Conjugate Hermite Function Expansion (1) Can And Sexual

There are abundant results for Hermite expansion over the past several decades, such as the convergence, the suminability and Possion integral. In 1989, S. Thangavelu proved that the critical index for (C, a) is ~. B. Muckenhoupt introduced conjugate Hertnite expansions in 1969. Since then, J. Gosselin and K. Stempak did further research about this. We prove that the (C, a) summability of the conjugate Hermite expansions is (C, 1) summable in this paper. The first chapter contains preparatory material about Hermite polynomial and Herniite function expansions . The (C, a) sumnaability, convergence in 12 , Poisson and conjugate Poisson integral of these two expansions are give as well. Chapter II is the main part of the paper and divided into four sections . In section 1, we give a integral representation of (C, a) kernel S~j, z) for conjugate Hermite expansions. In section 2, the estimate of the derivative of (C, a) Kernel is obtained .We evaluate conjugate (C, a) in section 3. Using (C, a) means recurring formual for conjugate Jacobi series derived by Zh.-k. Li in 1997, we express conjugate (C, a) kernel as infinite series. Thus ,together with the result obtained in section 2, we can easily get the estimate of conjugate (C, a) kernel. In section 4, we show that (C, 1) means of conjugate Hermite expansion converges in L~ norm and a]most everwhere.