On the Lebesgue constants of Fourier-Laplace series by Riesz Means

An asymptotic formula for the Lesbegue constant of the Riesz means of Fourier-Laplace series on the sphere obtained in this paper.


INTRODUCTION
Let us define σ α n f (x) the Cesaro means of order α of the partial sums of Fourier-Laplace series on unite sphere S N as where the kernel Investigations on the behaviour of the Cesaro means σ α n f (x) can be found in works [4] - [5] and [11] - [16]. The different aspects of the convergence and summability can be also found in the book [18]. Since σ α n f (x) is an integral operator the precise estimation of its kernel Θ α (x, y, n) is essential for the study. First estimations of the kernel Θ α (x, y, n) obtained by Gronwall [6] for the case of Legendre polynomials and Kogbetliantz [8] for the Gegenbauer polynomials.
The Lebesgue constant is L 1 norm of the kernel above. Note, that estimations of the Lebesgue constants of the Cesaro means studied by Khocholava [8], Akhobadze [2] and Macharashvili [10]. The Lebesgue constants of multiple Fourier series studied in [1] and [9].
This article focuses on Lebesque constants related to Fourier-Laplace series of the Laplace-Beltrami operator:

MAIN RESULT.
In the present paper we consider the Riesz means instead the Cezaro means of the partial sums of Fourier-Laplace series. The Riesz means of the partial sums will also be an integral operator and its kernel can be represented by where P ν k (t) denote the Gegenbauer polynomials as follows By this representation it is evident that Θ α (x, y, n) depends only on the spherical distance between x and y hence, allows the Riesz means of the spectral function to be written as Θ α (x, y, n) = Θ α n (cos γ). The Riesz means of the kernel is studied in the works [3] and [17].
The main goal of the paper is to obtain the estimation of the Lebesque constant ( L 1 norm of the kernel). Let us use the same notation L α n for the Lebesque constant as in (1.1) . Then following theorem is valid.

PROOF OF MAIN RESULT.
To estimate L α n , we first denote (1.1) as follows, Let us divide the integral on the right hand side of (3.1) into three parts as follows

Estimation from below
The next step is to obtain the lower bound of the Lebesgue constant. We proceed by first estimating I ′ 2 from below: (2 sin γ) is introduced and substituted into the chain of inequalities: we can now estimate I ′ 2 as Applying the change of variable β = t + τ π, we obtain Once more, the following 3 cases are considered: (3.21) Applying the reverse triangle inequality, |a − b| ≥ |a| − |b| with the Riesz mean of the spectral function from (3.5), gives the following (3.22) Given the inequalities (3.21), (3.14) and (3.15) of (3.22), I 2 is bounded as follows and by manner of (3.2), L α n ≥ I 2 . Consequently, gives the following estimates (3.24) Combination of the estimated upper bound of the Lebesgue constant in (3.17) and lower bound in (3.24), provides estimate of the Lebesgue constant, L α n in form of Theorem 2.1 is proved.

ACKNOWLEDGEMENT.
This paper has been supported by the Intenational Islamic University Malaysia under the postdoctoral scheme, Fundamental Research Grant Shceme, Grant Number: FRGS 14-142-0383.