Dear Vladimir,

I would advise you to kindly refer to the following PDF as a startup 

http://people.ricam.oeaw.ac.at/m.fornasier/CSFornasierRauhut.pdf

It's well explained :)

Dear Vladimir,

I think you can start with this following link 

http://statweb.stanford.edu/~markad/publications/ddek-chapter1-2011.pdf

http://www.math.tu-berlin.de/fileadmin/i26_fg-kutyniok/Kutyniok/Papers/SurveyCS.pdf.

or you can go for any book, I think "Compressed Sensing: Theory and Applications by Yonina C. Eldar, Gitta Kutyniok " will be a good book for you 

I hope that these books will help you

1. Donoho, D.L. ; Dept. of Stat., Stanford Univ., CA

Compressed sensing: http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1614066&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D1614066

2. The Elements of Statistical Learning

Data Mining, Inference, and Prediction

Authors: Trevor Hastie, Robert Tibshirani, Jerome Friedman

ISBN: 978-0-387-84857-0 (Print) 978-0-387-84858-7 (Online)

http://link.springer.com/book/10.1007/978-0-387-84858-7

Best regards, Leonid

You can have a look on the following, useful especially for beginners :

1. https://sites.google.com/site/igorcarron2/cs

2. http://dsp.rice.edu/cs

All the best