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51. CJM 2000 (vol 52 pp. 3)

Aizenberg, Lev; Vidras, Alekos
 On Small Complete Sets of Functions Using Local Residues and the Duality Principle a multidimensional variation of the completeness theorems by T.~Carleman and A.~F.~Leontiev is proven for the space of holomorphic functions defined on a suitable open strip $T_{\alpha}\subset {\bf C}^2$. The completeness theorem is a direct consequence of the Cauchy Residue Theorem in a torus. With suitable modifications the same result holds in ${\bf C}^n$. Categories:32A10, 42C30

52. CJM 1998 (vol 50 pp. 1236)

Kalton, N. J.; Tzafriri, L.
 The behaviour of Legendre and ultraspherical polynomials in $L_p$-spaces We consider the analogue of the $\Lambda(p)-$problem for subsets of the Legendre polynomials or more general ultraspherical polynomials. We obtain the best possible'' result that if $2 Categories:42C10, 33C45, 46B07 53. CJM 1998 (vol 50 pp. 1273) Lubinsky, D. S.  Mean convergence of Lagrange interpolation for exponential weights on$[-1,1]$We obtain necessary and sufficient conditions for mean convergence of Lagrange interpolation at zeros of orthogonal polynomials for weights on$[-1,1]$, such as $w(x)=\exp \bigl(-(1-x^{2})^{-\alpha }\bigr),\quad \alpha >0$ or $w(x)=\exp \bigl(-\exp _{k}(1-x^{2})^{-\alpha }\bigr),\quad k\geq 1, \ \alpha >0,$ where$\exp_{k}=\exp \Bigl(\exp \bigl(\cdots\exp (\ )\cdots\bigr)\Bigr)$denotes the$k$-th iterated exponential. Categories:41A05, 42C99 54. CJM 1998 (vol 50 pp. 605) Guzmán-Partida, Martha; Pérez-Esteva, Salvador  Hardy spaces of conjugate systems of temperatures We define Hardy spaces of conjugate systems of temperature functions on${\bbd R}_{+}^{n+1}$. We show that their boundary distributions are the same as the boundary distributions of the usual Hardy spaces of conjugate systems of harmonic functions. Categories:42B30, 42A50, 35K05 55. CJM 1998 (vol 50 pp. 29) Ding, Yong; Lu, Shanzhen  Weighted norm inequalities for fractional integral operators with rough kernel Given function$\Omega$on${\Bbb R^n}$, we define the fractional maximal operator and the fractional integral operator by$$M_{\Omega,\alpha}\,f(x)=\sup_{r>0}\frac 1{r^{n-\alpha}} \int_{|\,y|1)$, homogeneous of degree zero. Categories:42B20, 42B25

56. CJM 1997 (vol 49 pp. 1010)

Lorente, Maria
 A characterization of two weight norm inequalities for one-sided operators of fractional type In this paper we give a characterization of the pairs of weights $(\w,v)$ such that $T$ maps $L^p(v)$ into $L^q(\w)$, where $T$ is a general one-sided operator that includes as a particular case the Weyl fractional integral. As an application we solve the following problem: given a weight $v$, when is there a nontrivial weight $\w$ such that $T$ maps $L^p(v)$ into $L^q(\w )$? Keywords:Weyl fractional integral, weightsCategories:26A33, 42B25

57. CJM 1997 (vol 49 pp. 708)

Duran, Antonio J.; Lopez-Rodriguez, Pedro
 Density questions for the truncated matrix moment problem For a truncated matrix moment problem, we describe in detail the set of positive definite matrices of measures $\mu$ in $V_{2n}$ (this is the set of solutions of the problem of degree $2n$) for which the polynomials up to degree $n$ are dense in the corresponding space ${\cal L}^2(\mu)$. These matrices of measures are exactly the extremal measures of the set $V_n$. Categories:42C05, 44A60

58. CJM 1997 (vol 49 pp. 175)

Xu, Yuan
 Orthogonal Polynomials for a Family of Product Weight Functions on the Spheres Based on the theory of spherical harmonics for measures invariant under a finite reflection group developed by Dunkl recently, we study orthogonal polynomials with respect to the weight functions $|x_1|^{\alpha_1}\cdots |x_d|^{\alpha_d}$ on the unit sphere $S^{d-1}$ in $\RR^d$. The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator. Keywords:Orthogonal polynomials in several variables, sphere, h-harmonicsCategories:33C50, 33C45, 42C10
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