Generating Functions for the Mean Value of a Function on a Sphere and Its Associated Ball in R

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

Antonela Toma

Department of Mathematics II, University Politehnica of Bucharest, Splaiul Independent¸ei 313, 060042 Bucharest, Romania

Correspondence should be addressed to Antonela Toma,[email protected] 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 inRⁿ 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.

The properties of the generating functions are established using the formality concerning the calculus for higher order derivatives for functions of several variables.

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,Δ^h.

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∈C^{2m 1}Ω.

We denote bySra {x, x ∈ Rⁿ,|x−a| r}the sphere of radiusr, centered ina 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 thatBra ⊂ Ω. We define the functions

φ:−R, R−→R, φr

B1af

a rx−a

dx, 2.1

ψ:−R, R−→R, ψr

S1af

a rx−a

dS1, 2.2

wheredx,dS1 represent the volume element and area element for the unit ballB1aand for the unit sphere, respectively.

The mean valuesM^s_rfandM^b_rfforf :Ω→R overSra⊂Ωand overBra⊂Ω, respectively, are given by the following expressions:

M^s_rf 1 Sra

SrafxdS, 0≤r ≤R, 2.3

M_r^bf 1 Bra

Brafxdx, 0≤r≤R, 2.4

wheresee6, page 22|Sra| 2π^n/2/Γn/2rⁿ⁻¹represents the area of the sphereSra and |Bra| 2π^n/2/Γn/2rⁿ/n represents the volume of the ballBra, 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,

2.5

where

ρ≥0, θi∈0, π, i1, n−2, θn−1∈0,2π. 2.6 The Jacobian of this punctual transform is

J

ρ, θ1, . . . , θn−1 ∂x

∂y ρⁿ⁻¹sinⁿ⁻²θ1sinⁿ⁻³θ2sinⁿ⁻⁴θ3· · ·sinθn−2. 2.7 The volume elementdywritten in spherical coordinates has the expression

dyJ

ρ, θ1, . . . , θn−1

dρ dθ1· · ·dθn−1 2.8

and the area element forSρais dSρρⁿ⁻¹y^∗

θ1, . . . , θn−2

dθ1dθ2· · ·dθn−1, 2.9 where

J^∗

θ1, . . . , θn−2 J

ρ, θ1, . . . , θn−1

ρⁿ⁻¹ sinⁿ⁻²θ1sinⁿ⁻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 functionsφ, ψ :−R, R → R defined in2.1and2.2, there is the relation

φr ₁

0uⁿ⁻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 and taking into account2.14, the relation2.12is proved.

Concerning the dependence between the functionsφ, ψ and the mean valuesM_r^sf, M_r^bfof a functionfover a sphere and over the associated ball, respectively, we can state the following.

Proposition 2.2. Between the functionsφ, ψ defined by2.1and2.2, respectively, and the mean valuesM^frf,M^b_rfdefined by2.3and2.4, there are the following relations:

M^b_rf rⁿ

Braφr nΓn/2

2π^n/2 φr, 0≤r≤R, 2.16

M^srf rⁿ⁻¹

Sraψr nΓn/2

2π^n/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

Brafxdx 2π^n/2

nΓn/2M^b_rf

2.18

and so,2.16is proved.

Using the spherical coordinates, we have M^s_rf 1

Sra

Srx0fxdx 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

M^s_rf rⁿ⁻¹

Sraψr Γn/2

2π^n/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 valuesM^b_rfandM^s_rfand permit the calculus of these two quantities.

These relations justify the introduction of the following.

Definition 2.3. The functionsφ, ψ:−R, R→R defined by2.1and2.2are called generating functions for the mean valuesM_r^bfandM^s_rfof a functionf:Ω→R over the ballBra⊂Ω and over the sphereSra⊂Ω, respectively. Particularly, iff≥0 onΩ⊂Rⁿ, then this function can be considered as the mass density onSraand onBra, respectively.

Consequently, taking into account2.3and2.17, the total massm^s_rof the sphereSra is given by the expression

m^sr

SrafxdSSraM^s_rf rⁿ⁻¹ψr, 0≤r≤R. 2.21 In this case,M^s_rfrepresents the mean value of the mass density onSra. Similarly, from 2.4and2.16we have the following expression for the total mass onBra:

m^b_r

BrafxdxBraM^b_rf rⁿφr, 0≤r ≤R. 2.22 The expressions2.21and 2.22 prove that the generating functionsφ 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 functionsψandφ, 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 · · · mnxn2k

dx1· · ·dxn

π^n−1/2 Γk 1/2 Γn/2 k 1

m²₁ · · · m^r_nk

.

2.23

Denotingm m1, m2, . . . , mn∈Rⁿand “ ” the scalar product,2.23becomes

S

B10 m, x^2kπ^n−1/2 Γk 1/2

Γn/2 k 1 m, m^k. 2.24

We mention that this result can be justifiedsee3, page 365on the basis of Dirichlet formula

D

B10x^2u₁ ¹· · ·x^2un ⁿdx1· · ·dxnΓ

μ1 1/2

· · ·Γ

μn 1/2

Γn/2 k 1 , 2.25

whereμi∈N0,i1, nandkμ1 · · · μn.

Usingμ μ1 · · · μn∈Nⁿ₀, where|μ|μ1 · · · μnk, Dirichlet formula becomes

D

B10x^2μdx Γ

μ1 1/2

· · ·Γ

μn 1/2

Γn/2 k 1 . 2.26

Making the substitutionsui xi−ai,i 1, nand using Ju1, . . . , um ∂x/∂u 1 the

Jacobian of the transform, we have S

B1a m, x−a^2kdx

B10 m, x^2kdxπ^n−1/2 Γk 1/2

Γn/2 k 1 m, m^k, 2.27

D

B1ax−a^2μdx

B10x^2μdxΓ

μ1 1/2

· · ·Γ

μn 1/2

Γn/2 k 1 , 2.28 whereB1arepresents the unit ball centered ina∈Rⁿ.

Let us consider the integrals S^∗

S1a m, x−a^2kdS1, D^∗

S1ax−a^2μdS1, 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 2π^n−1/2Γk 1/2

Γk n/2 m, m^k, 2.30

D^∗ 2k nD 2Γ

μ1 1/2

· · ·Γ

μn 1/2

Γk n/2 . 2.31 Proof. Using the spherical coordinates2.5, we have

x−ah

h1, h2, . . . , hn

. 2.32

Taking into account2.11, Sonin’s integral2.27becomes S

B1a m, x−a^2kdx

₁

0

Δ m, ρh^2kρⁿ⁻¹dρ dS1

θ1, . . . , θn−1 1

2k n

Δ m, h^2kdS1

θ1, . . . , θn−1 1

2k n

S1a m, x−a^2kdS1,

2.33

whereΔ 0, π × · · · ×0, π

n−2

×0,2π. We obtain

S^∗

S1a m, x−a^2kdS1

2k nS

2k n Γk 1/2

Γn/2 k 1π^n−1/2 m, m^k 2Γk 1/2

Γn/2 kπ^n−1/2 m, m^k.

2.34

Using the same procedure, Dirichlet’s integral2.28becomes

D

B1ax−a^2μdx

₁

0

Δ m, ρh^2μρⁿ⁻¹dρ dS1

θ1, . . . , θn−1 1

2k n

Δh^2μdS1

θ1, . . . , θn−1 1

2k n

S1ax−a^2μdS1.

2.35

Hence

D^∗

S1ax−a^2μdS1

2k nD 2k nΓ

μ1 1/2

· · ·Γ

μn 1/2 Γk n/2 1 2Γ

μ1 1/2

· · ·Γ

μn 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 valuesM^s_rfandM_r^bf.

Letf∈C^pΩwithB1a⊂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/∂x²₁ · · · ∂²/∂x²_nthe Laplace operator inRⁿ;

3

|m|^2k m, m^k−→ ∇,∇^kfa Δ^kfa, 2.39

where Δ^k Δ ·Δ· · ·Δ

k

∂²/∂x²₁ · · · ∂²/∂x_n^2k 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−a^2k−→ ∇, x−a^2kfa d²ⁿfa, 2.41

where d^2k ∂/∂x1x1−a1 · · · ∂/∂xnxn−an^2krepresents the differential operator of order 2k.

Using these correspondences and taking into account2.23,2.27,2.29, and2.30, we obtain

B1ad^2kfadxπ^n−1/2 Γk 1/2

Γn/2 k 1Δ^kfa, 2.42

S1ad^2kfadS12π^n−1/2Γk 1/2

Γn/2 kΔ^kfa. 2.43 We note that the expressions2.42and2.43represent the correspondings for2.27 and2.30. We will prove the availability of these relations.

Proposition 2.5. Forf∈C^pΩwithB1a⊂Ωand 2k≤pthe relations2.42and2.43held.

Proof. We have

B1ad^2kfadx

B1a

∂

∂x1

x1−a1

· · · ∂

∂xn

xn−an

^2k

fadx1· · ·dxn

2μ1 ··· 2μn2k

2k!

2μ1

!· · · 2μn

! ∂²

∂x²_{1 μ1}

· · · ∂²

∂x²n

_μn fa·

B1ax−a^2μdx.

2.44 On the basis of2k!2^kk!2k−1!!2^2kk!Γk 1/2/√π, we obtain

2k!

2μ1

!· · · 2μn

! k!

μ1!· · ·μn!

π^n−1/2Γk 1/2 Γ

μ1 1/2

· · ·Γ

μn 1/2. 2.45

Taking into account2.45and Dirichlet Formulas2.43, the expression2.44becomes

B1ad^2kfadxπ^n−1/2 Γk 1/2 Γn/2 k 1

μ1 μ2 ··· μnk

k!

μ1!· · ·μn! ∂²

∂x²₁ μ1

· · ·

∂²

∂x²n

μn

fa, 2.46

so that

B1ad^2kfadxπ^n−1/2 Γk 1/2 Γn/2 k 1

∂²

∂x²₁ · · · ∂²

∂xn²

k

fa π^n−1/2 Γk 1/2

Γn/2 k 1Δ^kfa.

2.47

Using the same method, we obtain

S1ad^2kfadx

S1a

∂

∂x1

x1−a1

· · · ∂

∂xn

xn−an

^2k

fadS1

2μ1 ··· 2μn2k

2k!

2μ1

!· · · 2μn

! ∂²

∂x²_{1 μ1}

· · · ∂²

∂x²n

_μn fa·

S1ax−a^2μdS1.

2.48

On the basis of2.45and2.31, we have

S1ad^2kfadS1

2π^n−1/2Γk 1/2 Γn/2 k

μ1 ··· μnk

k!

μ1!· · ·μn! ∂²

∂x²_{1 μ1}

· · · ∂²

∂x²n

_μn fa 2π^n−1/2Γk 1/2

Γn/2 k ∂²

∂x²₁ · · · ∂²

∂xn²

_k fa 2π^n−1/2Γk 1/2

Γn/2 kΔ^kfa

2.49

and so the proposition is proved.

For the generating functionsφandψwe can state the following.

Proposition 2.6. LetΩ ∈Rⁿbe a bounded set andf : Ω→ R,f ∈C^{2m 1}Ω. Then the functions φ, ψ:−R, R→R defined by2.1and2.2, whereRmax|x−a|such thatBra⊂Ω, have the following properties:

1φ,ψare even functions andφ, ψ∈C^{2m 1}−R, R, 2φ^k0 ψ^k0 0 forkodd,k≤2m 1, 3ψ^2k0

S1ad^2kadS12π^n−1/2Γk 1/2/Γn/2 kΔ^kfa, k≤m, 4φ^2k0 1/n 2kψ^2k0 π^n−1/2Γk 1/2/Γn/2 k 1Δ^kfa, k≤m, 5φ^pr ₁

0u^{n p−1}ψ^prudu, p≤2m 1, 6ψ^pr

S1ad^pfa rx−adS1, p≤2m 1, 7ψr 2π^n/2_m

k01/k!Γn/2 kr/2^2kΔ^kfa R2m 1r, where

R2m 1r ψ^{2m 1}θr

2m 1! r^{2m 1} r^{2m 1} 2m 1!

S1ad^{2m 1}f

a θrx−a

dS1, 0< θ <1, 2.50

8φr π^n/2_m

k01/k!Γn/2 k 1r/2^2kΔ^kfa R^∗_{2m 1}r, where

R^∗_{2m 1}r φ^{2m 1}θr

2m 1!r^{2m 1} r^{2m 1} 2m 1!

₁

0

S1au^{n 2m}d^{2m 1}f

a θrux−a

du 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

S1af

a rx−a dS1

ΔJ^∗

θ1, . . . , 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 ∈ C^{2m 1}Ω, using the differentiation rule for integrals depending on a parameter, it results inψ∈C^{2m 1}−R, R.

In order to prove thatψis 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 in the expression2.52.

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^∗

u1, . . . , un−2^π

−πf^∗

−r, u1, . . . , 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^∗

u1, . . . , 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−rsoψis an even function.

Consequently,ψ^2k is an even function andψ^{2k 1} is odd, it means thatψ^{2k 1}0 0.

From 2.12 we have φ is even, φ ∈ C^{2m 1}−R, R, φ^2k is even and φ^{2k 1} is odd, so φ^{2k 1}0 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

S1af

a rx−a

dS1, 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 Tpⁿ named polare transform. Thus we writeψT_pⁿf. From2.2applying the differentiation rule for composite functions and denotingyarx−a, we obtain

ψ^pr

S1a

∂p

∂r^pf

a rx−a dS1

S1a

∂

∂yi

∂yi

∂r p

fydS1

S1a

∂

∂y1

x1−a1

· · · ∂

∂yn

xn−an

^p

fydS1

S1ad^pf

a rx−a dS1.

2.58

Forp2k≤2mandr0 we have ψ^2k0

S1a

∂

∂x1

x1−a1

· · · ∂

∂xn

xn−an

^2k

fadS1

S1ad^2kfadS1. 2.59 Taking into account2.43, we have

ψ^2k0

S1ad^2kfadS12π^n−1/2Γk 1/2

Γn/2 kΔ^kfa. 2.60 On the other hand, from2.12and2.60we obtain

φ^pr ₁

0

u^{n p−1}ψ^prudu, |r| ≤R, p≤2m 1,

φ^2k0 1

n 2kψ^2k0 2π^n−1/2

n 2k

Γk 1/2 Γn/2 kΔ^kfa π^n−1/2 Γk 1/2

Γn/2 k 1Δ^kfa.

2.61

The formulas2.60and2.61justify the properties3,4,5,6from the proposition.

In order to obtain the formula7we will apply Mac-Laurin formulas. Thus ψ being even andψ∈C^{2m 1}−R, R, on the basis of2.58, we can write

ψr ^m

0

ψ^2k0

2k! r^2k R2m 1r, 2.62 where

R2m 1r ψ^{2m 1}θr

2m 1! r^{2m 1} r^{2m 1} 2m 1!

S1ad^{2m 1}f

a θrx−a

dS1, 0< θ <1. 2.63

Since2k!2^2kk!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 andf is analytic, then the remainders R2m 1t → 0, R^∗_{2m 1}r→0 form→ ∞.

We obtain the following Mac-Laurin Series forψ, φ:−R, R→R:

ψr 2π^n/2

k0

1 k!Γn/2 k

r 2

_2k

Δ^kfa, |r| ≤R, 2.64

φr π^n/2

k0

1 k!Γn/2 k 1

r 2

2k

Δ^kfa, |r| ≤R, 2.65

The results established inProposition 2.6permit to give the corresponding representa- tions for the mean valuesM_r^sfandM^b_rfdefined in2.3and2.4, by using2.16and 2.17.

Thus, for 0≤r≤R, taking into account8,2.16, and2.64we can write M^b_rf 1

Bra

Brafxdx n

2Γ n

2 _m

k0

1 k!Γn/2 k 1

r 2

_2k

Δ^kfa

n 2π^n/2Γ

n 2

r^{2m 1} 2m 1!

₁

0

S1au^{n 2m}d^{2m 1}f

a θrux−a

du dS1, 0< θ <1,

M^brf n 2Γ

n 2

k0

1 k!Γn/2 k 1

r 2

2k

Δ^kfa.

2.66 Similarly, from7,2.62, and2.17we have

M^s_rf 1 Sra

SrafxdS Γ

n 2

m k0

1 k!Γn/2 k

r 2

2k

Δ^kfa Γn/2

2π^n/2 · r^{2m 1} 2m 1!

S1ad^{2m 1}f

a θrx−a

dS1, 0< θ <1,

2.67

M^s_rf Γ n 2

k0

1 k!Γn/2 k

r 2

2k

Δ^kfa. 2.68 We remark that the expression of the remainder forM^s_rfgiven in2.67

R^∗∗_{2m 1}r Γn/2 2π^n/2

r^{2m 1} 2m 1!

S1ad^{2m 1}f

a θrx−a

dS1, 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!/2^{2k 1}k!√

π, from 2.68, we obtain Pizzetti’s Formula1:

M^s_rf

k0

r^2k

2k 1!Δ^kfa. 2.70 Similarly, since

Γ3

2 k 1

3

2 k

Γ3 2 k

2k 32k 1!

2^{2k 2}k!

√π, 2.71

we have

M_r^bf 3

k0

r^2k

2k 1!2k 3Δ^kfa. 2.72 This last formula constitutes the similarity of Pizzetti’s formula for the mean value M_r^bfover the ballBra⊂R³.

References

1P. 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.

2R. Courant and D. Hilbert, Methoden der Mathematischen Physik, Springer, Berlin, Germany, 1937.

3G. M. Fichtenholz, Differential-und Integralrechnung. III, vol. 63 of Hochschulb ¨ucher f ¨ur Mathematik, VEB Deutscher Verlag der Wissenschaften, Berlin, Germany, 1977.

4I. S. Gradstein and I. M. Ryshik, Tables of Series, Products and Integrals. Vol. 1, 2, Harri Deutsch, Thun, Switzerland, 1981.

5G. E. Shilov, Generalized Functions and Partial Differential Equations, vol. 7 of Mathematics and Its Applications, Gordon and Breach, New York, NY, USA, 1968.

6S. G. Mihlin, Ecuat¸ii liniare cu derivate part¸iale, Editura S¸tiint¸ific˘a s¸i Enciclopedic˘a, Bucures¸ti, Romania, 1983.

7W. W. Kecs, Teoria distribit¸iilor s¸i aplicat¸ii, Editura Academiei Romˆane, Bucures¸ti, Romania, 2003.

8M. N. Olevski˘ı, “An explicit expression for the remainder in the Pizetti formula,” Functional Analysis and Its Applications, vol. 23, no. 4, pp. 331–333, 1989.