Fourier-jacobi Transform The Poisson Integral And Conjugate Poisson Integrals

In this paper, we mainly discuss Poisson and conjugate Poisson integral of Fourier-Jacobi transform, the corresponding conjugate function and the generalized Hubert transform. The paper consists of four chapters.In the first chapter, we give a brief introduction of Jacobi function, Jacobi transform and present the latest developments about Jacobi series.In chaper II, we associate Jacobi transform to the 揾armonic?function (Poisson integral) a2u 82u [(2a?)cothy+U(a^, y), which satisfies the generalized Laplace equations = a2(2³ + 1) tanhy] = 0. The evaluation of the Poisson kernel is given and for half integers more precise asymptotic estimate is obtained. We also prove that the Poisson maximal function is dominated by the maximal function defined through Jacobi translation.In chapter III, we introduce conjugate Poisson integral VQr, y), which is essential to define the conjugate function f related to f. V(z, y) and UQr, y) satisfy the generalized CauchyRiemann equationsUy-v1⁼0,I U + l4 + [(2a+ 1)cothy+ (2³+ 1)tanhy]V = 0.Moreover, we give a integral representation of the conjugate Poisson kernel QQr, y, t).It is well known that for an even function f defined in R, its Hilbert transform can be??-iexpressed as Hf(y) = lim I /_<sub><sub><sub><sub>JC^<sup> ^慸t. We generalize this to Jacobi transformtin chapter IV. More precisely, H^f(y) = lirn f Ttf(y)G(t)dt, where T and G(t) are the generalized difference operator and singular integral kernel respectively. When a, /3 楓 ?1/2, Ttf(y) 梸 ^{f(y ?t) ?f(y + t)}, and G(t) ? The integral expression of Tt is obtained. Forhalf integers a, /3 , using the asymptotic estimate obtained in chapter II, we get the asymptotic expression of G(t). To reach this goal, we introduce the generalization of Fourier-sine transform, the conjugate Jacobi transform and the inverse transform. At last, we point out that for f € C^, H^f = f.