**Mathematical Surveys and Monographs**

Volume: 17;
1980;
160 pp;
Hardcover

MSC: Primary 41;

Print ISBN: 978-0-8218-1517-5

Product Code: SURV/17

List Price: $60.00

AMS Member Price: $48.00

MAA Member Price: $54.00

**Electronic ISBN: 978-1-4704-1244-9
Product Code: SURV/17.E**

List Price: $60.00

AMS Member Price: $48.00

MAA Member Price: $54.00

# Approximation by Polynomials with Integral Coefficients

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*Le Baron O. Ferguson*

Results in the approximation of functions by polynomials with
coefficients which are integers have been appearing since that of Pál in
1914. The body of results has grown to an extent which seems to justify this
book. The intention here is to make these results as accessible as
possible.

The book addresses essentially two questions. The first is the question of
what functions can be approximated by polynomials whose coefficients are
integers and the second question is how well are they approximated (Jackson
type theorems). For example, a continuous function \(f\) on the interval
\(-1,1\) can be uniformly approximated by polynomials with integral
coefficients if and only if it takes on integral values at \(-1,0\) and
\(+1\) and the quantity \(f(1)+f(0)\) is divisible by
\(2\). The results regarding the second question are very similar to the
corresponding results regarding approximation by polynomials with arbitrary
coefficients. In particular, nonuniform estimates in terms of the modules of
continuity of the approximated function are obtained.

Aside from the intrinsic interest to the pure mathematician, there is the
likelihood of important applications to other areas of mathematics; for example,
in the simulation of transcendental functions on computers. In most computers,
fixed point arithmetic is faster than floating point arithmetic and it may be
possible to take advantage of this fact in the evaluation of integral
polynomials to create more efficient simulations. Another promising area for
applications of this research is in the design of digital filters. A central
step in the design procedure is the approximation of a desired system function
by a polynomial or rational function. Since only finitely many binary digits of
accuracy actually can be realized for the coefficients of these functions in
any real filter the problem amounts (to within a scale factor) to approximation
by polynomials or rational functions with integral coefficients.