*Robert Alicki and Mark Fannes*

- Published in print:
- 2001
- Published Online:
- February 2010
- ISBN:
- 9780198504009
- eISBN:
- 9780191708503
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504009.003.0012
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Quantum systems with a discrete dynamical spectrum, such as finite dimensional systems or particles confined to a finite volume, cannot exhibit true ergodic behaviour. Nevertheless, scaling limits ...
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Quantum systems with a discrete dynamical spectrum, such as finite dimensional systems or particles confined to a finite volume, cannot exhibit true ergodic behaviour. Nevertheless, scaling limits can be considered. This chapter focuses on the behaviour of entropy for small value of Planck's constant, i.e. how the classical limit is reached. It also considers other classical limits such as the limit of large angular momentum for the kicked top. Numerical computations are needed to compute the actual value of the total entropy produced up to a finite time, but its saturation behaviour for long times can be established analytically. The operational partitions that eventually lead to the largest entropy can also be determined. As a complementary tool for the analysis of randomizing behaviour, the chapter sketches the use of the Gram matrix to study the statistics of return times.Less

Quantum systems with a discrete dynamical spectrum, such as finite dimensional systems or particles confined to a finite volume, cannot exhibit true ergodic behaviour. Nevertheless, scaling limits can be considered. This chapter focuses on the behaviour of entropy for small value of Planck's constant, i.e. how the classical limit is reached. It also considers other classical limits such as the limit of large angular momentum for the kicked top. Numerical computations are needed to compute the actual value of the total entropy produced up to a finite time, but its saturation behaviour for long times can be established analytically. The operational partitions that eventually lead to the largest entropy can also be determined. As a complementary tool for the analysis of randomizing behaviour, the chapter sketches the use of the Gram matrix to study the statistics of return times.

*Marek A. Kowalski, Krzysztof A. Sikorski, and Frank Stenger*

- Published in print:
- 1995
- Published Online:
- November 2020
- ISBN:
- 9780195080599
- eISBN:
- 9780197560402
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/9780195080599.003.0005
- Subject:
- Computer Science, Mathematical Theory of Computation

In this chapter we acquaint the reader with the theory of approximation of elements of normed spaces by elements of their finite dimensional subspaces. The theory of best approximation was ...
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In this chapter we acquaint the reader with the theory of approximation of elements of normed spaces by elements of their finite dimensional subspaces. The theory of best approximation was originated between 1850 and 1860 by Chebyshev. His results and ideas have been extended and complemented in the 20th century by other eminent mathematicians, such as Bernstein, Jackson, and Kolmogorov. Initially, we present the classical theory of best approximation in the setting of normed spaces. Next, we discuss best approximation in unitary (inner product) spaces, and we present several practically important examples. Finally, we give a reasonably complete presentation of best uniform approximation, along with examples, the Remez algorithm, and including converse theorems about best approximation. The goal of this section is to present some general results on approximation in normed spaces.
Less

In this chapter we acquaint the reader with the theory of approximation of elements of normed spaces by elements of their finite dimensional subspaces. The theory of best approximation was originated between 1850 and 1860 by Chebyshev. His results and ideas have been extended and complemented in the 20th century by other eminent mathematicians, such as Bernstein, Jackson, and Kolmogorov. Initially, we present the classical theory of best approximation in the setting of normed spaces. Next, we discuss best approximation in unitary (inner product) spaces, and we present several practically important examples. Finally, we give a reasonably complete presentation of best uniform approximation, along with examples, the Remez algorithm, and including converse theorems about best approximation. The goal of this section is to present some general results on approximation in normed spaces.

*Marek A. Kowalski, Krzystof A. Sikorski, and Frank Stenger*

- Published in print:
- 1995
- Published Online:
- November 2020
- ISBN:
- 9780195080599
- eISBN:
- 9780197560402
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195080599.003.0004
- Subject:
- Computer Science, Mathematical Theory of Computation

In this chapter we acquaint the reader with the theory of approximation of elements of normed spaces by elements of their finite dimensional subspaces. The theory of ...
More

In this chapter we acquaint the reader with the theory of approximation of elements of normed spaces by elements of their finite dimensional subspaces. The theory of best approximation was originated between 1850 and 1860 by Chebyshev. His results and ideas have been extended and complemented in the 20th century by other eminent mathematicians, such as Bernstein, Jackson, and Kolmogorov. Initially, we present the classical theory of best approximation in the setting of normed spaces. Next, we discuss best approximation in unitary (inner product) spaces, and we present several practically important examples. Finally, we give a reasonably complete presentation of best uniform approximation, along with examples, the Remez algorithm, and including converse theorems about best approximation. The goal of this section is to present some general results on approximation in normed spaces.
Less

In this chapter we acquaint the reader with the theory of approximation of elements of normed spaces by elements of their finite dimensional subspaces. The theory of best approximation was originated between 1850 and 1860 by Chebyshev. His results and ideas have been extended and complemented in the 20th century by other eminent mathematicians, such as Bernstein, Jackson, and Kolmogorov. Initially, we present the classical theory of best approximation in the setting of normed spaces. Next, we discuss best approximation in unitary (inner product) spaces, and we present several practically important examples. Finally, we give a reasonably complete presentation of best uniform approximation, along with examples, the Remez algorithm, and including converse theorems about best approximation. The goal of this section is to present some general results on approximation in normed spaces.