We study bijections between algebras of smooth functions preserving certain parts of its structure. In particular, we show that multiplicative bijections are implemented by diffeomorphisms and they are automatically algebra isomorphisms. This confirms a conjecture by Mrčun and Šemrl.

In this paper, we propose a time-frequency analysis method to obtain instantaneous frequencies and the corresponding decomposition by solving an optimization problem. In this optimization problem, the basis that is used to decompose the signal is not known a priori. Instead, it is adapted to the signal and is determined as part of the optimization problem. In this sense, this optimization problem can be seen as a dictionary adaptation problem, in which the dictionary is adaptive to one signal rather than a training set in dictionary learning. This dictionary adaptation problem is solved by using the augmented Lagrangian multiplier (ALM) method iteratively. We further accelerate the ALM method in each iteration by using the fast wavelet transform. We apply our method to decompose several signals, including signals with poor scale separation, signals with outliers and polluted by noise and a real signal. The results show that this method can give accurate recovery of both the instantaneous frequencies and the intrinsic mode functions.

We discuss operators of the type H=-\Delta+ V (x)-lpha\delta (x-\Sigma) with an attractive interaction, H=-\Delta+ V (x)-lpha\delta (x-\Sigma) , in H=-\Delta+ V (x)-lpha\delta (x-\Sigma) , where H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is an infinite surface, asymptotically planar and smooth outside a compact, dividing the space into two regions, of which one is supposed to be convex, and H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is a potential bias being a positive constant H=-\Delta+ V (x)-lpha\delta (x-\Sigma) in one of the regions and zero in the other. We find the essential spectrum and ask about the existence of the discrete one with a particular attention to the critical case, H=-\Delta+ V (x)-lpha\delta (x-\Sigma) . We show that H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is then empty if the bias is supported in theexterior'region, while in the opposite case isolated eigenvalues may exist.

We introduce a class of tri-linear operators that combine features of the bilinear Hilbert transform and paraproduct. For two instances of these operators, we prove boundedness property in a large range D = { ( p 1 , p 2 , p 3 ) 3 : 1 < p 1

We study the super-resolution (SR) problem of recovering point sources consisting of a collection of isolated and suitably separated spikes from only the low frequency measurements. If the peak separation is above a factor in (1,2) of the Rayleigh length (physical resolution limit), L 1 minimization is guaranteed to recover such sparse signals. However, below such critical length scale, especially the Rayleigh length, the L 1 certificate no longer exists. We show several local properties (local minimum, directional stationarity, and sparsity) of the limit points of minimizing two L 1 based nonconvex penalties, the difference of L 1 and L 1 norms ( L 1 ) and capped L 1 (C L 1 ), subject to the measurement constraints. In one and two dimensional numerical SR examples, the local optimal solutions from difference of convex function algorithms outperform the global L 1 solutions near or below Rayleigh length scales either in