Analysis.
2nd ed.

*(English)*Zbl 0966.26002
Graduate Studies in Mathematics. 14. Providence, RI: American Mathematical Society (AMS). xxii, 346 p. (2001).

See the review of the first edition (1996) in Zbl 0873.26002.

From the Preface to the second edition:

“This second edition contains corrections and some fresh items. Chief among these is Chapter 12 in which we explain several topics concerning eigenvalues of the Laplacian and the Schrödinger operator, such as the min-max principle, coherent states, semiclassical approximation and how to use these to get bounds on eigenvalues and sums of eigenvalues. But there are other additions, too, such as more on Sobolev spaces (Chapter 8) including a compactness criterion, and Poincaré, Nash and logarithmic Sobolev inequalities. The latter two are applied to obtain smoothing properties of semigroups.

Chapter 1 (Measure and integration) has been supplemented with a discussion of the more usual approach to integration theory using simple functions, and how to make this even simpler by using ‘really simple functions’. Egoroff’s theorem has also been added. Several additions were made to Chapter 6 (Distributions) including one about the Yukawa potential.

There are, of course, many more exercises as well”.

From the Preface to the second edition:

“This second edition contains corrections and some fresh items. Chief among these is Chapter 12 in which we explain several topics concerning eigenvalues of the Laplacian and the Schrödinger operator, such as the min-max principle, coherent states, semiclassical approximation and how to use these to get bounds on eigenvalues and sums of eigenvalues. But there are other additions, too, such as more on Sobolev spaces (Chapter 8) including a compactness criterion, and Poincaré, Nash and logarithmic Sobolev inequalities. The latter two are applied to obtain smoothing properties of semigroups.

Chapter 1 (Measure and integration) has been supplemented with a discussion of the more usual approach to integration theory using simple functions, and how to make this even simpler by using ‘really simple functions’. Egoroff’s theorem has also been added. Several additions were made to Chapter 6 (Distributions) including one about the Yukawa potential.

There are, of course, many more exercises as well”.

##### MSC:

26-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions |

28-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration |

42-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces |

46-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis |

49-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control |

26D10 | Inequalities involving derivatives and differential and integral operators |

26D15 | Inequalities for sums, series and integrals |

31B05 | Harmonic, subharmonic, superharmonic functions in higher dimensions |

31B15 | Potentials and capacities, extremal length and related notions in higher dimensions |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

46F05 | Topological linear spaces of test functions, distributions and ultradistributions |

46F10 | Operations with distributions and generalized functions |

49R50 | Variational methods for eigenvalues of operators (MSC2000) |

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |