View source for Approximation of functions, direct and inverse theorems

Theorems and equalities establishing a connection between the difference-differential properties of the function to be approximated and the magnitude (and behaviour) of the error of approximation, by various methods. Direct theorems give an estimate of the error of approximation of a function in terms of its smoothness properties (the existence of derivatives of a given order, the modulus of continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a0129601.png" /> or of some of its derivatives, etc.). In the case of best approximation by polynomials, direct theorems are also known as Jackson-type theorems [[#References|[1]]], together with their many generalizations and refinements (see [[Jackson inequality|Jackson inequality]] and [[Jackson theorem|Jackson theorem]]). Inverse theorems characterize difference-differential properties of functions depending on the rapidity with which the errors of best, or any other, approximations tend to zero. The problem of obtaining inverse theorems in the approximation of functions was first stated, and in some cases solved, by S.N. Bernstein [S.N. Bernshtein] [[#References|[2]]]. A comparison of direct and inverse theorems allows one sometimes to characterize completely the class of functions having specific smoothness properties using, for instance, sequences of best approximations.

The connection between direct and inverse theorems is most simple in the periodic case. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a0129602.png" /> be the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a0129603.png" />-periodic continuous functions on the whole real axis with norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a0129604.png" /></td> </tr></table>

let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a0129605.png" /></td> </tr></table>

be the best approximation of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a0129606.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a0129607.png" /> by the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a0129608.png" /> of trigonometric polynomials of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a0129609.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296010.png" /> be the modulus of continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296011.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296013.png" /> be the set of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296015.png" />) that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296016.png" /> times continuously differentiable on the whole real axis. A direct theorem states: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296017.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296019.png" /></td> </tr></table>

where the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296020.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296021.png" />. A stronger assertion is: It is possible to indicate a sequence of linear methods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296023.png" /> associating a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296024.png" /> to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296025.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296026.png" /> and such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296027.png" /> the error <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296028.png" /> is bounded by the right-hand side of (1). An inverse theorem states that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296029.png" />

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296031.png" /> is an absolute constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296032.png" /> is the integer part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296033.png" />. And if for some positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296034.png" /> the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296035.png" /></td> </tr></table>

converges, then it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296036.png" /> and that, analogous to (2), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296037.png" /> can be estimated in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296039.png" /> (see [[#References|[4]]], [[#References|[8]]] and [[#References|[12]]]). These estimates imply, in particular, that if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296040.png" /></td> </tr></table>

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296041.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296042.png" /> satisfy for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296043.png" /> the Hölder inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

whereas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296045.png" /> they satisfy the Zygmund inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

Denoting this class of functions by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296047.png" />, one obtains a constructive characteristic for it: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296048.png" /> if and only if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296049.png" /></td> </tr></table>

A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296050.png" />-periodic function is infinitely differentiable on the whole real axis if and only if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296051.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296052.png" />.

Similar properties hold for the approximation of periodic functions in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296053.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296054.png" />), and also for (not necessarily periodic) functions defined on the whole real axis when being approximated by entire functions of finite order (see [[#References|[7]]] and [[#References|[8]]]). Direct and inverse theorems are known for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296055.png" />; as a difference-differential characteristic the modulus of smoothness <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296056.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296057.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296058.png" /> (or of some of its derivatives) is used (see [[#References|[4]]] and [[#References|[8]]]).

The situation is different in the case of approximation on a finite interval. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296059.png" /> be the space of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296060.png" />, with norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296061.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296062.png" /> be the set of the functions that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296063.png" /> times continuously differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296065.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296066.png" /> be the class of functions defined by the inequalities (3) and (4) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296067.png" />. For the best approximation,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296068.png" /></td> </tr></table>

of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296069.png" /> by the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296070.png" /> of algebraic polynomials of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296071.png" /> one has an estimate of the form (1) in terms of the modulus of continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296072.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296073.png" />, but the converse, analogous to that of the periodic case (with an inequality of the form (2)), now only applies for intervals contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296074.png" />. For instance, if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296075.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296076.png" /></td> </tr></table>

then one can only assert that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296077.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296078.png" /> defined by the inequalities (3) (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296079.png" />) merely on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296080.png" />; the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296081.png" /> depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296082.png" />, and can increase unboundedly as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296083.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296084.png" />. There exist functions not belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296085.png" /> for which nevertheless

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296086.png" /></td> </tr></table>

For instance

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296087.png" /></td> </tr></table>

although <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296088.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296089.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296090.png" />. It turned out that algebraic polynomials, retaining on the whole interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296091.png" /> the best order of approximation of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296092.png" />, can yield at the end points of the interval a substantially better approximation. This phenomenon was first discovered by S.M. Nikol'skii (see [[#References|[3]]]). If, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296093.png" />, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296094.png" /> there is a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296095.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296096.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296097.png" /></td> </tr></table>

where the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296098.png" /> depends neither on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a01296099.png" /> nor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a012960100.png" />. Contrary to (5), this assertion has a converse: If for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a012960101.png" /> there exists a sequence of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a012960102.png" /> such that for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a012960103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a012960104.png" /> (6) is satisfied, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a012960105.png" />. Direct and inverse theorems for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012960/a012960106.png" /> are known, stated in terms of the modulus of continuity and moduli of smoothness (see [[#References|[4]]] and [[#References|[8]]]).

Direct theorems giving order estimates for the error of approximation in terms of difference-differential properties of the approximated function were established for many concrete approximation methods (see [[#References|[6]]], [[#References|[8]]] and [[#References|[9]]], in particular for splines, splines of best approximation and interpolation [[#References|[10]]], cf. [[Spline|Spline]]).

Direct and inverse theorems for approximation in a Hausdorff metric are known (see [[#References|[13]]]). Some peculiarities arise here; in particular, the characterization of classes of functions by their best Hausdorff approximation is connected not only with the order of this approximation but also with the magnitude of the constant in the relevant inequality. For direct and inverse theorems in the multi-dimensional case, see [[Approximation of functions of several real variables|Approximation of functions of several real variables]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Jackson,   "Ueber die Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung" , Göttingen  (1911)  (Thesis)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.N. Bernshtein,   "On the best approximation of continuous functions by polynomials of a given degree (1912)" , ''Collected works'' , '''1''' , Moscow  (1952)  pp. 11–104</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,   ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''10''' :  4  (1946)  pp. 295–317</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.K. Dzyadyk,   "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.P. Korneichuk,   "Extremal problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.M. Tikhomirov,   "Some problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,   "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.F. Timan,   "Theory of approximation of functions of a real variable" , Pergamon  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  P.P. Korovkin,   "Linear operators and approximation theory" , Hindushtan Publ. Comp.  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  S.B. Stechkin,   Yu.N. Subbotin,   "Splines in numerical mathematics" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  I.K. Daugavet,   "Introduction to the theory of approximation of functions" , Leningrad  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  S.A. Stechkin,   ''Izv. Akad. Nauk SSSR Ser.'' :  3  (1951)  pp. 217–242</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  B. Sendov,   "Hausdorff approximations" , Sofia  (1979)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  G.G. Lorentz,   "Approximation of functions" , Holt, Rinehart &amp; Winston  (1966)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.P. Natanson,   "Constructive function theory" , '''1–3''' , F. Ungar  (1964–1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.A. DeVore,   "The approximation of continuous functions by positive linear operators" , Springer  (1972)</TD></TR></table>