On Parallel Hexagon's Summation Methods Of Double Fourier Series And Linear Summation Methods Of Neumann-Bessel Series

Linear summation methods are very old important methods to improve the convergent properties of series, which have been used widely in the fields of engineeringand technology. In the past years, researchers paid more and more attentionto the Linear summation methods of the double Fourier series. But, up to now, only two methods have been presented for the research of linear summation problem of the partial sums of double Fourier series, namely, spheral summation methods and rectangular summation methods. The two methods of summation have been discussed detailedly, on which some results have been obtained.In the paper, we present two kinds of methods of the parallel hexagon's summation.Our research work deals mainly with the parallel hexagon's summation methods of double Fourier series associating with three-directional coordinates. Until now, no treatise on the parallel hexagon's summation methods of double Fourier series associating with three-directional coordinates has been found.In R², we take and let e₃ = -e₁- e₂. ThenDefine the normal vectors of e₁, e₂, and e₃ as n₁, n₂, and n₃, respectively:And then, a new system of three-direction coordinates R² = O(n₁,n₂,n₃) is set up as follow: A rule of one-one correspondence is established between a point x = (x₁,x₂) in R² and the point t = (t₁,t₂, t₃) in the new three-direction coordinates (see Fig.1), whereNamely,Obviously, the three-direction coordinates (t₁, t₂, t₃)∈R² of for any point P on the total plane must satisfy the following identity Using the relation of three-direction coordinates and Descartes coordinates of any point on the plane, we obtainWe take the following parallel hexagonΩ, drown in Fig.1, as our basic domainThe domain can be denoted byunder Descartes coordinates.For a functionf(t)∈e L(Ω), the related double Fourier series are defined as[1]where j = (j₁,j₂,j₃), t = (t₁,t₂,t₃), g_j(t) = e^{2Ï€i/3 j·t}, and c_Ω- area of the basic hexagonΩ.A partial sum of double Fourier series(1) of f(t) is defined as respectivelyS_n(f; t) is called truncations on parallel hexagon.Writing P = (t₁,t₂,t₃) and Q = (s₁,s₂,s₃). S_n(f;t) can be denoted by the following integral form where G_n(P) = sum from |j|=0 to 2n gj(t).Given a factor matrixwhere j = (j₁,j₂,j₃).The linear mean H_n(f;t) of truncations on parallel hexagon is called a parallelhexagon's summation methods, whereWritingω=e^{2Ï€/3 i}, we get g_j(t) =ω^j·tt.It may be found from reference [1] that S_n(f;t) can't converge uniformly to any continuous function f(t)∈C_*(Ω), where C_*(Ω) denotes the space of all continuous function f(t) on with periodΩ. To improve convergence property of S_n(f; t), the linear summation methods are applied to the truncations on parallel hexagon in the first part of this dissertation. We present the first kind of methods of parallel hexagon's summation. Namely, a convergence factor is established as:where j = (j₁,j₂,j₃);j₁,j₂,j₃= -n, -(n - 1),…, -1,0,1,…,n; n = 1,2,…;r₁, r₂ are arbitrary odd natural numbers, and h = 3/2n.If g_j(t) =ω^{j₂(t₂-t₃)+j₁(t₁-t₃)},then{j₁,J₃}=j₁,{j₂,j₃}=-j₂; If g_j(t) =ω^{j₃(t₃-t₁)+j₂(t₂-t₁)},then{j₁,J₃}=-j₃,{j₂,j₃}=0;If g_j(t) =ω^{j₁(t₁-t₂)+j₃(t₃-t₂)},then{j₁,J₃}=0,{j₂,j₃}=j₃.The linear integral operator with the convergence factorρ_jⁿ is defined asOn the operator W_n,r1,r2(f; t), we obtain the convergent theorem as follow :Theorem A If f(t)∈C_*(Ω), thenis valid uniformly onΩ.In the second part of this dissertation, the partial sums of double Fourier series(1)under system of three-directional coordinates is redefined asObviously, S_n(f;t) is also a kind of truncation on parallel hexagon.The partial sums can be rewritten in terms of integration formwhereIt may be found from reference [1] that S_n(f; t) can't still converge uniformly to any continuous function f(t)∈C_*(Ω). Therefore, the linear summation methodsare applied to the truncations on parallel hexagon S_n(f; t). We present the second kind of method of parallel hexagon's summation. Namely, we find a convergencefactorj_e,j_d = -n, -(n-1),…, -1, 0,1,…, n; n = 1, 2,…for[e, d] = [1, 2], [2, 3], [3, 1].where r₁, r₂ and r₃ are all arbitrary odd natural numbers, and h = 3/2(2n + 1). Thus, the linear integral operator with the convergence factorρ_je,jdⁿ, [e,d] = [1, 2], [2, 3] and [3,1] is defined asOn the operator W_n,r1,r2,r3(f; t), the main theorems are as follows :Theorem B If f(t)∈C_*(Ω), thenis valid uniformly onΩ.Theorem C Let f(t) = g(x₁,x₂), and f(t)∈C_*(Ω), thenwhere " O " is independent of n, f,ωⁿ(g,·) is r-order spheral smooth modulus of g, and h = 3/2(2n + l). Theorem D Letf(t) = g(x₁,x₂), and f(t)∈C_*(Ω). Further, let g(x₁,x₂)∈C(Ω) exists r-order(0≤r≤min(r₁,r₂,r₃}) continuous partial derivative, thenwhere " O " is independent of n and f, h = 3/2(2n + 1), andω(g,·) is the spheral modulus of continuity of g.In the third part of this dissertation, linear summation problems of Neumann-Bessel series on the unit circle are discussed mainly. Neumann-Bessel series have been applied widely in the field of engineering and technology, such as physics, mechanics, and wireless and so on. A lot of foreign scholars and Chinese scholars study Neumann-Bessel series. In order to meet the need of this dissertation, we state only two results.Let J_n(z) be Bessel functions and let Q_n(z) be Neumann polynomials:where[a]is the maximal integer part no more than number a. Q_n(z)is a n+1-degree polynomials about 1/z.Let the curveΓbe the unit circle |z| = 1. It is well known that both the sequence of functions {J_n^z} and the sequence of functions {Q_n(z)} are orthogonal on the unit circleΓ.Definition A Suppose that f(z) is Lebesgue-integrabel onΓ. Writing The seriesare said to be Neumann-Bessel series on the unit circleΓ.Let S_n^N,B (f; z) denote the n-th partial sums of Neumann-Bessel series, i.e.The partial sum can be rewritten in terms of integrality form.whereare called the kernel functions.On the kernel functions K_n^N,B(z,ζ), Mu Lehua [2] gave the following estimatein 1996:For z = e^iθ,ζ= e^is, he gotwhere the bound of " O " is an absolute constant, and he gave the convergent theorem on S_n^N,B (f;z) as follows:Theorem E Suppose that f(z) is Lebesgue-integrabel onΓ, and (?)(θ)= f(e^iθ). Let s_n^â–³((?);θ) be the partial sums of Fourier series of (?)>(θ), thenis valid uniformly onΓ. It may be found from Theorem E that S_n^N,B(f;z) can't converge uniformlyto each continuous function f(z) on the unit circleΓ. In 1999, Mu Lehua[3]eonsidered Feje'r sums of f(z):On Fejer, he obtained an asymptotic formula as below.Theorem F Let f(z) be a function of bounded variation onΓand let z₀ be a point inΓ. If the two one-side derivatives f₊^'(z₀) and f_<sup>'(z₀) of f(z) at z₀ exist, then we havewhereTheorem E and F show that the partial sums S_n^N,B (f; z) of Neumann-Bessel series can't converge uniformly to each continuous function f(z) on the unit circleΓ, while the Feje'r sums can converge uniformly to each continuous function f(z) on the unit circleΓ. But, the convergence order ofσ_n^N,B (f; z) does not reach the order of the best approximation for any fixed continuous function f(z)onΓ. As we know, linear summation methods are important means to improve the convergent properties of series, and have found applications in various series besides Fourier series in order to improve the convergent properties of the partial sums of series. Inl996, Tang Ping investigated the linear summation problem of Fourier-Laplace. In the dissertation, linear summation methods are applied to the partial sums of Neumann-Bessel series on the unit circle. Using the partial sums of Neumann-Bessel series, we construct a new integral operator, which converges to any fixed continuous function f(z) onΓand has the host approximation order for f(z) onΓ.The operator H_n,r^N,B(f; z) is determined as follows:Let r be an arbitrary odd natural number and let h =Ï€/n + 1, thenDefinition B Let f(z) be a continuous function on the unite circle |z|=1, then the modulus of continuity of f(z)is given byWe have the following results concerning H_n,r^N,B (f; z).Theorem G Let f(z) be a continuous function onΓ, thenwhere " O " is independent of n, andω(f,δ) is the modulus of continuity of f(z) onΓ.Theorem H Let f(z) be a continuous function onΓ, thenis valid uniformly onΓ.