Talk:Jackson's inequality

L_2 vers. L_\infty

Hi, thanks for the additions to Jackson's thm. Could you please define E_n(f)? Also, why did you focus on the L_2 norm - it is also true in $L_{\infty }$ norm; for L_2 a weaker condition is sufficient (some average modulus of cont.) Thanks, Sasha 12:37, 7 February 2007 (UTC)

In this topic $L_{2}$ and $L_{\infty }$ are extremely different spaces, and they have different Jackson inequalities proved by absolutely different methods with different sharp constants (${1}/{\sqrt {2}}$ and $1$) found by different persons (N. I. Chernych and N. P. Korneychuk respectively). So one cannot say here that one norm is weaker than another. They are simply different.

$L_{p}$ spaces are closer to $L_{2}$ but AFAIK sharp constants are still not found. Stechkin proved constant ${\frac {3}{2}}$ for all of them ($p\in [1,\infty )$), it is very good estimation but not sharp at least at $p=2$. For a brief history see Russian page on the inequality. Mir76 11:57, 8 February 2007 (UTC)

P.S. $E_{n}(f)$ usualy means value of best approximation by polynomials of degree n-1 (or, more general, by linear space of dimension n), i.e. $\inf\{\|f-p\|:p\in P_{n-1}\}$. This is a common abbreviation. Mir76 11:57, 8 February 2007 (UTC)

You are right of course (that $L_{2}$ and $L_{\infty }$ ineqs. are incomparable), but on a finite interval (e.g. for periodic f-ns), $\|\cdot \|_{2}\leq C\|\cdot \|_{\infty }$. Therefore, if on the right-hand side of Jackson's inequality you write the usual modulus of cont., the $L_{\infty }$ form implies the L_2 form (up to some constant). If you wish to write a more precise L_2 ineq, you should replace the modulus of cont. by the so-called L_2 modulus of continuity. Sasha 15:54, 8 February 2007 (UTC)

Actually such "mixed norms" Jackson's inequalites usually are not of much interest since functions should be approximated in their spaces where its modulus of continuity are defined for sure - for example usual (uniform) modulus of continuity doesn't have much sense in $L_{2}$. So weaker is a really wrong word here.

And Stechkin in mentioned 3/2-inequality surely used $L_{p}$-modulus.

Have you read history which I posted to russian page. Or should I translate it? Mir76 17:14, 8 February 2007 (UTC)

Thanks, your survey is very interesting. Here are a few comments:

• regarding the L_2 - L_\infty discussion: it seems I understand what you mean, I agree. Just, please write the precise formulation in the article (if you think it's important enough), with all the definitions.
• I'am certainly not an expert on the subject, almost all I know is from Akhiezer's book (first edition in 1947, 2-nd - in 1965). If you look through Chapter 5 (on harmonic approximation), you see that already then it was a beautiful and well-developed theory (including L_\infty, L_p and other theorems), to which many people contributed. Therefore I think Stechkin deserves a part of the name no more than Kornneichuk, or Bernstein, or even de la Vallee Poussin, or Nagy (or btw Akhiezer himself).
• Especially, the contribution of Bernstein is much more than what you wrote. Bernstein was the first to suggest that approximability is related to properties such as modulus of continuity, and he proved Bernstein's theorem (the reverse of Jackson's thm.), chronologically even before Jackson. Many other ideas in the "constructive theory of functions" originate in his works. Therefore the collective name "Jackson--Bernstein theorems" that is used sometimes is not so bad, to my taste.

Best, Sasha

I have removed the "personal part" about Stechkin, after deciding that it's irrelevant. Sasha 00:58, 10 February 2007 (UTC)