In a recent article, we showed that trigonometric shearlets are able to detect directional step discontinuities along edges of periodic characteristic functions. In this paper, we extend these results to bivariate periodic functions which have jump discontinuities in higher order directional derivatives along edges. In order to prove suitable upper and lower bounds for the shearlet coefficients, we need to generalize the results about localization- and orientation-dependent decay properties of the corresponding inner products of trigonometric shearlets and the underlying periodic functions.
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Figure 2. Left: Schematic visualization of the function from Figure 1 with colored boundary lines where the function has directional jump discontinuities of different orders. Right: Magnitudes of L(i),maxℓ and L(i),minℓ from (17) as functions of the orientation angles θ(i)10,ℓ
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Left: Cartoon-like function with jump discontinuities in the zeroth (red), first (blue) and second (black) order directional derivative on a circle with radius
Left: Schematic visualization of the function from Figure 1 with colored boundary lines where the function has directional jump discontinuities of different orders. Right: Magnitudes of
Left: Star-like set