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1.
2.
Summary The existence of optimal nodes with preassigned multiplicities is proved for the Hardy spacesH
p
(1<p<). This is then used to show that the exact order of convergence for the optimal qudrature formula withN nodes (including multiplicity) is
where 1/p+1/q=1 and 1p. 相似文献
3.
Palle E. T. Jorgensen 《Integral Equations and Operator Theory》1999,35(2):125-171
This paper is devoted to an approximation problem for operators in Hilbert space, that appears when one tries to study geometrically thecascade algorithm in wavelet theory. Let
be a Hilbert space, and let be a representation ofL
(
) on
. LetR be a positive operator inL
(
) such thatR(1) =1, where1 denotes the constant function 1. We study operatorsM on
(bounded, but noncontractive) such that
where the * refers to Hilbert space adjoint. We give a complete orthogonal expansion of
which reduces such thatM acts as a shift on one part, and the residual part is
() =
n
[M
n
], where [M
n
] is the closure of the range ofM
n
. The shift part is present, we show, if and only if ker (M
*){0}. We apply the operator-theoretic results to the refinement operator (or cascade algorithm) from wavelet theory. Using the representation , we show that, for this wavelet operatorM, the components in the decomposition are unitarily, and canonically, equivalent to spacesL
2(E
n
) L
2(), whereE
n , n=1,2,3,..., , are measurable subsets which form a tiling of ; i.e., the union is up to zero measure, and pairwise intersections of differentE
n
's have measure zero. We prove two results on the convergence of the cascale algorithm, and identify singular vectors for the starting point of the algorithm.Terminology used in the paper
the one-torus
-
Haar measure on the torus
-
Z
the Zak transform
-
X=ZXZ
–1
transformation of operators
-
a given Hilbert space
-
a representation ofL
(
) on
-
R
the Ruelle operator onL
(
)
-
M
an operator on
-
R
*,M
*
adjoint operators
Work supported in part by the U.S. National Science Foundation. 相似文献
4.
We obtain upper and lower bounds for Christoffel functions for Freud weights by relatively new methods, including a new way to estimate discretization of potentials. We then deduce bounds for orthogonal polynomials on thereby largely resolving a 1976 conjecture of P. Nevai. For example, let W:=e
–Q, whereQ: is even and continuous in, Q" is continuous in (0, ) andQ
'>0 in (0, ), while, for someA, B,
相似文献
5.
Let M
f(r) and f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality
be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +. 相似文献
6.
We establish a criterion for the existence of a solution of the interpolation problem f(
n
) = b
n in the class of functions f analytic in the unit disk and satisfying the relation
7.
Guillermo López Lagomasino 《Constructive Approximation》1989,5(1):199-219
Letd be a finite positive Borel measure on the interval [0, 2] such that >0 almost everywhere; andW
n be a sequence of polynomials, degW
n
=n, whose zeros (w
n
,1,,w
n,n
lie in [|z|1]. Let d
n
<> for eachnN, whered
n
=d/|W
n
(e
i
)|2. We consider the table of polynomials
n,m such that for each fixednN the system
n,m,mN, is orthonormal with respect tod
n
. If
|