Further Estimates for Certain Integrals of Six Bessel Functions

Abstract

In this thesis we study integrals involving a sixfold product of Bessel functions of the first kind and integer order. We establish good asymptotic estimates with precise error bounds for a certain one-parameter subclass of integrals. In the general case of six arbitrary integer orders we conjecture a formula for the asymptotics of the integral that is consistent with the special case. Testing this formula on numerical reference data suggests that it approx- imates the integral very accurately if at least one of the integers is larger than 20.<br /> The interest in these kinds of Bessel integrals stems from the study of a sharp Fourier restriction inequality on the circle and a program to characterize its extremizers by looking at those integrals. This program is proposed in [14] and [1] and also includes the works [12], [13] and [15]. This thesis lines up here and builds upon the work of Oliveira e Silva and Thiele in [1]. We improve and extend their methods and generalize the results to a much wider range of integrals.<br /> The thesis consists of seven chapters.<br /> In Chapter 1 we discuss the connection between the study of extremizers in the Fourier restriction theory on the circle and integrals of a sixfold product of Bessel functions. We review the existing results on asymptotic bounds for those integrals and state our main theorem. We then outline the four major parts of the proof and give a summary of the used techniques. Each of the Chapters 2, 3, 4 and 5 elaborates on one of the parts of the proof.<br /> In Chapter 2 we expand a product of four Bessel functions into a power series of finite length. If all four functions are equal we provide an expression for the remainder term.<br /> In Chapter 3 we replace four of the six Bessel functions in our integral with the power series we derived in Chapter 2 and deduce an alternative representation of the initial in- tegral in terms of sums of quotients of gamma functions and hypergeometric functions. We identify those parts of this representation that carry the asymptotic information of the integral and those that contribute to the error. In the case of six different Bessel functions of arbitrary integer orders we only establish the main asymptotic term without a proof.<br /> Chapters 4 and 5 are entirely dedicated to the one-parameter subclass of integrals such that the orders of the Bessel functions form the six vector (n, n, n, n, 2n, 2n) for integers n. More precisely, in Chapter 4 we prove upper bounds on all components of the decomposition of the integral we deduced in Chapter 3. The analytical approach of this chapter fails for finitely many integrals. Those are estimated numerically in Chapter 5.<br /> In Chapter 6 we take a closer look at the conjectured formula for the asymptotics of the general Bessel integral and test its quality on numerical data for selected subfamilies of integrals. Those numerical values have been calculated for the paper [13]. To demonstrate the excellent performance of our formula we use it to reproduce some of the findings of [13]. We then sketch open problems that have to be solved in order to turn the conjecture into a theorem and list some interesting questions for further research on this topic.<br /> Chapter 7 provides results from the theory of gamma, Bessel and hypergeometric func- tions, as well as some other useful inequalities and identities, that are frequently used throughout this thesis.