A New Take on Spherical, Whittaker and Bessel Functions
نویسنده
چکیده
0. Introduction 5 0.1. Objectives and main results 5 0.1.1. Affine Satake isomorphisms 5 0.1.2. Whittaker functions 6 0.1.3. The setting of the paper 7 0.2. Dunkl operators via DAHA 7 0.2.1. Families of Dunkl operators 8 0.3. The technique of spinors 9 0.3.1. Connections to AKZ 9 0.3.2. Isomorphism theorems 10 0.3.3. The localization functor 10 0.3.4. The Whittaker limit 11 0.4. On Langlands’ program 11 0.4.1. Key connection 12 0.4.2. Discussion 13 0.4.3. The category Whit 14 0.5. Acknowledgements 15 0.5.1. My Harvard lectures 15 0.5.2. Special thanks 15 1. P-adic theory revisited 16 1.1. Affine Weyl group 16 1.1.1. Root systems 16 1.1.2. Affine root systems 17 1.1.3. The length function 18 1.1.4. Twisted affinization 18 1.2. AHA and spherical functions 19 1.2.1. Affine Hecke algebras 19 1.2.2. Matsumoto functions 20
منابع مشابه
Infinite integrals of Whittaker and Bessel functions with respect to their indices
We obtain several new closed-form expressions for the evaluation of a family of infinite-domain integrals of the Whittaker functions W , x and M , x and the modified Bessel functions I x and K x with respect to the index . The new family of definite integrals is useful in a variety of contexts in mathematical physics. In particular, the integral involving K x represents a new example of the Kon...
متن کاملPoint Processes and the Infinite Symmetric Group. Part V: Analysis of the Matrix Whittaker Kernel
The matrix Whittaker kernel has been introduced by A. Borodin in Part IV of the present series of papers. This kernel describes a point process — a probability measure on a space of countable point configurations. The kernel is expressed in terms of the Whittaker confluent hypergeometric functions. It depends on two parameters and determines a J-symmetric operator K in L(R+)⊕ L(R+). It turns ou...
متن کاملLaplace transform of certain functions with applications
The Laplace transform of the functions tν(1+ t)β, Reν > −1, is expressed in terms of Whittaker functions. This expression is exploited to evaluate infinite integrals involving products of Bessel functions, powers, exponentials, and Whittaker functions. Some special cases of the result are discussed. It is also demonstrated that the famous identity ∫∞ 0 sin(ax)/xdx =π/2 is a special case of our ...
متن کاملA Whittaker-shannon-kotel’nikov Sampling Theorem Related to the Dunkl Transform
A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Dunkl transform on the real line is proved. To this end we state, in terms of Bessel functions, an orthonormal system which is complete in L2((−1, 1), |x|2α+1 dx). This orthonormal system is a generalization of the classical exponential system defining Fourier series.
متن کاملGenerating bessel functions in mie scattering calculations using continued fractions.
A new method of generating the Bessel functions and ratios of Bessel functions necessary for Mie calculations is presented. Accuracy is improved while eliminating the need for extended precision word lengths or large storage capability. The algorithm uses a new technique of evaluating continued fractions that starts at the beginning rather than the tail and has a built-in error check. The conti...
متن کامل