By Birkhoff G., Bellman R., Lin C.C. (eds.)
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Extra resources for Hydrodynamic instability: proceedings of the 13th symposium in applied mathematics
Sample text
A condition that is easier to remember is derived by noting that s ≤ N and m ≤ N (otherwise, we are not in the range of interest for compressive sensing). 22) implies recovery by 1 -minimization with probability at least 1 − N −γ ln 3 (N ) . 28 Holger Rauhut E. Candès and T. 22) in case of the random partial Fourier matrix with an exponent 6 instead of 4 at the ln(N ) term. M. Rudelson and R. 1 below. 22) with exponent 4 and super-polynomially decreasing failure 3 probability N −γ ln(N ) is presently the best known result.
3) k=1 with coefficients x1 , . . , xN ∈ C. Let t1 , . . , tm ∈ D be some points and suppose we are given the sample values N xk ψk (t ) , y = f (t ) = k=1 = 1, . . , m. 20 Holger Rauhut Introducing the sampling matrix A ∈ Cm×N with entries A ,k = ψk (t ) , = 1, . . , m, k = 1, . . 4) the vector y = (y1 , . . 3). Our task is to reconstruct the polynomial f — or equivalently its vector x of coefficients — from the vector of samples y. We wish to perform this task with as few samples as possible.
Such a condition excludes for instance that the functions ψj are very localized in small regions of D. Expressed differently, the quotients ψj ∞ / ψj 2 should be uniformly bounded in j (in case that the functions ψj are not yet normalized); or at least grow only very slowly. (b) It is not essential that D is a (measurable) subset of Rd . This assumption was only made for convenience. In fact, D can be any measure space endowed with a probability measure ν. 3) k=1 with coefficients x1 , . . , xN ∈ C.