Asymptotic behavior of integrals connected with spectral functions for hypoelliptic operators.

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Published **1967**
by Almqvist & Wiksell in Stockholm .

Written in English

- Polynomials.,
- Spectral theory (Mathematics),
- Integrals.,
- Hypoelliptic operators.

**Edition Notes**

Bibliography: p. 298.

Series | Arkiv för matematik,, bd. 7, nr. 20 |

Classifications | |
---|---|

LC Classifications | QA3 .A7 bd. 7, nr. 20 |

The Physical Object | |

Pagination | 283-298 p. |

Number of Pages | 298 |

ID Numbers | |

Open Library | OL5509475M |

LC Control Number | 73363928 |

Project Euclid - mathematics and statistics online. Ark. Mat. Volume 7, Number 3 (), Asymptotic behavior of integrals connected with spectral functions for hypoelliptic operatorsCited by: 3. Abstract. This paper is part of a series papers devoted to geometric and spectral theoretic ap-plications of the hypoelliptic calculus on Heisenberg manifolds. More speciﬁcally, in this paper we make use of the Heisenberg calculus of Beals-Greiner and Taylor to analyze the spectral theory of hypoelliptic operators on Heisenberg manifolds. 4. Gortjakov, V. N., On the asymptotic behaviour of the spectral function of a class of hypoelliptic operators. Doklady Ak. Nauk,3, p. – ().. Google ScholarCited by: Asymptotic behavior of integrals connected with spectral functions for hypoelliptic operators Authors. Jöran Friberg; Content type: OriginalPaper; Published: 01 December ; Explore Volumes and issues. Advertisement. Over 10 million scientific documents at your fingertips. Switch Edition.

To this end, we build upon recent work of Eckmann & Hairer [14,12,13], Hèrau & Nier [21] and Helffer & Nier [20] on the spectral properties of hypoelliptic operators, and show that the generator. A useful asymptotic awkward to derive from Stirling’s formula for (s), but easy to obtain from Watson’s lemma, is an asymptotic for Euler’s beta integral[6] B(s;a) = Z 1 0 xs 1 (1 x)a 1 dx = (s)(a) (s+ a) Fix awith Re(a) >0, and consider this integral as a function of s. Setting x= e u gives an integrand tting Watson’s lemma, B(s;a. Chapter 1 covers the asymptotic theory of real Laplace-type integrals. The computation of the coeﬃcients appearing in the asymptotic expansions are de-scribed completely in this chapter. In Chapter 2, we discuss two methods from the asymptotic theory of complex Laplace-type integrals: the Method of Steepest Descents and Perron’s Method. Asymptotic expansions of integrals 29 Chapter 4. Laplace integrals 31 Laplace’s method 32 Watson’s lemma 36 Chapter 5. Method of stationary phase 39 Chapter 6. Method of steepest descents 43 Bibliography 49 Appendix A. Notes 51 A Remainder theorem 51 A Taylor series for functions of more than one variable 51 A

involving the target function etc. Green’s function technology expresses the solution of a diﬀer-ential equation as a convolution integral etc. Integrals are also important because they provide the simplest and most accessible examples of concepts like asymptoticity and techniques such as asymptotic matching. The Airy function. have reduced the problem of nding the asymptotic expansion of the spectral function to computing certain rather complicated integrals. In the diagonal case these integrals can be computed with the brute force, whereas in the non-degenerate o -diagonal case these integrals can be computed (or rather approximated) using the stationary phase method. In this paper we consider the asymptotic behaviour of the spectral function of an elliptic differential (pseudodifferential) equation or system of equations. For the case of differential operators this problem has been widely studied, and various methods have been developed for its solution (see [1] and [2] for a survey of these methods). In this paper we shall obtain the best possible estimates for the remainder term in the asymptotic formula for the spectral function of an arbitrary elliptic (pseudo-)differential operator. This is achieved by means of a complete description of the singularities of the Fourier transform of the spectral function .

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