QUANTUM APPROXIMATION ON ANISOTROPIC SOBOLEV AND HÖLDER-NIKOLSKII CLASSES

We estimate the quantum query error of approximation to functions from the anisotropic Sobolev class ℬ ( W p r ( [ 0 ,   1 ] d ) ) and the Hölder-Nikolskii class ℬ ( H p r ( [ 0 ,   1 ] d ) ) in theLq ([0, 1] d ) norm for all 1 ≤p, q≤ ∞. It turns out that for the class ℬ ( W p r ( [ 0 ,   1 ] d ) )...

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Published inTaiwanese journal of mathematics Vol. 16; no. 1; pp. 71 - 88
Main Author Ye, Peixin
Format Journal Article
LanguageEnglish
Published Mathematical Society of the Republic of China 01.02.2012
Subjects
Approximation
Banach space
Coordinate systems
Cubes
Determinism
Embeddings
Mathematical functions
Quantum computers
Sobolev spaces
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Abstract We estimate the quantum query error of approximation to functions from the anisotropic Sobolev class ℬ ( W p r ( [ 0 ,   1 ] d ) ) and the Hölder-Nikolskii class ℬ ( H p r ( [ 0 ,   1 ] d ) ) in theLq ([0, 1] d ) norm for all 1 ≤p, q≤ ∞. It turns out that for the class ℬ ( W p r ( [ 0 ,   1 ] d ) ) (r∈ ℕ d ), whenp<q, the quantum algorithms can essentially improve the rate of convergence of classical deterministic and randomized algorithms; while for the class ℬ ( H p r ( [ 0 ,   1 ] d ) ) and ℬ ( W p r ( [ 0 ,   1 ] d ) )   ( r ∈ ℝ + d ) , whenp≥q, the optimal convergence rate is the same for all three settings. 2010Mathematics Subject Classification: 41A63, 65D15, 65Y20. Key words and phrases: Quantum approximation, Anisotropic classes, Minimal query error.
AbstractList We estimate the quantum query error of approximation to functions from the anisotropic Sobolev class ℬ ( W p r ( [ 0 ,   1 ] d ) ) and the Hölder-Nikolskii class ℬ ( H p r ( [ 0 ,   1 ] d ) ) in theLq ([0, 1] d ) norm for all 1 ≤p, q≤ ∞. It turns out that for the class ℬ ( W p r ( [ 0 ,   1 ] d ) ) (r∈ ℕ d ), whenp<q, the quantum algorithms can essentially improve the rate of convergence of classical deterministic and randomized algorithms; while for the class ℬ ( H p r ( [ 0 ,   1 ] d ) ) and ℬ ( W p r ( [ 0 ,   1 ] d ) )   ( r ∈ ℝ + d ) , whenp≥q, the optimal convergence rate is the same for all three settings. 2010Mathematics Subject Classification: 41A63, 65D15, 65Y20. Key words and phrases: Quantum approximation, Anisotropic classes, Minimal query error.
Author Ye, Peixin
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10.1006/jcom.2001.0629
10.1007/978-3-642-65711-5
10.1016/j.jco.2003.08.002
10.1016/j.jat.2007.04.013
10.1007/s10208-002-0074-6
10.1016/j.jco.2003.08.003
10.1016/j.jco.2007.01.007
10.1137/0717033
10.1016/j.jco.2006.04.005
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Snippet We estimate the quantum query error of approximation to functions from the anisotropic Sobolev class ℬ ( W p r ( [ 0 ,   1 ] d ) ) and the Hölder-Nikolskii...
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SubjectTerms Approximation
Banach space
Coordinate systems
Cubes
Determinism
Embeddings
Mathematical functions
Quantum computers
Sobolev spaces
Title QUANTUM APPROXIMATION ON ANISOTROPIC SOBOLEV AND HÖLDER-NIKOLSKII CLASSES
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Volume 16
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