A subspaceY of a Banach spaceX is called a Chebyshev one if for everyx∈X there exists a unique elementPY(x) inY of best approximation. In this paper, necessary and sufficient conditions are obtained in order that certain classes of subspacesY of the Hardy spaceH1=H1 (|z|<1) be Chebyshev ones, and also the properties of the operatorPY are studied. These results show that the theory of Chebyshev subspaces inH1 differs sharply from the corresponding theory inL1(C) of complex-valued functions defined and integrable on the unit circleC:|z|=1. For example, it is proved that inH1 there exist sufficiently many Chebyshev subspaces of finite dimension or co-dimension (while inL1(C) there are no Chebyshev subspaces of finite dimension or co-dimension). Besides, it turned out that the collection of the
Chebyshev subspacesY with a linear operatorPY inH1 (in contrast toL1(C)) is exhausted by that minimum which is necessary for any Banach space.
Authors:P. López-de-Alba, S. García González, and J. Gómez Lara
The precipitation of uranyl ion with 2-hydroxy-1-naphthaldehyde /2H–1N=HL/ was studied. The solid complex /orange crystals/ was characterized by IR, UV-Vis spectra. Uranium was determined as U3O8 after calcination of the complex at 850°C /37.78% U experimental, 36.64% U calculated for C22H14O6U, UO2L2/. Using a statistical experimental design, the best conditions for quantitative precipitation were obtained. A gravimetric method for the determination of UO
is proposed by weighing the complex after drying at 110°C.
Summary A multivariate Hausdorff operator H = H(µ, c, A) is defined in terms of a s-finite Borel measure µ on Rn, a Borel measurable function c on Rn, and an n × n matrix A whose entries are Borel measurable functions on rn and such that A is nonsingular µ-a.e. The operator H*:= H (µ, c | det A-1|, A-1) is the adjoint to H in a well-defined sense. Our goal is to prove sufficient conditions for the boundedness of these operators on the real Hardy space H1(Rn) and BMO (Rn). Our main tool is proving commuting relations among H, H*, and the Riesz transforms Rj. We also prove commuting relations among H, H*, and the Fourier transform.