Hilbert transforms for thep-adic andp-series fields

Classical Wavelet Transforms over Finite Fields

This article introduces a systematic study for computational aspects of classical wavelet transforms over finite fields using tools from computational harmonic analysis and also theoretical linear algebra. We present a concrete formulation for the Frobenius norm of the classical wavelet transforms over finite fields. It is shown that each vector defined over a finite field can be represented as...

LINEAR SERIES OVER REAL AND p-ADIC FIELDS

We note that the degeneration arguments given by the author in [5] to derive a formula for the number of maps from a general curve C of genus g to P with prescribed ramification also yields weaker results when working over the real numbers or p-adic fields. Specifically, let k be such a field: we see that given g, d, n, and e1, . . . , en satisfying P i (ei − 1) = 2d − 2 − g, then there exists ...

p-ADIC FORMAL SERIES AND PRIMITIVE POLYNOMIALS OVER FINITE FIELDS

In this paper, we investigate the Hansen-Mullen conjecture with the help of some formal series similar to the Artin-Hasse exponential series over p-adic number fields and the estimates of character sums over Galois rings. Given n we prove, for large enough q, the Hansen-Mullen conjecture that there exists a primitive polynomial f(x) = xn − a1xn−1 + · · ·+ (−1)an over Fq of degree n with the m-t...

Two-dimensional Hilbert Transforms'

Uniform Bounds for the Bilinear Hilbert Transforms

It is shown that the bilinear Hilbert transforms Hα,β(f, g)(x) = p.v. ∫ R f(x− αt)g(x− βt) dt t map Lp1(R) × Lp2(R) → Lp(R) uniformly in the real parameters α, β when 2 < p1, p2 < ∞ and 1 < p = p1p2 p1+p2 < 2. Combining this result with the main result in [9], we deduce that the operators H1,α map L2(R)×L∞(R) → L2(R) uniformly in the real parameter α ∈ [0, 1]. This completes a program initiated...