have imT ∩ imP = {0}. Indeed, let u ∈ imT ∩ imP = {0} then u = Pw = Tv

and

u = Pw = PTrPw = PTrTv = 0.

Theorem 1. (Mean value formula) Set imT ⊕ imP =: Y1 ⊂ Y. There is a unique linear operator

L with D(L) = Y1 and L : D(L) → X, such that

u = PTru + TLu.

Proof. Let u ∈ D(L). Then u permits the representation

u = Pv + Tw,



CUBO
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On a new notion of holomorphy and its applications 147

with v ∈ imTr ∩ Y, and w ∈ X. Applying PTr from the left it follows

PTru = PTrP v + PTrT w = P v.

In this way the first item of the desired formula is obtained. In order also to obtain the second

item we have to use the injectivity of the operator T . On the linear set im T there exists a linear

operator L̃ with

L̃Tw = w.

The operator L̃ can be extended to an linear operator L on Y1 setting

Lz := L̃z1,

where z = z1 + z2 with z1 ∈ im T and z2 ∈ im P . The additivity follows from

L(z + z′) = L(z1 + z2 + z
′
1

+ z′
2
) = L̃(z1 + z

′
1
) = L̃z1 + L̃z

′
1

= Lz + Lz′.

The monogeneity with a real constant λ is also fulfilled. Indeed, we have

L(λz) = L̃(λz1) = λL̃z1 = λLz.

Now we obtain easily Lu = LPTru + LT w = w and our decomposition formula is completely

proved. The uniqueness follows from

TLu − TL1u = 0 leads to Lu = L1u,

where L1 is another linear operator which has to fulfil the decomposition formula. #

Corollary 1. The following relations between the operators L,P and T are valid:

(i) The operator L is the left-inverse to the operator T, i.e. LT = I.

(ii) Set R := TL then R is a projection onto Y1 with imR = imT.

(iii) It holds kerL = imPTr.

Proof. The relation (i) follows by the definition of L. Indeed, let v ∈ X, then

LTv = L̃Tv = v.

(ii) Obviously, TL fulfils the idempotential property and so we have R2 = R. It is immediately

clear that

im R ⊂ im T.

Conversely, let v ∈ im T then v = Tw and

Rv = RTw = TLTw = Tw = v,



148 Wolfgang Sproessig and Le Thu Hoai CUBO
11, 1 (2009)

i.e. im T ⊂ im R. To prove the relation (iii) we have to argue as follows: Let u ∈ ker L, then

u = PTru + TLu = PTru ∈ im PTr.

On the other hand it follows from u ∈ im PTr that u = PTrv with v ∈ Y and

u = PTrv = PTrPTrv + TLu = PTrv + TLu,

which leads to TLu = 0 and because of the injectivity of the operator T : X → im T we conclude

Lu = 0, i.e. u ∈ ker L. #

Definition 1. Elements u ∈ ker L∩Y are called L-holomorphic. The operator L is called algebraic

derivative.The operator PTr is called the initial projection and the operator T is denoted as general

Teodorescu transform. From the point of view of a general operator theory T is also called algebraical

integral.

Corollary 2. Set Pr := TrP : imTr ∩ Y → Z and Qr := I − Pr. The following properties are

valid:

(i) The operators Pr,Qr are idempotent, i.e. we have P
2
r

= Pr and Q
2
r

= Qr and furthermore

QrPr = PrQr = 0.

(ii) An element ξ ∈ Z is the generalized trace of an element u from kerL if and only if Prξ = ξ.

(iii) We have Qrξ = TrTLu.

Proof. (i). It is sufficient to show

P 2
r
ξ = TrPTrPξ = TrPξ = Prξ,

with ξ ∈ Z. In order to prove (ii) let ξ = Tru ∈ Z and u ∈ ker L. Then we have

u = PTru + TLu = PTru = Pξ.

It now follows ξ = Tru = TrP ξ = Prξ. Conversely, let us assume ξ = Prξ, then

Tru = ξ = Prξ = TrPξ = TrPTru.

On the other hand Theorem 1 yields

Tru = TrPTru + TrTLu.

Hence TrTLu = 0. Because of imT ∩kerTr = {0} follows TLu = 0 and such Lu = 0, i.e. u ∈ ker L.

For (iii) we have

Tru = TrPTru + TrTLu.

Therfore, it holds

TrTLu = Tru − TrPTru = ξ − TrP ξ = ξ − Prξ = Qrξ. #

Denotation The operators Pr,Qr are called general Plemelj projections.



CUBO
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On a new notion of holomorphy and its applications 149

Remark 2. The condition imTL ∩ kerTr = {0} can be seen as a very general formulation of a

maximum principle.

3 Types of L–holomorphy

3.1 L-holomorphy in R1

A trivial example is given by consideration of all functions u ∈ C1[0, 1] with

L :=
d

dt
, T :=

t∫

0

·dτ ,

P := I and Tr : C1[0, 1] → R1 with Tru = u(0). Then we get the well-known mean-value theorem:

u(t) = u(0) +

t∫

0

u̇(τ)dτ = PTru + TLu.

This is just the main-theorem of differential-integral calculus. The class of all L-holomorphic

functions consist of all real constants.

Also a slightly modification of the trace operator and the generalized Teodorescu transfrom

does not change the triviality of the class of L-holomorphic functions. Indeed, let u ∈ C1[0, 1],

take L := d
dt
, P := I and Tru := 1

2
[u(0) + u(1)], then

(Tu)(t) :=

t∫

0

u(τ)dτ −
1

2

1∫

0

u(τ)dτ .

Because of imPTr = ker L we have again the space of all constants for the class of L-holomorphic

functions.

By using the so-called Riemann-Liouville integral of order α (cf. [14],[9]) we obtain a more

interesting example. For this reason let u ∈ C[0, 1], 0 < α < 1. We consider the absolut continuous

function

(Iα
a+
u)(t) :=

1

Γ(α)

t∫

0

1

(t − τ)1−α
u(τ)dτ ,

which has almost everywhere a derivative in L1[0, 1]. Take now

(Lu)(t) :=
1

Γ(1 − α)

d

dt
(I

1−α
a+

u)(t) , (Tu)(t) := (Iα
a+
u)(t)



150 Wolfgang Sproessig and Le Thu Hoai CUBO
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and with n = [α] + 1

(PTru)(t) :=

n−1∑

k=0

(t − a)α−k−1

Γ(α − k)

dn−k−1

dtn−k−1
I
n−α
a+

u(t).

(Iα
a+
u)(t) is called Riemann-Liouville fractional integral and (Dα

a+
u)(t) is denoted by Riemann–

Liouville fractional derivative. The main-value theorem holds again.

3.2 Notions of holomorphy in the complex plane

The original notion of the holomorphy forms in natural way a class of L-holomorphic function. We

have only to set

L := ∂z .

In more detailed we have the following: Let G ⊂ C be a bounded domain with sufficient smooth

boundary curve then the mean-value formula is written as

1

2πi

∫

Γ

u(t)

t − z
dt −

1

2πi

∫

G

1

t − z
(∂u)(t)dξ dη =

{
u(z) , z ∈ G

0 , z ∈ C \ G
.

We have only to identify

L := ∂ =
1

2
(∂ξ + i∂η) (tξ + iη) ,

T := −
1

2πi

∫

G

1

t − z
· dξ dη , P :=

1

2πi

∫

Γ

1

t − z
· dΓt .

The trace operator Tr is defined as non-tangential limit from inner points tending to the boundary

Γ.

Remark 3. It is quite curious that the initial projection acts on the boundary. It seems that

”initial values” are ”smudged” over the surface.

Another example in the complex plane can be given by

L := ∂ , (T ·)(z) = −
1

2πi

∫

G

[
1

t − z
−

1

t + z

]
· dξ dη

and (P ·)(z) = −
1

2πi

∫

Γ

[
1

t − z
−

1

t + z

]
· dΓt .

The trace operator is definded as before. This model goes back to J. Ryan (cf. [8]).



CUBO
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On a new notion of holomorphy and its applications 151

3.3 L-holomorphy models generated by matrices

A further model for L-holomorphy is given by: Let {Ei}
n

i=1
be a family of orthogonal matrices of

order n with entries 0, 1,−1 as well as the property

E∗
i
Ej + E

∗
j
Ei = 0 (i 6= j)

Furthermore, set E(a) =
n∑
i=1

Eiai , a = (a1, ...,an)
T and E∗(a) =

n∑
i=1

E∗
i
ai and ∇ = (∂1, ...,∂n)

T .

Take L := D(∇) , T := 1
σn

∫
G

D
∗
(y−x)

|y−x|n · dy and P :=
−1
σn

∫
Γ

D
∗
(y−x)

|y−x|n · dΓy then it holds

(Pu)(x) + TL(∇)u(x) =

{
u(x) , x ∈ G

0 , x ∈ Rk \ G
.

Here σn denotes the area of the n-dimensional unit sphere. (cf. [13],[15]).

3.4 Dzuraev’s model

Also Dzuraev’s model from 1982 [3] is worthy of being mentioned:

Let u := (u1,u2), z = x2 + ix3 and
∂

∂z
:=

1

2
(
∂

∂x2
+ i ∂

∂x3
) , y = y2 + iy3. Further, let

∂x =

(
∂

∂x1
2
∂

∂z

−2 ∂
∂z

∂

∂x1

)
, E(y − x) =

−1

|y − x|3

(
y1 − x1 −(y − z)

y − z y1 − x1

)

and n(y) =

(
n1 n2 − in3

−(n2 + in3) n1

)
. Then take

L := ∂x , T :=
1

σ3

∫

G

E(y − x) · dy and P :=
1

σ3

∫

Γ

E(y − x)n(y) · dΓy .

The trace operator Tr means in both cases the non-tangential limit to the boundary Γ from inside

of G.

4 Quaternionic holomorphic functions

Real Quaternions: The algebra of real quaternions H is defined by the basis elements

e0 = 1 , e1,e2,e3,



152 Wolfgang Sproessig and Le Thu Hoai CUBO
11, 1 (2009)

which obey the arithmetic rules:

e2
0

= 1 , e1e2 = −e2e1 = e3 , e2e3 = −e3e2 = e1 , e3e1 = −e1e3 = e2 .

Each quaternion a ∈ H permits the representation

a =

3∑

k=0

akek (ak ∈ R ; k = 0, 1, 2, 3) .

Addition and multiplication in H turn it into a non-commutative number field. The main-involution

in H is called quaternionic conjugation and defined by

e0 = e0 , ek = −ek (k = 1, 2, 3) .

which can be extended onto H by R-linearity. Therefore we have

a = a0 −

3∑

k=1

akek = a0 − a.

Note that

aa = aa =

3∑

k=1

a2
k

=: |a|2
H
.

If a ∈ H \ {0} then the quaternion

a−1 :=
a

|a|2

is the inverse to a. For a,b ∈ H we have abba.

Complex quaternions: The set of complex quaternions, which we also need, is denoted by

H(C) and consist of all elements of the form

a =

3∑

k=0

akek (ak ∈ C ; k = 0, 1, 2, 3) .

By definition we state: iek = eki, k = 0, 1, 2, 3. Here i denotes the usual imaginary unit in C.

Elements of H(C) can also be represented in the form

a = a1 + ia2 (ak ∈ H; k = 1, 2).

Notice that the quaternionic conjugation acts only on the quaternionic units and not on the pure

complex number i.



CUBO
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On a new notion of holomorphy and its applications 153

Let X = Wk
p
(G),Y = Wk+1

p
(G),Z = W

k−(1/p)+1
p (Γ); k = 0, 1, 2, ...; 1 < p < ∞. Further, let

L := D =

3∑

i=1

∂iei (Dirac operator (mass zero)),

(Tu)(x) := −
1

σ3

∫

G

e(x − y)u(y)dy (Teodorescu transform),

(Pu)(x) := (FΓu)(x) =
1

σ3

∫

Γ

e(x − y)n(y)u(y)dΓy (Cauchy − Fueter operator),

(Tru)(ξ) := n.t. − lim
z→ξ∈Γ

z∈G

u(z),

with e(x) = D 1|x| and n =
∑

3

i=1
eini the outward pointing unit vector of the normal.

The class of L-holomorphic functions are just the solutions of the Mosil–Teodorescu system.

We now consider so called Dirac operators with mass. We will use the same spaces as above.

Then the general operators L,T and P are given by

L := D + iα (Dirac operator with mass),

(Tu)(x) := −
1

σ3

∫

G

eiα(x − y)u(y)dy (Teodorescu type transform),

(Pu)(x) :=
1

σ3

∫

Γ

eiα(x − y)n(y)u(y)dΓy (Cauchy − Fueter − typeoperator),

(Tru)(ξ) := n.t. − lim
z→ξ∈Γ

z∈G

u(z).

For the description of the kernel function of this new Teodorescu transform we have to use Bessel-

functions of third kind so called MacDonald functions. We have

eiα(x) := −

(
iα

2π

)(3/2) [
|x|−1/2K3/2(iα|x|)ω − K1/2(iα|x|)

]
,

where ω ∈ S2 and Kµ(t) denotes.

5 Discrete quaternionic holomorphic functions

One advandage of our notion of L-holomorphy is its applicability also on lattices. We will present

a calculus which was obtained by K. Guerlebeck in 1988 [4] (cf. also [6]). For this reason we have

to represent the domain on the lattice and to define what are inner and outer points relatively

to the ”discrete boundary” and to say what the discrete boundary means. This boundary has to



154 Wolfgang Sproessig and Le Thu Hoai CUBO
11, 1 (2009)

approximate the original domain. It is necessary to disdinguish between a right and a left parts of

the boundary. The approximating discrete domain is here always an axes–parallel polyeder with

side faces, edges and corner points. More exactly holds

R
3
h

:= {(ih,jh,kh) : i,j,k integer, h > 0}, Gh := G ∩ R
3
h
,

Γh := {x ∈ Gh : dist(x, coGh) ≤
√

3h}.

Let V ±
i,h
x the translation of x by ±h in xi-direction, then

Γh,ℓ(r) := {x ∈ Γh : ∃i : V
±
i,h
x /∈ Gh} (left(right) side planes),

Γh,ℓ(r);i := {x ∈ Γh : V
±
i,h
x /∈ Gh},

Γh,ℓ(r);i,j := Γh,ℓ(r);i ∩ Γh,ℓ(r);j (left(right) edges),

Γh,ℓ(r);i,j,k := Γh,ℓ(r);i,j ∩ Γh,ℓ(r);k (left(right) corners).

Let be X = W 1
2,h

(Gh), Y = L2,h(Gh), Z = W
1
2

2,h
(Gh). Then

(Lu)(x) := (D
±
h
u)(x) =

3∑

i=1

ei[u(V
±
i,h
x) − u(x)]

1

h
(discr. Dirac operator),

(Tu)(x) := (T
±
h
u)(x) (discrete Teodorescu transform)

=




∑

intGh∪Γh,ℓ(r)
+

∑

left(right) corners

−
∑

left(right) edges


e±

h
(x − y)u(y)h3,

where e±
h

are the discrete fundamental solutions of D±
h

. The discrete Cauchy–Fueter operator is

introduced as follows

(Pu)(x) := (F
±
h
u)(x) =

3∑

i=1


−

∑

si

+

∑

sij

−
∑

sijk


e±

h
(x − V ∓

i,h
y)n(y)u(y)h2

+

3∑

i=1

∑

y∈Γh,ℓ(r);m,j,k
m 6=j 6=k

h±(x − y)eiu(y)h
2,

where si = Γh,ℓ;i ∪ Γh,r;i, sij := Γh,ℓ;j − V
+

i,h
Γh,ℓ, sijk := Γh,ℓ;j,k − V

+

i,h
Γh,ℓ;i,k.

The corresponding mean value formulae are given as follows

u(x) = (F
±
h
u)(x) + T

±
h
D

±
h
u(x)

Much more complicated is to find a suitable discrete fundamental solution, which is given by Eh(x)

as solution of a suitable difference equation

−∆hEh(x) = −
3∑

i=1

D
−
i,h
D

+

i,h
Eh(x) = δh(x) =

{
h−3,x = 0

0,x ∈ R3
h
\ {0}



CUBO
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On a new notion of holomorphy and its applications 155

expressed by using the Fourier-Transform we have

Eh(x) =
1

√
2π

3
RhF

(
1

d2

)
.

The function d is defined as follows

d2 =
4

h2

(
sin

2
hξ1

2
+ sin

2
hξ2

2
+ sin

2
hξ3

2

)

and Rhu is the restriction of the continuous function u onto the lattice R
3
h
. We have |Eh| ≤ C|x|

m

with a certain m > 0 depending on the properties of the difference operator

e
±
h

(x) := D
∓
j,h
Eh(x).

6 L-holomorphy on the sphere

Meanwhile is also existing the notion of holomorphy on the sphere. A good reference is doctoral

thesis of P. Van Lancker [17] The following operators has to be used ΓS + α α ∈ C \ N ∪ (−N).

Lα : = ω(ΓS + α) (Günter’s gradient),

Tα : = −

∫

Ω

Eα(ω,ξ) · dS(ω) (Teodorescu transform),

PC,α : = −

∫

−C

Eα(ω,ξ)n(ω) · dC(ω) (Cauchy-Fueter type operator).

A corresponding Borel-Pompeiu formula is given by

PC,αu + TαDαu =

{
u in Ω

0 in S \ Ω
.

We will consider the fundamental solution of Günter’s gradient. Let α ∈ C \ N ∪ {−2 − N}. Then

Eα(ω,ξ) =
π

σ3 sin πα
Kα(−ξ,ω)ω,

where σ3 is the surface area of the unit sphere. Further, we define

Kα(−ξ,ω)ω = C
3/2
α

(ω · ξ) + ξωC
3/2

α−1(ω · ξ),

with the so-called Gegenbauer polynomials Cµ
α
(t).

Using Kummer’s function 2F1(a,b; c; z) we get the representation

C3/2
α

(z) =
Γ(α + 3)

Γ(α + 1)

1

4
2F1(−α,α + 3; 2;

1 − z

z
) z ∈ C \ {−∞, 1}.



156 Wolfgang Sproessig and Le Thu Hoai CUBO
11, 1 (2009)

Kummer’s function is for |z| < 1 defined by

2F1(a,b; c; z) :=

∞∑

k=0

(ak)(ak)

(c)k

zk

k!
, (a)k =

Γ(α + k)

Γ(α)
.

Solutions of Dαu = 0 in Ω are called inner spherical holomorphic functions of order α in Ω.

We have

DαEα(ω,ξ) = δ(ξ − ω) .

A good reference for this topic is [1]. Further we introduce a singular integral operator of Bitzadse’s

type

(SC,αu)(ξ) : = 2 lim
ε→0

∫

C\Bε(ξ)

Eα(ω,ξ)n(ω)u(ω)dS(ω)

= 2v.p.

∫

C

Eα(ω,ξ)n(ω)u(ω)dS(ω).

One can prove the algebraical identity S2
C,α

= I. Let Ω+ := Ω , Ω− := coΩ. Applying the general

trace operator as non-tangential limit on the sphere towards the boundary C we get Plemelj-

Sokkotzkij-type formulae.

n.t. − lim
t→ξ

t∈Ω±

(FC,αu)(t)
1

2
[±I + SC,α]u(ξ) =:

{
PC,αu(ξ), t ∈ Ω

+

−QC,αu(ξ), t ∈ Ω
− .

The operators

QC,α :=
1

2
[I − SC,α], PC,α :

1

2
[I + SC,α]

are called Plemelj projections. The space L2(Γ) is now decomposed into the Hardy spaces

L2(C) = HS
α
(Ω

+
) ⊕ HSα(Ω−)

↑ ↑

PC,α QC,α

(cf. [12]).

7 Taylor type formula

Using ideas of the theory of right invertible operators (cf. D. Przeworska-Rolewicz, [10]) one has

with Ym = D(L
m

) ⊂ Y (m is a natural number) the operators

Lj : Ym → Xm−j, P : Zm−j → Ym−j, PTr : Ym−j → Ym−j,

Tj : Xm−j → Ym (0 ≤ j ≤ m − 1).

Here we have Ym ⊆ . . . ⊆ Y2 ⊆ Y1 and L
0

= T 0 = I.



CUBO
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On a new notion of holomorphy and its applications 157

Proposition 1. The following properties are fulfiled

(i) The operators TjPTrLj (0 ≤ j ≤ m − 1) are projections on Ym.

(ii) The projections TjPTrLj (0 ≤ j ≤ m−1) are complementary on Ym, i.e. (T
jPTrLj)(TkPTrLk) =

(TkPTrLk)(TjPTrLj) = 0 for all 0 ≤ j,k ≤ m − 1 and k 6= j.

Proof. (i) Indeed, using the assumption PTrP = P and corollary 1 we obtain

(TjPTrLj)(TjPTrLj) = TjPTrLjTjPTrLj = TjPTrPTrLj = TjPTrLj,

i.e. TjPTrLj are projections on Ym. To prove property (ii) we also use corollary 1. It is im-

mediately clear that LjTj = I from LT = I. Because of PTrT = 0 and LjTj = I follows for

j < k:

(TjPTrLj)(TkPTrLk) = TjPTrLjTkPTrLk = TjPTrTk−jPTrLk = 0,

i.e.

(TjPTrLj)(TkPTrLk) = 0 (0 ≤ j < k ≤ m).

Taking into account relation in the corollary from above, the commutative property is obtained.

Indeed, from property LPTr = 0 we have

(TkPTrLk)(TjPTrLj) = TkPTrLkTjPTrLj = TkPTrLk−jPTrLj = 0,

i.e.

(TkPTrLk)(TjPTrLj) = 0 (0 ≤ j < k ≤ m).

Hence all TjPTrLj(0 ≤ j ≤ m) are complementary on Ym. #

Then the next corollary is clear.

Corollary 3. The operator

Pm :=

m−1∑

j=0

TjPTrLj = T 0PTrL0 ⊕ T 1PTrL1 ⊕ . . . ⊕ Tm−1PTrLm−1

is a projection on Ym−1.

Corollary 4. The operators Pm,T
m and Lm have the following relations

(i) The operator Tm is the right-inverse to the operator Lm, i.e. LmTm = I.

(ii) The operators Lm,Pm satisfy the property L
mPm = 0.



158 Wolfgang Sproessig and Le Thu Hoai CUBO
11, 1 (2009)

(iii) It holds PmT
m

= 0.

Proof. The relation (i) is simple to be obtained from corollary 1. To prove (ii), one use

assumption LPTr = 0 and LjTj = I for 0 ≤ j ≤ m − 1 as mentioned above then

LmPm =

m−1∑

j=0

LmTjPTrLj =

m−1∑

j=0

Lm−jPTrLj = 0.

The same for relation (iii) with assumption PTrT = 0:

PmT
m

= Pm :=

m−1∑

j=0

TjPTrLjTm = Pm :=

m−1∑

j=0

TjPTrTm−j = 0.

Theorem 2. (The Taylor type formula) Let L be a right invertible operator that defined from an

injection T and an initial operator P. Then for m = 1, 2, ... the following identity holds on Ym

u =

m−1∑

j=0

TjPTrLju + TmLmu.

Proof. We have ker Tm = {0} by assumption T is an injection and im Tm ⊂ Ym = D(L
m

).

Corollary 3 shows that Pm is a projection and PmT
m

= 0. Furthermore, it is simple to show that

im Tm ∩ im Pm = {0}. Indeed, let u ∈ im T
m ∩ im Pm then

u = Pmv = T
mw, (v ∈ Ym−1,w ∈ X).

Since PmT
m

= 0 we get

u = Pmv = PmPmv = PmT
mw = 0.

Let B be the (unique) right inverse to Tm then (from the mean value formula)

u = Pmu + T
mBu with D(B) := imTm ⊕ imPm.

Now we will show that Lm also satisfies above formula. By applying the mean value formula for

Lju we get

Lju = PTrLju + TLj+1u (0 ≤ j ≤ m − 1)

Rewrite in more detail and acting operators Tj (0 ≤ j ≤ m − 1) to both sides we have

T 0L0u = T 0PTrL0u + TLu,

TLu = TPTrLu + T 2L2u,

· · ·

Tm−1Lm−1u = Tm−1PTrLm−1u + TmLmu.



CUBO
11, 1 (2009)

On a new notion of holomorphy and its applications 159

Sum up all equabilities we obtain

u = T 0L0u = T 0PTrL0u + TPTrLu + . . . + Tm−1PTrLm−1u + TmLmu

= Pmu + T
mLmu.

Then the property of uniqueness of right inverse operator leads to

B = Lm.

This completes the proof of our theorem.

Example 15. (Realisation in R1) We continue the first example in section 3.1.For all functions

u ∈ C1[0, 1], recall that

L :=
d

dt
, T :=

t∫

0

·dτ ,

P := I and Tr : C1[0, 1] → R1 with Tru = u(0). Then we have

TjPTr(Lju)(t) = (Lju)(0)
tj

j!

and

(Tmu)(t) =

t∫

0

(t − τ)m−1

(m − 1)!
u(τ)dτ .

Hence the theorem 2 yields the classical Taylor’s formula

u(t) =

m−1∑

j=0

(Lju)(0)
tj

j!
+

t∫

0

(t − τ)m−1

(m − 1)!
(Lmu)(τ)dτ .

Example 16. (Taylor formula for fractional operators) In [9] J.D. Munkhammar gave Taylor’s

formula based on fractional caculus. Let u(t) ∈ C1([a,b]) then the Riemann-Liouville fractional

integral of order α is

(Tu)(t) := Iα
a+
u(t) =

1

Γ(α)

t∫

a

u(s)

(t − s)1−α
ds ,

and the Riemann–Liouville fractional derivative of order α as follow

(Lu)(t) := Dα
a+
u(t) =

1

Γ(1 − α)

d

dt

t∫

a

u(s)

(t − s)α
ds



160 Wolfgang Sproessig and Le Thu Hoai CUBO
11, 1 (2009)

where α ∈]0, 1[ and Γ is a well known Gamma function. Hence

Dα
a+
Iα
a+

= I.

Let α > 0, m ∈ Z+ and u(t) ∈ C[α]+m+1([a,b]), the Taylor formula is

u(t) =

m−1∑

k=−m

D
α+k
a+

u(t0)

Γ(α + k + 1)
(t − t0)

α+k
+ I

α+m
a+

D
α+m
a+

u(t)

for all a ≤ t0 < t ≤ b.

8 Taylor-Gontcharov’s formula for high order genaralized Dirac

operators

Corollary 5. (The Taylor-Gontcharov’s formula) A generalization of the Taylor formula leads

to

u =

m−1∑

j=0

T0T1...TjPjLj...L1L0u + T1...TmLm...L1u

with L0 = T0 = I.

Example 17. (Realisation on a lattice) Let Gh be the lattice of the bounded domain G and

∆h = D
+

h
D

−
h

be the discretized Laplace operator. We consider the following problem

∆hu = f on Gh,

trΓPΓhu = g0 on Γh,

trΓhD
−
h
u = g1 on Γh.

Γh is the "‘numerical"’ boundary of G for a meshwidth h. The unique solution is then given by

u = F
−
h
g0 + T

−
h
F

+

h
(trΓhT

−
h
F

+

h
)
−1T−

h
D

−
h
g1 + T

−
h

QhT
+

h
f

with Bergman projection

Ph = F
+

h
(trΓhT

−
h
F

+

h
)
−1trΓhT

−
h

The operators in Taylor-Gontcharov’ s formula are chosen as follows

L1 := D
−
h
, L2 := D

+

h
, P1 := F

−
h
, P2 := F

+

h
, T1 := T

−
h
, T2 := T

+

h

Received: April 2008. Revised: August 2008.



CUBO
11, 1 (2009)

On a new notion of holomorphy and its applications 161

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