Left-Inverses of Fractional Laplacian and Sparse Stochastic Processes

The fractional Laplacian (-)/2 commutes with the primary coordination transformations in the Euclidean space Rd: dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < < d, its inverse is the classical Riesz potential I which is dilationinvariant and translation-invariant. In this work, we investigate the functional properties (continuity, decay and invertibility) of an extended class of differential operators that share those invariance properties. In particular, we extend the definition of the classical Riesz potential I to any non-integer number larger than d and show that it is the unique left-inverse of the fractional Laplacian (-)/2 which is dilation-invariant and translation-invariant. We observe that, for any 1 p and d(1 - 1/p), there exists a Schwartz function f such that If is not p-integrable. We then introduce the new unique leftinverse I,p of the fractional Laplacian (-)/2 with the property that I,p is dil...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Qiyu Sun, Michael Unser
• Keywords: Classical Riesz, CORR 2010, Education, Primary Coordination Transformations, Riesz Potential