Introduction

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The Illinois Applied Harmonic Analysis Seminar was held at the University of Illinois on December 2, 2006.

Open Problem Session

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The open problems discussed at the end of the seminar include the following. Names attached to problems or constructions are intended to be helpful, but there is not space here to give a full history or credit for each problem. There is also no guarantee of completeness! Corrections and amplifications to the notes are welcome.

1. Directional wavelets/frames in n-dimensions (n ≥ 2): existence and construction.
   • Some known constructions include 2-D directional wavelets (Antoine et al.), complex wavelets (Kingsbury, Selesnick), contourlets (Do and Vetterli), curvelets (Candes et al., http://www.curvelet.org), surfacelets (Lu and Do), shearlets (Labate, Weiss et al., http://www.shearlet.org), composite dilation wavelets (Weiss, Wilson et al.), wavepackets (de Hoop, Smith).

   • One wants compact support of the wavelets in space (that is, time), along with reasonable localization in frequency.

     Note: Wilson's talk described "Some simple Haar--type wavelets in higher dimensions", work with Krishtal, Robinson and Weiss, which gives compact support in space for the quincunx dilation. But the wavelet is not continuous (?) and so its Fourier transform decays slowly, thus has poor frequency localization.

   • Sampling issues: time, bandwidth? Slepian's prolate spheroidal wave functions. (Allen and He)

   • Low redundancy of the frame. The wavepacket frames of de Hoop have redundancy of about 4. Can one do better? Get an orthonormal basis while maintaining good frequency localization? (Lu)

   • Robustness - coding, correlations? (Allen)

2. Existence of wavelets w.r.t. group actions (dilations), measurable cross-sections, non-ergodic. (Wilson)

3. Can one characterize those affine systems that span L^p? For example, Yves Meyer asked whether the dyadic Mexican hat system (generated by ψ(x)=(1-x²)e^-x2/2) spans L^p for 1 < p < ∞. It does for p = 2, where it is a frame. (Laugesen)

4. Optimal sampling for acquisition systems. (Bresler)
   • Tomography, MRI. What is the sampling condition for X-ray transforms in the 3-D case?

   • Related to finding lower bound of an infinite matrix.

5. Cardinality of frames « dimension of space. (He)

6. Spectral radius of generalized Henkel matrix. (He)

7. Existence and construction of non-random measurement matrices for compressed sensing. (Do)

8. Data ® "structure" ® frame ® sparse representation. (Duchkov)

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