Volume 2008, Article ID 656329,14pages doi:10.1155/2008/656329
Research Article
Generating Functions for the Mean
Value of a Function on a Sphere and Its
Associated Ball in
R
n
Antonela Toma
Department of Mathematics II, University Politehnica of Bucharest, Splaiul Independent¸ei 313, 060042 Bucharest, Romania
Correspondence should be addressed to Antonela Toma,antonela2222@yahoo.com Received 20 April 2008; Accepted 22 May 2008
Recommended by Patricia Wong
We define two functions which determine the properties and the representation of the mean value of a function on a ball and on its associated sphere. Using these two functions, we obtain Pizzetti’s formula inRnas well as a similar formula for the mean value of a function on the ball associated to
the sphere. We also give the expressions of the remainders in these two formulas, using the surface integral on a sphere.
Copyrightq2008 Antonela Toma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The mean function over values of a sphere and over its associated ball are very important in the study of some mathematical-physics problems as well as in the theory of a potential and in partial differential equations.
A representation for the mean values of a function over a sphere in Rn was given by Pizzetti1using polyharmonic operators.
In2, volume II chapter IV, Section 3, page 258, using the second Green’s formula, it is given the proof for Pizzetti’s formula and then its generalization inRn.
In this paper, we define two functions which determine the properties and the representations of the mean values of a function over a sphere and over its associated ball.
These functions are called generating functions of these two averages and using these functions we obtain Pizzetti’s formula inRnas well as a new formula for the mean values of a
function over a ball.
There are given the expression of the remainders using an integral over a sphere.
For this purpose we use two formulas of N. Ya. Sonin and of Dirichlet3, page 365,4, page 671, respectively.
There are defined the corresponding quantities of some scalar quantities using the differential operators∇,Δ,Δh.
This fact allows us to prove two new formulas which determine the properties of the generating functions.
It is very important to mention that in this paper the way of deducting Pizzetti’s formula is totally different from the way used in3, page 73as well as in5, page 104.
2. General results
LetΩ⊂Rnbe a bounded set andf:Ω→R,f ∈C2m 1Ω.
We denote bySra {x, x ∈ Rn, |x−a| r} the sphere of radiusr, centered in a
a1, a2, . . . , an,Bra {x, x ∈Rn,|x−a| ≤r}the ball of radiusr and centered inawhich is
associated toSra.
We will also denote by R, R max{r |x−a|n} for that Bra ⊂ Ω. We define the
functions
φ:−R, R−→R, φr
B1a
fa rx−adx, 2.1
ψ :−R, R−→R, ψr
S1a
fa rx−adS1, 2.2
wheredx,dS1represent the volume element and area element for the unit ball B1aand for
the unit sphere, respectively. The mean valuesMs
rfandMbrfforf : Ω → RoverSra⊂Ωand overBra⊂Ω,
respectively, are given by the following expressions:
Msrf 1 Sra
Sra
fxdS, 0≤r≤R, 2.3
Mrbf 1 Bra
Bra
fxdx, 0≤r ≤R, 2.4
wheresee6, page 22|Sra| 2πn/2/Γn/2rn−1represents the area of the sphere Sra
and |Bra| 2πn/2/Γn/2rn/n represents the volume of the ball Bra, Γ being
beta-function.
Between the cartesian coordinates x x1, x2, . . . , xn and spherical coordinates y
ρ, θ1, . . . , θn−1centered ina a1, a2, . . . , anthere are the relations
x1a1 ρsinθ1sinθ2· · ·sinθn−2sinθn−1a1 ρh1,
x2a2 ρsinθ1sinθ2· · ·sinθn−2cosθn−1a1 ρh2,
x3a3 ρsinθ1sinθ2· · ·sinθn−3cosθn−2a3 ρh3,
.. .
xn−2an−2 ρsinθ1sinθ2cosθ3an−2 ρhn−2,
xn−1an−1 ρsinθ1cosθ2an−1 ρhn−1,
xnan ρcosθ1an ρhn,
where
ρ≥0, θi∈0, π, i1, n−2, θn−1∈0,2π. 2.6
The Jacobian of this punctual transform is
Jρ, θ1, . . . , θn−1
∂x
∂y ρ
n−1sinn−2θ
1sinn−3θ2sinn−4θ3· · ·sinθn−2. 2.7
The volume elementdywritten in spherical coordinates has the expression
dyJρ, θ1, . . . , θn−1
dρ dθ1· · ·dθn−1 2.8
and the area element forSρais
dSρ ρn−1y∗
θ1, . . . , θn−2
dθ1dθ2· · ·dθn−1, 2.9
where
J∗θ1, . . . , θn−2
J
ρ, θ1, . . . , θn−1
ρn−1 sin n−2θ
1sinn−3θ2· · ·sin2θn−3sinθn−2. 2.10
From2.8and2.9we have
dy dρ dSρρn−1dρ dS1
θ1, . . . , θn−1
, 2.11
where dS1θ1, . . . , θn−1 represents the area element of the unit sphere S1a in spherical
coordinates.
Proposition 2.1. Between the functions φ, ψ : −R, R → Rdefined in2.1and2.2, there is the
relation
φr
1
0
un−1ψrudu, |r| ≤R. 2.12
Proof. Using the spherical coordinates and the relations 2.9 and 2.11 we obtain the following:
ψr
Δf
. . . , ai rhi, . . .
J∗θ1, . . . , θn−2
dθ1· · ·dθn−1
Δf
. . . , ai rhi, . . .
dS1
θ1, . . . , θn−1
,
2.13
φr
1
0
Δf
. . . , ai ruhi, . . .
un−1du dS1
θ1, . . . , θn−1
1
0
un−1
Δf
. . . , ai ruhi, . . .
dS1
θ1, . . . , θn−1
du,
2.14
whereΔ 0, π× · · · ×0, π×0,2π
n−2 times
. From2.13we note that
ψru
Δf
. . . , ai ruhi, . . .
dS1
θ1, . . . , θn−1
2.15
Concerning the dependence between the functions φ, ψ and the mean values Ms rf,
Mb
rfof a functionf over a sphere and over the associated ball, respectively, we can state the
following.
Proposition 2.2. Between the functions φ, ψ defined by 2.1and2.2, respectively, and the mean
valuesMfrf,Mbrfdefined by2.3and2.4, there are the following relations:
Mb rf
rn
Braφr nΓn/2
2πn/2 φr, 0≤r ≤R, 2.16
Ms rf
rn−1
Sraψr nΓn/2
2πn/2 ψr, 0≤r ≤R. 2.17
Proof. For 0≤ r ≤ R, from2.14makingthe substitutionur ρand taking into account2.4
we obtain
φr 1 rn
r
0
ρn−1
Δf
. . . , ai ρhi, . . .
dS1
θ1, . . . , θn−1
dρ
1
rn
Bra
fxdx 2πn/
2
nΓn/2M
b rf
2.18
and so,2.16is proved.
Using the spherical coordinates, we have
Msrf
1
Sra
Srx0
fxdx
1
Sra
Δf
. . . , ai rhi, . . .
dSr
θ1, . . . , θn−1
rn−1
Sra
Δf
. . . , ai rhi, . . .
dS1
θ1, . . . , θn−1
.
2.19
Taking into account2.13we have
Ms rf
rn−1
Sraψr Γn/2
2πn/2 ψr 2.20
which is2.17.
The relations2.16and2.17show that for 0≤r ≤R, the functionsφ, ψ:−R, R→ R, determine the properties of the mean valuesMbrfandMsrfand permit the calculus of these
two quantities.
These relations justify the introduction of the following.
Definition 2.3. The functionsφ, ψ:−R, R→Rdefined by2.1and2.2are called generating functions for the mean valuesMb
rfandMsrfof a functionf:Ω→Rover the ballBra⊂Ω
and over the sphereSra⊂Ω, respectively. Particularly, iff ≥0 onΩ⊂Rn, then this function
Consequently, taking into account2.3and2.17, the total massms
r of the sphereSra
is given by the expression
ms r
Sra
fxdSSraMsrf rn−1ψr, 0≤r≤R. 2.21
In this case, Msrf represents the mean value of the mass density onSra. Similarly, from
2.4and2.16we have the following expression for the total mass onBra:
mb r
Bra
fxdxBraMbrf rnφr, 0≤r≤R. 2.22
The expressions 2.21and 2.22 prove that the generating functions φ andψ, when f ≥ 0 and 0≤ r ≤ R, have a mechanical meaning, allowing the calculus of the mass forSra
and forBrawith the densityρx fx,x∈Ω⊂Rn.
Next, for the study of the properties of the generating functionsψ andφ, we will use M. Ya. Sonin Formula.
Letmi ∈R,i 1, nbe real numbers andk ∈N0. Thensee3, page 365,4, page 671
we have Sonin formula
S
B10
m1x1 · · · mnxn
2k
dx1· · ·dxn
πn−1/2ΓΓk 1/2 n/2 k 1
m21 · · · mrnk.
2.23
Denotingm m1, m2, . . . , mn∈Rnand “ ” the scalar product,2.23becomes
S
B10
m, x2k πn−1/2ΓΓk 1/2
n/2 k 1 m, m
k. 2.24
We mention that this result can be justifiedsee3, page 365on the basis of Dirichlet formula
D
B10 x2u1
1 · · ·x 2un
n dx1· · ·dxn
Γμ1 1/2
· · ·Γμn 1/2
Γn/2 k 1 , 2.25
whereμi∈N0,i1, nandkμ1 · · · μn.
Usingμ μ1 · · · μn∈Nn0, where|μ|μ1 · · · μn k, Dirichlet formula becomes
D
B10
x2μdx Γ
μ1 1/2
· · ·Γμn 1/2
Γn/2 k 1 . 2.26
Jacobian of the transform, we have
S
B1a
m, x−a2kdx
B10
m, x2kdxπn−1/2ΓΓk 1/2
n/2 k 1 m, m
k, 2.27
D
B1a
x−a2μdx
B10
x2μdx Γ
μ1 1/2
· · ·Γμn 1/2
Γn/2 k 1 , 2.28 whereB1arepresents the unit ball centered ina∈Rn.
Let us consider the integrals
S∗
S1a
m, x−a2kdS1, D∗
S1a
x−a2μdS1, 2.29
whereS1ais the unit sphere, centered ina∈Rn.
Proposition 2.4. Between the pairs of integralsS∗, SandD∗, D, there are the following relations:
S∗ 2k nS 2πn−1/2ΓΓk 1/2
k n/2 m, m
k, 2.30
D∗ 2k nD 2Γ
μ1 1/2
· · ·Γμn 1/2
Γk n/2 . 2.31
Proof. Using the spherical coordinates2.5, we have
x−ahh1, h2, . . . , hn
. 2.32
Taking into account2.11, Sonin’s integral2.27becomes
S
B1a
m, x−a2kdx
1
0
Δ m, ρh
2kρn−1dρ dS 1
θ1, . . . , θn−1
1
2k n
Δ m, h 2kdS
1
θ1, . . . , θn−1
1
2k n
S1a
m, x−a2kdS1,
2.33
whereΔ 0, π× · · · ×0, π
n−2
×0,2π. We obtain
S∗
S1a
m, x−a2kdS1
2k nS
2k nΓΓk 1/2 n/2 k 1π
n−1/2 m, mk
2Γk 1/2
Γn/2 kπ
n−1/2 m, mk.
Using the same procedure, Dirichlet’s integral2.28becomes
D
B1a
x−a2μdx
1
0
Δ m, ρh
2μρn−1dρ dS 1
θ1, . . . , θn−1
1
2k n
Δh 2μdS
1
θ1, . . . , θn−1
1
2k n
S1a
x−a2μdS1.
2.35
Hence
D∗
S1a
x−a2μdS1
2k nD
2k nΓ
μ1 1/2
· · ·Γμn 1/2
Γk n/2 1
2Γ
μ1 1/2
· · ·Γμn 1/2
Γn/2 k .
2.36
So the proposition is proved.
Next, using the formulas concerning the calculus of the higher order differential for functions of several variables, we will define the correspondings for some scalar quantities which appear in the expressions ofSandS∗, respectively2.27and2.29.
The corresponding scalar quantities are defined using some differential operators, this fact leads us to new expressions similar to2.24and2.30.
On the basis of these formulas, we will establish the properties of the generating functionsψandφand of the mean valuesMsrfandMbrf.
Letf ∈CpΩwithB
1a⊂Ωand 2k≤p. We will define the following correspondings:
1
m m1, . . . , mn
−→ ∇fa
∂ ∂x1, . . . ,
∂ ∂xn
fa, 2.37
where∇ ∂/∂x1, . . . , ∂/∂xnrepresents the operator “nabla”;
2
|m|2 m, m −→ ∇,∇fa Δfa, 2.38
where| · |represents the norm of a vector andΔ ∇,∇∂2/∂x2
1 · · · ∂2/∂x2nthe
3
|m|2k m, mk −→ ∇,∇kfa Δkfa, 2.39
where Δk Δ ·Δ· · ·Δ
k
∂2/∂x21 · · · ∂2/∂x2nk represents the polyharmonic
operator of orderk;
4
m, x−a −→ ∇, x−afa
∂ ∂x1
x1−a1
· · · ∂
∂xn
xn−an
fa dfa,
2.40
whered ∂/∂x1x1−a1 · · · ∂/∂xnxn−anrepresents the differential operator;
5
m, x−a2k −→ ∇, x−a2kfa d2nfa, 2.41
where d2k ∂/∂x
1x1−a1 · · · ∂/∂xnxn−an2k represents the differential
operator of order 2k.
Using these correspondences and taking into account 2.23, 2.27, 2.29, and 2.30, we obtain
B1a
d2kfadxπn−1/2ΓΓk 1/2 n/2 k 1Δ
kfa, 2.42
S1a
d2kfadS12πn−1/2ΓΓk 1/2
n/2 kΔ
kfa. 2.43
We note that the expressions 2.42and 2.43 represent the correspondings for 2.27
and2.30. We will prove the availability of these relations.
Proposition 2.5. Forf ∈CpΩwithB
1a⊂Ωand2k≤pthe relations2.42and2.43held. Proof. We have
B1a
d2kfadx
B1a
∂ ∂x1
x1−a1
· · · ∂
∂xn
xn−an
2k
fadx1· · ·dxn
2μ1 ··· 2μn2k
2k!
2μ1
!· · ·2μn
!
∂2
∂x2 1
μ1
· · ·
∂2
∂x2 n
μn
fa·
B1a
x−a2μdx.
2.44
On the basis of2k!2kk!2k−1!!22kk!Γk 1/2/√π, we obtain
2k!
2μ1
!· · ·2μn
! k! μ1!· · ·μn!
πn−1/2Γk 1/2
Γμ1 1/2
· · ·Γμn 1/2
Taking into account2.45and Dirichlet Formulas2.43, the expression2.44becomes
B1a
d2kfadxπn−1/2ΓΓk 1/2 n/2 k 1
μ1 μ2 ··· μnk
k! μ1!· · ·μn!
∂2 ∂x21
μ1
· · ·
∂2 ∂xn2
μn
fa,
2.46
so that
B1a
d2kfadxπn−1/2ΓΓk 1/2 n/2 k 1
∂2
∂x21 · · · ∂2
∂x2n
k
fa
πn−1/2ΓΓk 1/2 n/2 k 1Δ
kfa.
2.47
Using the same method, we obtain
S1a
d2kfadx
S1a
∂ ∂x1
x1−a1
· · · ∂
∂xn
xn−an
2k
fadS1
2μ1 ··· 2μn2k
2k!
2μ1
!· · ·2μn
!
∂2
∂x21
μ1
· · ·
∂2
∂x2n
μn
fa·
S1a
x−a2μdS1.
2.48
On the basis of2.45and2.31, we have
S1a
d2kfadS 1
2πn−1/2Γk 1/2
Γn/2 k
μ1 ··· μnk
k! μ1!· · ·μn!
∂2
∂x21
μ1
· · ·
∂2
∂x2n
μn
fa
2πn−1/2Γk 1/2
Γn/2 k
∂2
∂x21 · · · ∂2
∂xn2
k
fa
2πn−1/2ΓΓk 1/2 n/2 kΔ
kfa
2.49
and so the proposition is proved.
Proposition 2.6. LetΩ ∈ Rn be a bounded set andf : Ω → R,f ∈ C2m 1Ω. Then the functions
φ, ψ :−R, R→Rdefined by2.1and2.2, whereR max|x−a|such thatBra⊂Ω, have the following properties:
1φ,ψare even functions andφ, ψ∈C2m 1−R, R,
2φk0 ψk0 0forkodd,k≤2m 1,
3ψ2k0 S1ad
2kadS
1 2πn−1/2Γk 1/2/Γn/2 kΔkfa, k≤m,
4φ2k0 1/n 2kψ2k0 πn−1/2Γk 1/2/Γn/2 k 1Δkfa, k≤m,
5φpr 01un p−1ψprudu, p≤2m 1,
6ψpr S1adpfa rx−adS
1, p ≤2m 1,
7ψr 2πn/2mk01/k!Γn/2 kr/22kΔkfa R2m 1r,
where
R2m 1r ψ
2m 1θr
2m 1! r
2m 1 r2m 1
2m 1!
S1a
d2m 1fa θrx−adS1, 0< θ <1, 2.50
8φr πn/2m
k01/k!Γn/2 k 1r/22kΔkfa R∗2m 1r, where
R∗2m 1r φ
2m 1θr
2m 1!r
2m 1 r 2m 1
2m 1!
1
0
S1a
un 2md2m 1fa θrux−adu dS1, 0< θ <1.
2.51
Proof. Using the spherical coordinates given by 2.5 and denoting h h1, . . . , hn, the
expression forψgiven by2.2becomes
ψr
S1a
fa rx−adS1
ΔJ ∗θ
1, . . . , θn−2 2π
0
f∗r, θ1, . . . , θn−1
dθn−1
dθ1· · ·dθn−2,
2.52
whereΔ 0, π× · · · ×0, π
n−2
,f∗r, θ1, . . . , θn−1 fa rhandJ∗θ1, . . . , θn−2is given by the
formulas2.10.
Since f ∈ C2m 1Ω, using the differentiation rule for integrals depending on a
parameter, it results inψ ∈C2m 1−R, R.
In order to prove thatψ is an even function we will change the variablesθ1, . . . , θn−1→
u1, . . . , un−1by the relations
θiπ−ui, i1, n−2, θn−1π un−1,
ui∈0, π, i1, n−2, un−1∈−π, π
2.53
The Jacobian of this transform is defined by
J1
u1, . . . , un−1
∂
θ1, . . . , θn−1
∂u1, . . . , un−1
−1n−2 2.54
Having this change of variables,2.52becomes
ψr
ΔJ ∗u
1, . . . , un−2
π
−πf
∗−r, u
1, . . . , un−1
dun−1
du1· · ·dun−2. 2.55
Sincef∗−r, u1, . . . , un−1is periodical, with the period 2πwith respect to the variableun−1, the
expression2.55becomes
ψr
ΔJ ∗u
1, . . . , un−2 2π
0
f∗−r, u1, . . . , un−1
dun−1
du1· · ·dun−2. 2.56
Makingthe comparison between the expression from above and2.52we haveψr ψ−rsoψ is an even function.
Consequently,ψ2kis an even function andψ2k 1is odd, it means that ψ2k 10 0.
From 2.12 we have φ is even, φ ∈ C2m 1−R, R, φ2k is even and φ2k 1 is odd, so
φ2k 10 0.
So we proved the properties1and2.
Remark 2.7. In the case off :Rn→ R,f ∈ DRnwhereDRnis L. Schwartz space,f ∈CRn
and having a compact support, thenψ : R → R,ψr S1afa rx−adS1,ψr ∈ CR
with compact support.
Iff ∈ DRn, we can state that the formulas
ψr
S1a
fa rx−adS1, 2.57
r ∈R, define an integral transform which associatesf ∈ DRnto the even functionψ ∈ DR.
According to 7, page 61, this integral transform is denoted by Tn
p named polare
transform. Thus we writeψ Tn
pf. From2.2applying the differentiation rule for composite
functions and denotingyarx−a, we obtain
ψpr
S1a ∂p ∂rpf
a rx−adS1
S1a
∂ ∂yi
∂yi
∂r
p
fydS1
S1a
∂ ∂y1
x1−a1
· · · ∂
∂yn
xn−an
p
fydS1
S1a
dpfa rx−adS1.
Forp2k≤2mandr 0 we have
ψ2k0
S1a
∂ ∂x1
x1−a1
· · · ∂
∂xn
xn−an
2k
fadS1
S1a
d2kfadS1.
2.59
Taking into account2.43, we have
ψ2k0
S1a
d2kfadS12πn−1/2ΓΓk 1/2
n/2 kΔ
kfa. 2.60
On the other hand, from2.12and2.60we obtain
φpr
1
0
un p−1ψprudu, |r| ≤R, p≤2m 1,
φ2k0 1 n 2kψ
2k0
2π n−1/2
n 2k
Γk 1/2
Γn/2 kΔ
kfa
πn−1/2ΓΓk 1/2 n/2 k 1Δ
kfa.
2.61
The formulas2.60and2.61justify the properties3,4,5,6from the proposition. In order to obtain the formula 7 we will apply Mac-Laurin formulas. Thus ψ being even andψ ∈C2m 1−R, R, on the basis of2.58, we can write
ψr
m
0
ψ2k0
2k! r
2k R
2m 1r, 2.62
where
R2m 1r ψ
2m 1θr
2m 1! r
2m 1 r 2m 1
2m 1!
S1a
d2m 1fa θrx−adS1, 0< θ <1. 2.63
Since2k!22kk!Γk 1/2π−1/2and taking into account2.60, from2.62we obtain7.
Using the same method, we obtain8taking into account the results from4,5, and
6. So,Proposition 2.6is proved.
In particular, if f ∈ C−R, R and f is analytic, then the remainders R2m 1t → 0,
R∗2m 1r→0 form→ ∞.
We obtain the following Mac-Laurin Series forψ, φ:−R, R→R:
ψr 2πn/2
k0
1 k!Γn/2 k
r 2
2k
Δkfa, |r| ≤R, 2.64
φr πn/2
k0
1
k!Γn/2 k 1
r 2
2k
The results established inProposition 2.6permit to give the corresponding representa-tions for the mean values Ms
rf andMbrf defined in 2.3 and2.4, by using 2.16 and
2.17.
Thus, for 0≤r ≤R, taking into account8,2.16, and2.64we can write
Mbrf 1 Bra
Bra
fxdx
n 2Γ n 2 m k0 1
k!Γn/2 k 1
r 2
2k
Δkfa
n 2πn/2Γ
n 2
r2m 1
2m 1!
1
0
S1a
un 2md2m 1fa θrux−adu dS1, 0< θ <1,
Mbrf n 2Γ n 2 k0 1
k!Γn/2 k 1
r 2
2k
Δkfa.
2.66 Similarly, from7,2.62, and2.17we have
Msrf 1 Sra
Sra
fxdS
Γ n 2 m k0 1 k!Γn/2 k
r 2
2k
Δkfa
Γn/2 2πn/2 ·
r2m 1
2m 1!
S1a
d2m 1fa θrx−adS1, 0< θ <1,
2.67
Msrf Γ
n 2 k0 1 k!Γn/2 k
r 2
2k
Δkfa. 2.68
We remark that the expression of the remainder forMs
rfgiven in2.67
R∗∗2m 1r
Γn/2 2πn/2
r2m 1
2m 1!
S1a
d2m 1fa θrx−adS1, 0< θ <1 2.69
may have other forms as in2, volume II, chapter IV, Section 3, page 261and8.
Forn 3, sinceΓ3/2 k 2k 1!/22k 1k!√π, from 2.68, we obtain Pizzetti’s Formula1:
Msrf
k0
r2k
2k 1!Δ
kfa. 2.70
Similarly, since
Γ
3
2 k 1
3 2 k Γ 3 2 k
2k 32k 1!
22k 2k!
√
π, 2.71
we have
Mrbf 3
k0
r2k
2k 1!2k 3Δ
kfa. 2.72
References
1 P. Pizzetti, “Sulla media dei valori che una funzione dei punti dello spazio assume alla superficie di una sfera,”Rendiconti Lincei (5), vol. 18, pp. 309–316, 1909.
2 R. Courant and D. Hilbert,Methoden der Mathematischen Physik, Springer, Berlin, Germany, 1937. 3 G. M. Fichtenholz,Differential-und Integralrechnung. III, vol. 63 ofHochschulb ¨ucher f ¨ur Mathematik, VEB
Deutscher Verlag der Wissenschaften, Berlin, Germany, 1977.
4 I. S. Gradstein and I. M. Ryshik,Tables of Series, Products and Integrals. Vol. 1, 2, Harri Deutsch, Thun, Switzerland, 1981.
5 G. E. Shilov, Generalized Functions and Partial Differential Equations, vol. 7 of Mathematics and Its Applications, Gordon and Breach, New York, NY, USA, 1968.
6 S. G. Mihlin,Ecuat¸ii liniare cu derivate part¸iale, Editura S¸tiint¸ific˘a s¸i Enciclopedic˘a, Bucures¸ti, Romania, 1983.
7 W. W. Kecs,Teoria distribit¸iilor s¸i aplicat¸ii, Editura Academiei Romˆane, Bucures¸ti, Romania, 2003. 8 M. N. Olevski˘ı, “An explicit expression for the remainder in the Pizetti formula,”Functional Analysis
