Small Support Spline Riesz Wavelets in Low Dimensions

In [B. Han and Z. Shen, SIAM J. Math. Anal., 38 (2006), 530–556], a family of univariate short support Riesz wavelets was constructed from uniform B-splines. A bivariate spline Riesz wavelet basis from the Loop scheme was derived in [B. Han and Z. Shen, J. Fourier Anal. Appl., 11 (2005), 615–637]. Motivated by these two papers, we develop in this article a general theory and a construction method to derive small support Riesz wavelets in low dimensions from refinable functions. In particular, we obtain small support spline Riesz wavelets from bivariate and trivariate box splines. Small support Riesz wavelets are desirable for developing efficient algorithms in various applications. For example, the short support Riesz wavelets from [B. Han and Z. Shen, SIAM J. Math. Anal., 38 (2006), 530–556] were used in a surface fitting algorithm of [M.J. Johnson, Z. Shen and Y.H. Xu, J. Approx. Theory, 159 (2009), 197–223], and the Riesz wavelet basis from the Loop scheme was used in a very efficient geometric mesh compression algorithm in [A. Khodakovsky, P. Schröder and W. Sweldens, Proceedings of SIGGRAPH, 2000].