In
approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' wil ...
, Jackson's inequality is an inequality bounding the value of function's best approximation by
algebraic or
trigonometric polynomials In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...
in terms of the
modulus of continuity In mathematical analysis, a modulus of continuity is a function ω : , ∞
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
→ , ∞
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if and only if
:, ...
or
modulus of smoothness
In mathematics, moduli of smoothness are used to quantitatively measure smoothness of functions. Moduli of smoothness generalise modulus of continuity and are used in approximation theory and numerical analysis to estimate errors of approximation ...
of the function or of its derivatives.
Informally speaking, the smoother the function is, the better it can be approximated by polynomials.
Statement: trigonometric polynomials
For trigonometric polynomials, the following was proved by
Dunham Jackson
Dunham Jackson (July 24, 1888 in Bridgewater, Massachusetts – November 6, 1946) was a mathematician who worked within approximation theory, notably with trigonometrical and orthogonal polynomials. He is known for Jackson's inequality. He w ...
:
:Theorem 1: If
is an
times differentiable
periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to d ...
such that
::
:then, for every positive integer
, there exists a
trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...
of degree at most
such that
::
:where
depends only on
.
The
Akhiezer–
Krein–
Favard theorem gives the sharp value of
(called the
Akhiezer–Krein–Favard constant):
:
Jackson also proved the following generalisation of Theorem 1:
:Theorem 2: One can find a
trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...
of degree
such that
::
:where
denotes the
modulus of continuity In mathematical analysis, a modulus of continuity is a function ω : , ∞
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
→ , ∞
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if and only if
:, ...
of function
with the step
An even more general result of four authors can be formulated as the following Jackson theorem.
:Theorem 3: For every natural number
, if
is
-periodic continuous function, there exists a
trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...
of degree
such that
::
:where constant
depends on
and
is the
-th order
modulus of smoothness
In mathematics, moduli of smoothness are used to quantitatively measure smoothness of functions. Moduli of smoothness generalise modulus of continuity and are used in approximation theory and numerical analysis to estimate errors of approximation ...
.
For
this result was proved by Dunham Jackson.
Antoni Zygmund
Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century. ...
proved the inequality in the case when
in 1945.
Naum Akhiezer
Naum Ilyich Akhiezer ( uk, Нау́м Іллі́ч Ахіє́зер; russian: link=no, Нау́м Ильи́ч Ахие́зер; 6 March 1901 – 3 June 1980) was a Soviet and Ukrainian mathematician of Jewish origin, known for his works in appr ...
proved the theorem in the case
in 1956. For
this result was established by
Sergey Stechkin
Sergey Borisovich Stechkin (russian: Серге́й Бори́сович Сте́чкин) (6 September 1920 – 22 November 1995) was a prominent Soviet mathematician who worked in theory of functions (especially in approximation theory) and n ...
in 1967.
Further remarks
Generalisations and extensions are called Jackson-type theorems. A converse to Jackson's inequality is given by
Bernstein's theorem. See also
constructive function theory In mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation. It is closely related to approximation theory. The term was coined by Sergei Bernste ...
.
References
External links
*
*
Approximation theory
Inequalities
Theorems in approximation theory
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