Bernstein's theorem (approximation theory)




In approximation theory, Bernstein's theorem is a converse to Jackson's theorem.[1] The first results of this type were proved by Sergei Bernstein in 1912.[2]


For approximation by trigonometric polynomials, the result is as follows:


Let f: [0, 2π] → C be a 2π-periodic function, and assume r is a natural number, and 0 < α < 1. If there exists a number C(f) > 0 and a sequence of trigonometric polynomials {Pn}n ≥ n0 such that


degPn=n ,sup0≤x≤2π|f(x)−Pn(x)|≤C(f)nr+α ,{displaystyle deg ,P_{n}=n~,quad sup _{0leq xleq 2pi }|f(x)-P_{n}(x)|leq {frac {C(f)}{n^{r+alpha }}}~,} deg, P_n = n~, quad sup_{0 leq x leq 2pi} |f(x) - P_n(x)| leq frac{C(f)}{n^{r + alpha}}~,

then f = Pn0 + φ, where φ has a bounded r-th derivative which is α-Hölder continuous.



See also



  • Bernstein's lethargy theorem

  • Constructive function theory



References





  1. ^ Achieser, N.I. (1956). Theory of Approximation. New York: Frederick Ungar Publishing Co..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  2. ^ Bernstein, S.N. (1952). Collected works, 1. Moscow. pp. 11–104.









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