Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 69 / 4

Solution

Periodicity and the Hölder condition give

∣ f (x - t) - f (x) ∣ \leq M ∣ t ∣^{α}

for

∣ t ∣ \leq π

. The positive normalized Fejér kernel therefore yields

∥ σ_{n - 1} f - f ∥_{\infty} \leq \frac{2 M}{π} \int_{0}^{π} t^{α} F_{n - 1} (t) d t .

The exponential-sum representation of the kernel bounds

F_{n - 1} (t) \leq n /2

. Also

sin (t /2) \geq t / π

for

0 \leq t \leq π

, so

F_{n - 1} (t) \leq π^{2} / (2 n t^{2})

. Splitting at

1/ n

and using

0 < α < 1

gives

\int_{0}^{π} t^{α} F_{n - 1} (t) d t \leq \frac{n}{2} \int_{0}^{1/ n} t^{α} d t + \frac{π ^{2}}{2 n} \int_{1/ n}^{π} t^{α - 2} d t \leq (\frac{1}{2 ( α + 1 )} + \frac{π ^{2}}{2 ( 1 - α )}) n^{- α} .

∥ σ_{n - 1} f - f ∥_{\infty} \leq M (\frac{1}{π ( α + 1 )} + \frac{π}{1 - α}) n^{- α} .

The constant depends on the Hölder exponent and

M

, but not on

n

f

beyond its Hölder seminorm bound.

There are no discussions about this article yet.

There are currently no matching articles.