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51. CJM 2001 (vol 53 pp. 565)

Hare, Kathryn E.; Sato, Enji
 Spaces of Lorentz Multipliers We study when the spaces of Lorentz multipliers from $L^{p,t} \rightarrow L^{p,s}$ are distinct. Our main interest is the case when $s Keywords:multipliers, convolution operators, Lorentz spaces, Lorentz-improving multipliersCategories:43A22, 42A45, 46E30 52. CJM 2000 (vol 52 pp. 381) Miyachi, Akihiko  Hardy Space Estimate for the Product of Singular Integrals$H^p$estimate for the multilinear operators which are finite sums of pointwise products of singular integrals and fractional integrals is given. An application to Sobolev space and some examples are also given. Keywords:$H^p$space, multilinear operator, singular integral, fractional integration, Sobolev spaceCategory:42B20 53. CJM 2000 (vol 52 pp. 3) Aizenberg, Lev; Vidras, Alekos  On Small Complete Sets of Functions Using Local Residues and the Duality Principle a multidimensional variation of the completeness theorems by T.~Carleman and A.~F.~Leontiev is proven for the space of holomorphic functions defined on a suitable open strip$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$. Categories:32A10, 42C30 54. CJM 1998 (vol 50 pp. 1236) Kalton, N. J.; Tzafriri, L.  The behaviour of Legendre and ultraspherical polynomials in$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

55. 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

56. CJM 1998 (vol 50 pp. 605)

 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
 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)$, homogeneous of degree zero. Categories:42B20, 42B25 58. CJM 1997 (vol 49 pp. 1010) Lorente, Maria  A characterization of two weight norm inequalities for one-sided operators of fractional type In this paper we give a characterization of the pairs of weights$(\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 )$? Keywords:Weyl fractional integral, weightsCategories:26A33, 42B25 59. CJM 1997 (vol 49 pp. 708) Duran, Antonio J.; Lopez-Rodriguez, Pedro  Density questions for the truncated matrix moment problem For a truncated matrix moment problem, we describe in detail the set of positive definite matrices of measures$\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 60. 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