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Tractability of Tensor Product Linear Operators
- Source :
- Journal of Complexity. 13(4):387-418
- Publication Year :
- 1997
- Publisher :
- Elsevier BV, 1997.
-
Abstract
- This paper deals with the worst case setting for approximating multivariate tensor product linear operators defined over Hilbert spaces. Approximations are obtained by using a number of linear functionals from a given class of information. We consider the three classes of information: the class of all linear functionals, the Fourier class of inner products with respect to given orthonormal elements, and the standard class of function values. We wish to determine which problems are tractable and which are strongly tractable. The complete analysis is provided for approximating operators of rank two or more. The problem of approximating linear functionals is fully analyzed in the first two classes of information. For the third class of standard information we show that the possibilities are very rich. We prove that tractability of linear functionals depends on the given space of functions. For some spaces all nontrivial normed linear functionals are intractable, whereas for other spaces all linear functionals are tractable. In “typical” function spaces, some linear functionals are tractable and some others are not.
- Subjects :
- Discrete mathematics
Statistics and Probability
Pure mathematics
Numerical Analysis
Algebra and Number Theory
Control and Optimization
Rank (linear algebra)
General Mathematics
Applied Mathematics
Hilbert space
Tensor product of Hilbert spaces
Linear function
Continuous linear operator
symbols.namesake
Inner product space
Bra–ket notation
Linear form
symbols
Mathematics
Subjects
Details
- ISSN :
- 0885064X
- Volume :
- 13
- Issue :
- 4
- Database :
- OpenAIRE
- Journal :
- Journal of Complexity
- Accession number :
- edsair.doi.dedup.....54ea09ac1d27f2b337edcc2af3baaeb7
- Full Text :
- https://doi.org/10.1006/jcom.1997.0454