Asymptotic formula for the Riesz means of the spectralfunctions of Laplace-Beltrami operator on unit sphere

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Asymptotic formula for the Riesz means of the spectral functions of Laplace-Beltrami operator on unit sphere

To cite this article: Ahmad Fadly Nurullah Rasedee et al 2017 J. Phys.: Conf. Ser. 890 012117

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Asymptotic f ormula f or t he Riesz m eans o f t he s pectral f unctions o f Laplace-Beltrami o perator o n u nit s phere

Ahmad Fadly Nurullah Rasedee¹, Anvarjon Ahmedov² and Mohammad Hasan Abdul Sathar³

1Faculty of Economics and Muamalat, Islamic Science University of Malaysia, 78100 Nilai, Negeri Sembilan, Malaysia

2Faculty of Industrial Sciences and Technology, Pahang University of Malaysia, 26300 Gambang, Kuantan, Pahang, Malaysia

3Centre of Foundation Studies for Agricultural Science, Putra University of Malaysia, 43400 Serdang, Selangor, Malaysia

E-mail:[email protected]

Abstract. The mathematical models of the heat and mass transfer processes on the ball type solids can be solved using the theory of convergence of Fourier-Laplace series on unit sphere. Many interesting models have divergent Fourier-Laplace series, which can be made convergent by introducing Riesz and Cesaro means of the series. Partial sums of the Fourier-Laplace series summed by Riesz method are integral operators with the kernel known as Riesz means of the spectral function. In order to obtain the convergence results for the partial sums by Riesz means we need to know an asymptotic behavior of the latter kernel. In this work the estimations for Riesz means of spectral function of Laplace-Beltrami operator which guarantees the convergence of the Fourier-Laplace series by Riesz method are obtained.

1. Introduction

Kogbetliantz obtained an asymptotic formula for a multi dimensional sphere by Cesaro means. Pulatov in [1] then obtained an asymptotic formula for spectral function of the spectral expansions by Riesz meansα ∈ Z+. Finally, Ahmedov [2] extended the estimation for Riesz means for non integer values of the order using a new approach for summability of Laplace series, which is based on the spectral expansion properties of a self-adjoint Laplace-Beltrami operator.

For the properties on the equivalence of Riesz and Cesaro means, we refer to articles [3–8]. Hobson ( [5], pages 90-98) reproduced the proof for equivalence between Cesaro and Riesz means for any α ≥ 0. Since then, more fundamental proofs were obtained provided in Chandrasekharan and Minakshisundarams’Typical means[3].

Studies conducted in [7] established an equivalence between Cesaro and Riesz means by employing properties on the absolute summability of a series by Riesz means as provided in [6]. Ingham’s estimation in [7] was obtained by expressing the fundamental functions ofA^k_nandω^kin linear terms of each other.

The aim of this research is to establish the asymptotic behavior for Riesz means of the spectral function of the Laplace operator on unit sphere. We will show that the asymptotic behavior for ultra- spherical spectral expansion of Riesz means is similar to the Cesaro means obtained in [9] with the use of equivalent properties of the two means. With properties found in [4], [5] and [7], we are able to overcome previous restrictions and obtain the asymptotic formula for the Riesz means kernel of the

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2

spectral expansions of orderα >0, α∈ <. Some recent works on the Riesz means of spectral functions can be found in the following articles [10–19] and more.

2. CesaroandRiesz meansofthe spectral function LetS^N be unit sphere inR^N+1 :

S^N ={x= (x1, x2, ..., xN+1)∈R^N+1 :x²₁+x²₂+...+x²_N+1 = 1}.

The sphere S^N is naturally equipped with a positive measuredσ(x) and with an elliptic second order differential operator∆_s, which named the Laplace-Beltrami operator on the sphere. The contraction of a homogeneous polynomial of degreek= 0,1,2, ...to the unit sphereS^N is called a spherical harmonic of degreek.The space of spherical harmonics of degreekwe denote byH_k. The spherical harmonics Y(x) ∈ Hk are eigenfunctions of the Laplace-Beltrami operator ∆s corresponding to the eigenvalue k(k+N−1):

∆_sY(x) +k(k+N −1)Y(x) = 0, x∈S^N. Among the spherical harmonics of orderkit can be chosen a orthonormal basis:

{Y₁^(k)(x), Y₂^(k)(x), ..., Y_a^(k)_k (x)}, k= 0,1,2, ..., wherea_k=dim(H_k).

One of the main problems of harmonic analysis is the reconstruction of functions from their expansion:

f(x)∼

∞

X

k=0 ak

X

j=1

fjkY_j^(k)(x), (1)

wheref_jk =R

S^Nf(y)Y_j^(k)(y)dσ(y), j= 1,2, ..., a_k;k= 0,1,2, ....

We study the summability of the (1) by Riesz means E_n^αf(x) =

n

X

k=0 a_k

X

j=1

1−λk

λn

^α

fjkY_j^(k)(x), and Cesaro means

S_n^αf(x) =

n

X

k=0

A^α_n−k A^α_n

ak

X

j=1

f_jkY_j^(k)(x).

The Riesz means and Cesaro means are integral operators with the kernels Θ^α(x, y, n) and Φ^(α)(x, y, n)respectively:

Θ^α(x, y, n) =

n−1

X

k=0

1− λ_k

λn

α ak

X

j=1

Y_j^(k)(x)Y_j^(k)(y),

Φ^(α)(x, y, n) =

n

X

k=0

Γ(n+ 1)Γ(n−k+ 1 +α) Γ(n+α+ 1)Γ(n−k+ 1)

ak

X

j=1

Y_j^(k)(x)Y_j^(k)(y).

It was previously established that when α is a positive integer, the kernels Φ^(α)(x, y, n) and Θ^α(x, y, n)share common asymptotic behaviors. In this work, we will extend this result for any real α >0. To do this we need to establish some facts about the means of the series.

With the help of Gegenbauer polynomialsP_k^ν(t),which can be defined by P_k^ν(t) = (−2)^kΓ(k+ν)Γ(k+ 2ν)

Γ(ν)Γ(2(k+ν)) (1−t²)^−(ν−¹²⁾d^k dt^k

h

(1−t²)^k+ν−¹²i

, ν = N−1

2 , for the kernels of the Riesz means and Cesaro means we obtain:

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Lemma 2.1. For anyx, y∈S^N one has Θ^α(x, y, n) =

n−1

X

k=0

1− λ_k

λ_n α

(k+ν)^νP_k^(ν)(cosγ) (2) and

Φ^(α)(x, y, n) =

n

X

k=0

Γ(n+ 1)Γ(n−k+ 1 +α)

Γ(n+α+ 1)Γ(n−k+ 1)(k+ν)^νP_k^(ν)(cosγ), (3) whereγ(x, y)denotes a spherical distance betweenx∈S^N andy∈S^N.

3. PropertiesofRiesz means

Consider a functionf with support onR⁺. For a locally bounded variationf andα > −1, the Riesz means is defined as follows

f^α(t) = Z t

0

1−s

t α

df(s) =t^−α Z t

0

(t−s)^αdf(s), t >0

The meanf^α(t)is defined forα≥0except whenf is discontinuous att. Hence, makingf^αdefined for allt >0, by

f^α(t) = Z t−0

0

1−s

t α

df(s) and also denoted as

f^α(t) = α t

Z t

0

1−s

t α−1

f(t)dt, α >0 If we setα >0,

Rα(t), t >0; Rα(t) = 0, t≤0, where with the convolution

t^αf^α(t) =R_α∗f(t), which is more convenient to work with.

Note thatR_αhas a Fourier-Laplace transform analytic in the lower half plane Rb_α(ξ) =

Z ∞

0

αt^α−1e^−itξdt= Γ(α+ 1)(iξ)^−α

where(iξ)^−αis defined as the analytic function in the lower half plane with the argumentiξis taken as zero whenξ is on the negative imaginary axis. ByR⁻¹_α we denote the distribution with support on R+

with Laplace transform equal to (iξ)^α/Γ(α+ 1). The following property of the Riesz means will be applied to extend the estimation of the spectral function for non-integer values.

Theorem 3.1. Let κ be a complex number with Re(κ) > 0, and M0(t) and M1(t) are positive inreasing functions and in(0,+∞). Iff(t) = 0, t <0and for allt >0satisfied inequalities:

|f(t)| ≤M0(t) and |t^κf^κ(t)| ≤M1(t) (4) then for0< Re(α)< Re(κ)we have

|t^αf^α(t)| ≤C(1 +|α|)^Re(κ)+2 |α|

Re(α) + |κ−α|

Re(κ−α

M

Re(κ−α) Re(κ)

0 (t)M

Re(α) Re(κ)

1 (t) (5)

whereCis dependent only onκ.

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4

For details and proof of this theorem we refer the readers to the paper [20].

Let

Mα(t) = t^(N−1)/2

(sinγ)^(N−1)/2(sin^γ₂)^1+α + t^(N−3)/2

(sinγ)^(N+1)/2(sin^γ₂)^1+α + t⁻¹ (sin^γ₂)^1+N Ifκis an integer, then for the kernelΘ^κ(x, y, n)we have

|Θ⁰(x, y, n)| ≤M₀(t) and t^κ|Θ^κ(x, y, n)| ≤M_κ(t).

From theorem,

|t^αΘ^α(t)| ≤C(1 +|α|)^Reκ+2 |α|

Reα+ |κ−α|

Re(κ−α)

M

Re(κ−α) Reκ

0 (t)M

Reα Reκ

1 . (6)

This gives

t^αΘ^α(x, y, n)≤ t^(N−1)/2

(sinγ)^(N−1)/2(sin^γ₂)^1+α + t^(N−3)/2

(sinγ)^(N+1)/2(sin^γ₂)^1+α + t⁻¹ (sin^γ₂)^1+N

4. Main result

We can represent the Riesz means of the spectral function by Θ^α(x, y, n) =

n

X

k=0

1−λ_k

λn

αΓ k+^N₂⁻¹

Γ(k+N−1) π^N+1²

P

N−1 2

k (cosγ), withP_k^ν(t)denoting the Gegenbauer polynomials

P_k^ν(t) =(−2)^kΓ(k+ν)Γ(k+ 2ν)

Γ(ν)Γ(2(k+ν)) (1−t²)^−(ν−¹²⁾d^k dt^k

h

(1−t²)^k+ν−¹²i .

By this representation it becomes apparent thatΘ^α(x, y, n)depends only on the spherical distance be- tweenxandythus, can be denoted as :Θ^α(x, y, n) = Θ^α(cosγ).

For the kernelΘ^α(x, y, n) = Θ^α_n(cosγ), we prove the following theorem

Theorem 4.1. LetΘ^α(x, y, n)be the kernel of Riesz means of the spectral expansions. If

1)

^π₂ −γ(x, y)

< _2(n+1)^πn , n→ ∞,then

Θ^α(x, y, n) =n^N−1² ^−α(N−1) sin h

(n+^N₂ +^α₂)γ−π^(N^−1+2α)₄ i (2 sinγ)

N−1

2 2 sin^γ₂1+α + n^N−3² ^−αηn(γ)

(sinγ)^N+1² sin^γ₂1+α+(n+ 1)⁻¹εn(γ) sin^γ₂1+N

where|ηn(γ)|< C, |εn(γ)|< C; 2) 0< γ0≤γ≤π

|Θ^α(x, y, n)| ≤C4n^N^−1−α 3) 0≤γ≤π

|Θ^α(x, y, n)| ≤C5n^N.

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4.1.Proofof main result

For0<σ≤1asn→∞,wecandenoteRieszkernelofthespectralexpansionsas Θ^α(x, y, n) =

n

X

k=0

1− k

n+σ ^{α a}k

X

j=1

Y_j^(k)(x)Y_j^(k)(y)

rearranging the equations gives

Θ^α(x, y, n) = 1

n+σ α n

X

k=0

(n+σ−k)^αZk(x, y).

Letm=n−k, hence

Θ^α(x, y, n) = (n+σ)^−α

n

X

m=0

(m+σ)^αZn−m(x, y),

which can generally be written as,

Θ^α(x, y, n) = (n+σ)^−α

n

X

k=0

(k+σ)^αZ_n−k(x, y).

The function(n+σ)^αcan be expanded as (n+σ)^α=

h

X

r=0

α r

σ^rn^α−r+O(n^α−h−1),

and by implementing the combination of Taylor’s theorem for h+ 1terms and Lagrange remainder.

Let α be a real number with α 6= −1,−2,−3, . . ., and an integer h ≥ 0: This allows us to formulate a sequence of polynomials pr(σ) = pr(σ;α), r = 0,1,2, . . . and a set of polynomials Pr(σ) =Pr(σ;α, h), r= 0,1,2, . . .such that

(n+σ)^α=

h

X

r=0

p_r(σ)A^α−r_n +δ_n(σ) =

h

X

r=0

P_r(σ)A^α_n−r+δ_n(σ), (7)

where

δ_n(σ) =δ_n(σ;α, h) =O(n^α−h−1), (8) uniformly for0< σ≤1asn→ ∞and whenhis fixed andh > α.

For Cesaro means of orderαwe have the following property (see [5]), A^α_n=A^α+1_n −A^α+1_n−1,

and through mathematical induction, the Cesaro means of orderα−rcan be denoted as follows A^α−r_n =

r

X

k=0

(−1)^k r

k

A^α_n−k.

By substituting the equality above into Ph

r=0

p_r(σ)A^α−r_n yields

h

X

r=0

pr(σ)A^α−r_n =

h

X

r=0

"

pr(σ)

r

X

k=0

(−1)^k r

k

A^α_n−k

#

=

h

X

k=0

(−1)^kA^α_n−k

h

X

r=k

r k

pr(σ) =

h

X

r=0

Pr(σ)A^α_n−r,

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6

where

Pr(σ) = (−1)^r

h

X

j=r

j r

pj(σ). The polynomialsP_r(σ)are then substituted into the Riesz kernel giving

Θ^α(x, y, n) =(n+σ)^−α

n

X

k=0 h

X

r=0

Pr(σ)A^α_k−rZ_n−k(x, y) + (n+σ)^−α

n

X

k=0

δk(σ)Z_n−k(x, y)

=(n+σ)^−α

h

X

r=0

Pr(σ)

n

X

k=0

A^α_k−rZ_n−k(x, y) + (n+σ)^−α

n

X

k=0

δk(σ)Z_n−k(x, y)

=I_n^α+J_n^α.

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We know thatδ_n(σ) =O(n^α−h−1)where as the first part of (9) can be estimated by the following, I_n^α= (n+σ)^−α

h

X

r=0

Pr(σ)

n

X

k=0

A^α_k−r

a_k

X

j=1

Y_j^(n−k)(x)Y_j^(n−k)(y)

= (n+σ)^−α

h

X

r=0

P_r(σ)

n

X

k=0

A^α_k−r

ak

X

j=1

Y_j^(n−k)(x)Y(n−r−(k−r))

j (y).

Letν =k−r, then

I_n^α= (n+σ)^−α

h

X

r=0

Pr(σ)

n−r

X

ν=0

A^α_ν

ak

X

j=1

Y_j^(n−r−ν)(x)Y_j^(n−r−ν)(y)

= (n+σ)^−α

h

X

r=0

P_r(σ)A^α_n−r A^α_n−r

n−r

X

ν=0

A^α_ν

a_k

X

j=1

Y_j^(n−r−ν)(x)Y_j^(n−r−ν)(y)

= (n+σ)^−α

h

X

r=0

Pr(σ)A^α_n−rΦ^α_n−r(x, y).

By expanding the series,I_n^αcan be estimated by I_n^α=n^−α

P0A^α_nΦn(x, y) +P1A^α_n−1Φn−1(x, y) +. . .+PhA^α_n−hΦn−h(x, y)

≈n^−α

P0

n^α Γ(α+ 1)

1 +O

1 n

Φn+P1

(n−1)^α Γ(α+ 1)

1 +O

1 n

Φn−1+. . . +P_h(n−h)^α

Γ(α+ 1)

1 +O 1

n

Φ_n−h(x, y)

≈

Φ_n+O

1 n^N−1² ^−α+1

+ P₁Φ_n−1 Γ(α+ 1) +O

1 n^N−1² ^−α+1

+· · ·+ P_hΦ_n−h Γ(α+ 1)+O

1 n^N−1² ^−α+1

≈Φn+O

1 n^N−1² ^−α+1

.

Subsequently, this provides the following estimation of our kernel for Riesz means similiar to Kogbeliantz’s Cesaro means

Θ^α_n(cosγ) = (N−1)A

N−1

n2

A^α_n

sin

n+^N₂ +^α₂

γ−^π₂ ^N₂⁻¹+α

(2 sinγ)^N−1² 2 sin^γ₂α+1 + ξ_n^α(γ)n^N−3² ^−α

(sinγ)^N+1² sin^γ₂α+1 +(n+ 1)⁻¹η_n^α(γ) sin^γ₂^N+1 . For estimation purpose, we can approximateA^α_nandA

N−1

n2 (page 168, [8]) which gives A

N−1

n2

A^α_n ≈n^N−1² ^−α,

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hence, estimating the Riesz kernel in the form of Θ^α_n(cosγ) = (N−1)n^N−1² ^−αsin

n+^N₂ +^α₂

γ−^π₂ ^N₂⁻¹+α

(2 sinγ)^N−1² 2 sin^γ₂α+1 + ξ_n^α(γ)n^N−3² ^−α

(sinγ)^N+1² sin^γ₂α+1 +(n+ 1)⁻¹η^α_n(γ) sin^γ₂^N+1 . In the case when0< γ₀ ≤γ ≤πand0≤γ ≤π, it is not difficult to obtain the estimation:

|Θ^α(cosγ)| ≤C4n^N−1−α and |Θ^α(cosγ)| ≤C5n^N, respectively. This consequently proves Theorem 4.1.

Acknowledgments

This paper has been supported by University Sains Islam Malaysia (USIM) under its short term grant scheme and the Ministry of Education under its Fundamental Research Grant Scheme (FRGS).

References

[1] Pulatov, A K 1979 Ph.D. thesis Moscow State University Moscow, Russia [2] Ahmedov, A A 2011Applied Mathematics Letter24615–619

[3] Chandrasekharan, K and Minakshisundaram, S 1952Typical Means(Oxford: Oxford University Press.) [4] Hardy, G H 1949Divergent Series(Oxford: Clarendon Press)

[5] Hobson, E W 1958The theory of functions of a real variable and the theory of Fourier’s seriesvol II (New York: Dover Publications Inc)

[6] Hyslop, J M 1936Proceedings of the Edinburgh Mathematical Society(1vol 5) (Edinburgh Mathematical Society) pp 46–54

[7] Ingham, A E 1968Publ. Ramanujan Inst.1107–113

[8] Zygmund, A 1968Trigonometric series2nd ed vol I (London: Cambridge University Press.) [9] Kogbetliantz, E 1924J. Math. Pures Appl.9107–187

[10] Rakhimov A A 2011International training-seminars on mathematics

[11] Ahmedov, A A 2009Journal of Mathematical Analysis and Applications56310–321 [12] Alimov S A 2012Doklady Mathematics86597–599

[13] Ahmedov A A, Rasedee A F N and Rakhimov A 2013Journal of Physics: Conference Seriesvol 435 (IOP Publishing) p 012016

[14] Ahmedov A A, Rasedee A F N and Rakhimov A A 2013Malaysian Journal of Mathematical Sciences7315–326 [15] Ahmedov A A and Rasedee A F N 2015Malaysian Journal of Mathematical Sciences9337–348

[16] Rasedee A F N, Rakhimov A, Ahmedov A A, Ishak N and Hamzah S R 2017AIP Conference Proceedingsvol 1830 (AIP Publishing) p 070034

[17] Rasedee A F N, Rakhimov A and Ahmedov A A 2017AIP Conference Proceedingsvol 1830 (AIP Publishing) p 070006 [18] Rasedee A F N, Rakhimov A A, Ahmedov A A and Hassan T B H 2017Journal of Physics: Conference Seriesvol 819

(IOP Publishing) p 012018

[19] Rakhimov A A, Hassan T B H and Rasedee A F N 2017Journal of Physics: Conference Seriesvol 819 (IOP Publishing) p 012025

[20] Hormander L 1966Recent Advances in the Basic Sciences(Yeshiva University)