I am reading Rudin's Principle of mathematical analysis, in the section 'Stone-Weierstrass theorem', there is a steps that I don't understand:

$$ P_{n}(x)=\int_{-1}^{1} f(x+t)Q_{n}(t) dt $$ by changing variable, $$ P_{n}(x)=\int_{-x}^{1-x} f(x+t)Q_{n}(t) dt $$

I don't understand what Rudin did here, could anyone explain it to me?

  • 2
    $\begingroup$ There are assumptions on $f$ that you should also include here. (If I recall correctly, $f$ is supported on $[0,1]$ or something like that.) $\endgroup$ – BigbearZzz Nov 19 '15 at 16:12
  • $\begingroup$ $f$ is on [0,1] and $f(0)=f(1)=0$. Also $f$ is defined to be 0 outside [0,1]. I think (not sure) the last one is irrelevant here. $\endgroup$ – Cyriac Antony Apr 22 '18 at 6:47
  • $\begingroup$ $Q_n(t)=Q_n(-t)$ since $Q_n(s):=c_n(1-x^2)^n$. May be we need this. $\endgroup$ – Cyriac Antony Apr 22 '18 at 7:09
  • $\begingroup$ Already answered in another question. The answer(math.stackexchange.com/a/179822/120721) uses the fact that $f$ is defined to be 0 outside [0,1]. $\endgroup$ – Cyriac Antony Apr 22 '18 at 7:19

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