Sobol’s book from 1969

Recently a colleague forward me an email with a link to a scanned copy of Sobol’s book from 1969. The link is:

http://www.regmed.ru/getfile.asp?CatId=db&Name=IMSobol_1969_book.pdf The book can be cited using:

• @preamble{
“\def\cprime{$’$} ”
}
@book {Sobol1969,
AUTHOR = {{{\cyr{T}}sobol{\cprime}, I. M.}},
TITLE = {Mnogomernye kvadraturnye formuly i funktsii {K}haara},
PUBLISHER = {Izdat. “Nauka”, Moscow},
YEAR = {1969},
PAGES = {288}
}

The english translation of the book title is ‘Multidimensional quadrature formulae and Haar functions’.

Update 01/12/2010: Professor Sobol provided a response which is added at the end of this entry.

The book consists of two parts. In the first part, the author deals with Haar functions. Here the author considers functions defined on $[0,1].$ These functions are an important tool in the second part of the book which analyses numerical integration of functions defined on $[0,1]^n$ using Sobol sequences. Sobol’s sequences have become a staple tool for numerical integration of high dimensional functions defined on $[0,1]^n,$ where ${}n$ can be large. These are widely used in applications.

Update 01/12/2010 Below is a response from Professor Sobol:

Below is a very brief abstract for my 1969 book, and
I must inform you that the online link http://www.regmed.ru/getfile.asp?CatId=db&Name=IMSobol_1969_book.pdf
will be closed at 29 December 2010.

I.M.Sobol, Multidimensional Quadrature Formulas and Haar Functions, 1969, Moscow, Nauka, 288pp, in Russian.

Authors Summary

The first part of the book contains Haar functions, Fourier-Haar series and their applications in quadrature formulas and approximation problems.
The second part contains multiple Fourier-Haar series, Multidimensional quadrature formulas, exact error bounds for certain classes of integrands, $P_\tau$-nets and $LP_\tau$-sequences, infinite-dimensional quadrature formulas.

Remark. In 1987, $LP_\tau$-sequences got a second name: $(t,s)$-sequences in base ${}2$ (here $t=\tau$ and ${}s$ is the dimension). However, in most papers a third name is used: Sobol sequences. Now in 2010, such sequences having additional uniformity properties remain among the most popular quasi-random sequences.