Published online by Cambridge University Press: 07 September 2011
Introduction
In this chapter we construct and study associated homogeneous distributions (AHD) and quasi associated homogeneous distributions (QAHD) in the p-adic case. These results are based on the papers [16], [17], [223]. The results of this chapter will be used below in Chapters 9, 10 and 14.
In Appendix A, Sections A.2–A.7, we recall the theory of QAHDs in the better known case where the underlying field consists of the real numbers (“real QAHD”). By analogy with the theory of real QAHDs, in Sections 6.2–6.5 we develop the theory of p-adic associated and quasi associated homogeneous distributions. In Section 6.2, we recall the facts on p-adic homogeneous distributions from classical books [96, Ch.II, §2], [241, VIII]. In Section 6.3, Definition 6.2 of a p-adic QAHD is introduced. We prove Theorems 6.3.3 (Section 6.3) and 6.4.1 (Section 6.4) which give a description of all quasi associated homogeneous distributions and their Fourier transform respectively. In Section 6.5, a new type of p-adic Γ-functions is introduced. These Γ-functions are generated by QAHDs.
p-adic homogeneous distributions
Definition and characterization
Let πα be a multiplicative character of the field ℚp. We recall some facts from the theory of πα-homogeneous Bruhat–Schwartz p-adic distributions, where πα is a multiplicative character of the field ℚp [96, Ch.II, §2.3.], [241, VIII.1.].
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