Special functions on locally compact fields

Fourier series on the ring of integers in a 𝑝-series field

“…Then ƒ is radial and ƒ is locally constant at all x^O. Since ƒ is radial we see that (/*<E> 0 )(^) =ƒ ( A case of interest is the collection of potential kernels, ƒ«(#) = | #| ~a, 0<a<l. In [5] it was shown that f a (u) = 1/Ti(a)\u\ a -1 where ri(a) = (l-P a~l )/(l-p~a).…”

“…The linear spaces of test functions © and tempered distributions ©' and the Fourier transform on ©' are defined as in [5]. For details see [5, §2] and [4].…”

Classification of singular integrals over a local field

“…Then, as in [8], we say μ is homogeneous of degree σ if for all teK x , μ t = σ(t)μ, where ^ is that distribution defined by (μ t , <f) = (μ, \t l" 1^-^) . We take M to be M x U {0}, where M x is the group of roots of unity in K of order prime to p. Then M x is the unique cyclic group of order q -1 ([11], p. 16).…”

“…Chao [1] uses Theorem 4 of [8] to establish the conclusion of Lemma 4.1 for the case ω ramified of degree 1. However, he fails to compensate for the fact that he defines the Fourier transform as herein, i.e., ^f(y) = \ f(x)χ(xy)dx, while in [8] it is defined as \ f(%)X(xy)dx. Thus the result of [1] which corresponds to the conclusion of Lemma 4.4 above does not contain the necessary factor of ω(~l).…”

“…92-94. The space ^ of test functions on K and its topological dual the space of distributions, are defined as in [8]. Both are …”