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By Fornasier M. (ed.)

The current number of 4 lecture notes is the first actual contribution of this kind within the box of sparse restoration. Compressed sensing is among the vital points of the wider suggestion provided within the booklet, which via now has made connections with different branches resembling mathematical imaging, inverse difficulties, numerical research and simulation. This particular assortment might be of price for a extensive neighborhood and should function a textbook for graduate classes.

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Then, for n ∈ N,  2n M  E j j,k=1 k Bj,k 1/2n   ≤ 21/(2n) (2n)! 2n n! 33) S2n   1/2  M  ∗   × max Bj,k Bj,k   j,k=1  M ,  ∗ Bj,k Bj,k  , F S2n , j,k=1 S2n    1/2 S2n F S2n   ∗ M where F , F are the block matrices F = (Bj,k )M j,k=1 and F = (Bj,k )j,k=1 . 25) above. 25) twice. Taking scalars instead of matrices Bj,k results in a scalar Khintchine inequality for decoupled Rademacher chaos. 33) coincide and the third one is always dominated by the first. 22. Denote Cn = (2n)!

2. Let S ⊂ [N ] be of cardinality |S| = s and let = ( ) ∈S ∈ Cs be a sequence of independent random variables that take the values ±1 with equal probability. Alternatively, the may be uniformly distributed on the torus {z ∈ C, |z| = 1}. Let x be an s-sparse vector with support S and sgn(xS ) = . 2) for some constant K ≥ 1. Assume 26 Holger Rauhut that the random sampling points t1 , . . , tm are chosen independently and distributed according to the orthogonalization measure ν. 25. Set y = Ax. 12).

Then Bj,k Bj∗ ,k = 0 Bj,k Bj∗ ,k if k = k , if k = k , ∗ Bj,k Bj ,k = 0 ∗ B Bj,k j ,k if k = k , if k = k . 35) Since the singular values obey σk (A) = σk ((AA∗ )1/2 ), the Schatten class norm satisfies A S2n = (AA∗ )1/2 S2n . This allows us to verify that M j=1 =  Bj,k j 1/2  M k=1 k,k j,j S2n Bj,k Bj∗ ,k  j j S2n 1/2  =  1/2 Bj,k Bj∗ ,k  j j ∗ Hk ( )Hk ( ) = k j,j . k S2n S2n Similarly, we also get 1/2 M ∗ Hk ( ) Hk ( ) k = S2n M Bj,k j j=1 k=1 . 34) we can further estimate  2n M M  E ≤ Cn  E Bj,k j j=1 k=1 +E Bj,k j j=1 S2n  2n M M k=1  .

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