Dyadic wavelet frames on a half-line using the Walsh–Fourier transform

Abstract

This paper deals with a construction of dyadic wavelet frames on a positive half-line ℝ+ using the Walsh–Fourier transform. We prove necessary and sufficient conditions for the system to be a wavelet frame in L 2(ℝ+). These conditions are found to be better than those of Daubechies [Ten Lecture on Wavelets, NSF-CBMS Regional Conferences in Applied Mathematics, Vol. 61, SIAM, Philadelphia, PA, 1992], Chui and Shi [Inequalities of Littlewood–Paley type for frames and wavelets, SIAM J. Math. Anal. 24 (1993), pp. 263–277], Casazza and Christenson [Weyl–Heisenberg frames for subspaces of L 2(ℝ), Proc. Amer. Math. Soc. 129 (2001), pp. 145–154] and Christenson [Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bull. Amer. Math. Soc. 38 (2001), pp. 273–291; An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003].