2025/04/17 by Haar, Andrew
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2504.12783
In this paper, we investigate a spline frame generated by oversampling against the well-known Battle-Lemarié wavelet system of nonnegative integer order, n. We establish a characterization of the Besov and Triebel-Lizorkin (quasi-) norms for the smoothness parameter up to s < n+1, which includes values of s where the Battle-Lemarié system no longer provides an unconditional basis; we, additionally, prove a result for the endpoint case s=n+1. This builds off of earlier work by G. Garrigós, A. Seeger, and T. Ullrich, where they proved the case n=0, i.e. that of the Haar wavelet, and work of R. Srivastava, where she gave a necessary range for the Battle-Lemarié system to give an unconditional basis of the Triebel-Lizorkin spaces.