We study phaseless retrieval in spline spaces generated by B-splines with arbitrary knots. For real spline spaces, we give a necessary and sufficient condition for a sequence of sampling points to admit a local phaseless retrieval of functions from its phaseless sampled values. We also study phaseless retrieval in complex spline spaces and illustrate that phase retrieval is impossible in this case. Nevertheless, we show that phaseless retrieval is possible. For a function in a complex spline space, no mater it is separable or not, we show that the square of its modulus is uniquely determined and can be recovered linearly from its sampled values at a well chosen sequence of sampling points. We give necessary and sufficient conditions for such sequences.

We propose new approaches to the problems of recovering signals from the rearranged frame coefficients or frame coefficients with erasures at either known or unknown locations. These problems naturally arise from applications, where the encoded information needs to be transmitted, for example, in signal/image processing, information and coding theory, and communications. We show that with the appropriate choices of the frames that are used for encoding, the signal with erasures occurring at known locations can be easily recovered without inverting the (sub)frame operators each time. Our new easy to implement and cost-efficient algorithm provides perfect reconstruction of the original signal. To address the problem of recovering erased coefficients from unknown locations, we propose to use a class of frames that are almost robust with respect to m-erasures. We prove that every frame with uniform excess can be rescaled to an almost robust frame and the locations of erased data can be perfectly recovered for almost all the signals. Similar results are obtained for recovering the original order of a disordered (rearranged) set of frame coefficients. Numerical examples are presented to test the main results. Whenever the received data are noise free, we can recover the original signal exactly from frame coefficients with erasures at unknown locations or from a disordered set of frame coefficients.

Average sampling has been developed recently. In this letter, we study determining of averaging functions in an average sampling scheme by means of test functions. Explicit formulae are given.

In this correspondence, we give an average sampling theorem for bandlimited stochastic processes which shows that many average sampling theorems have their counterparts for stochastic signals.

The sampling theorem says that every band-limited signal is uniquely determined by its sampled values provided the sampling points satisfy certain conditions. However, sampled values obtained in practice may not be the exact values of a signal at sampling points, but only averages of the signal near these points. Grochenig (1992) proved that band-limited signals can be reconstructed exactly from local averages if the sampling density is large enough. We study the reconstruction of band-limited signals from local averages with symmetric averaging functions. We study the aliasing error arising when a non-band-limited signal is reconstructed from local averages and give explicit error bounds. Since the classical "point sampling" can be viewed as a limiting case of average sampling, we indeed give new aliasing error bounds for both regular and irregular sampling.