View source for Best approximations, sequence of

A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159102.png" /> of numbers, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159103.png" /> is the [[Best approximation|best approximation]] of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159104.png" /> of a normed linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159105.png" /> by elements of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159106.png" />-dimensional subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159107.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159108.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159109.png" />. Usually, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591010.png" /> is the linear span of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591011.png" /> elements of some fixed system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591012.png" /> of linearly independent elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591013.png" />.

In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591015.png" /> is the subspace of algebraic polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591016.png" />, sequences of best approximations were first considered in the 1850s by P.L. Chebyshev; the fact 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/b/b015/b015910/b01591017.png" /></td> </tr></table>

for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591018.png" /> was established in 1885 by K. Weierstrass. In the general case, the relation

<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/b/b015/b015910/b01591019.png" /></td> </tr></table>

is always satisfied when the union of the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591021.png" /> is everywhere dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591022.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/b/b015/b015910/b01591023.png" /></td> </tr></table>

(essentially, this is an equivalent statement). However, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591024.png" /> may converge to zero arbitrarily slowly. This follows from a theorem of Bernstein: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591025.png" /> is a sequence of subspaces of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591026.png" /> of a normed linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591027.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591029.png" />, then, for any monotone decreasing null sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591030.png" /> of non-negative real numbers, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591033.png" />. In the function spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591035.png" />, the rate at which a sequence of best approximations tends to zero depends both on the system of subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591036.png" /> and on the smoothness characteristics of the approximated function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591037.png" /> (the modulus of continuity, the existence of derivatives up to a specific order, etc.), and it can be estimated in terms of these characteristics. Conversely, knowing the rate of convergence to zero of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591038.png" />, one can draw conclusions with respect to the smoothness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591039.png" /> (see [[Approximation of functions, direct and inverse theorems|Approximation of functions, direct and inverse theorems]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.N. Bernshtein,   "Collected works" , '''2''' , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.L. Goncharov,   "The theory of interpolation and approximation of functions" , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.F. Timan,   "Theory of approximation of functions of a real variable" , Pergamon  (1963)  (Translated from Russian)</TD></TR></table>



====Comments====
Theorems inferring smoothness characteristics of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591040.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591041.png" /> from properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591042.png" /> were first given by D. Jackson in 1911 for algebraic or trigonometric approximation, see [[Jackson theorem|Jackson theorem]]. Theorems converse to these, i.e. inferring properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591043.png" /> from smoothness characteristics of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591044.png" />, have been proved by S.N. Bernstein [S.N. Bernshtein] and A. Zygmund, cf. [[Bernstein theorem|Bernstein theorem]]. See also [[#References|[a2]]], Chapt. 4, Sect. 6 and Chapt. 6, Sect. 3.

====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">  E.W. Cheney,   "Introduction to approximation theory" , Chelsea, reprint  (1982)  pp. 203ff</TD></TR></table>