## Tag Archives: fourier transform

### Partial sum operator of fourier series and Hilbert transform

$\mathbf{Problem:}$ Let $D_N=\sum\limits_{-N}^{N}e^{inx}$, the partial sum operator of fourier series can be represented by

$\displaystyle S_Nf(x)=\int_{-\pi}^\pi f(t)D_N(x-t)dt=f\ast D_N$

actually $\displaystyle D_N=\frac{\sin(N+\frac{1}{2})x}{\sin\frac{x}{2}}$

$\displaystyle S_Nf(x)=\int_{-\pi}^\pi f(t)\frac{\sin(N+\frac{1}{2})(x-t)}{\sin\frac{x-t}{2}}$

$\displaystyle =\int_{-\pi}^\pi f(t)\frac{\cos(N+\frac{1}{2})x\sin(N+\frac{1}{2})t-\sin(N+\frac{1}{2})x\cos(N+\frac{1}{2})t}{\sin\frac{x-t}{2}}$

$\displaystyle =P.V.\cos(N+\frac{1}{2})x\int_{-\pi}^\pi\frac{f(t)\sin(N+\frac{1}{2})t}{\sin\frac{x-t}{2}}+P.V.\sin(N+\frac{1}{2})x\int_{-\pi}^\pi\frac{f(t)\cos(N+\frac{1}{2})t}{\sin\frac{x-t}{2}}$

$\displaystyle =P.V.\frac{1}{\sin\frac{t}{2}}\ast f_N^1+P.V.\frac{1}{\sin\frac{t}{2}}\ast f_N^2$

Since $\displaystyle |\frac{1}{\sin\frac{t}{2}}-\frac{1}{\frac{t}{2}}|\leq ct$, for $t\in (-\pi,\pi)$, then $S_Nf$ behavior much like Hilbert transform

$\displaystyle Hf(x)=P.V.\int_{-\pi}^\pi\frac{f(x)}{x-t}$

$\mathbf{Thm:}$

(a) $||Hf||_p\leq c_p||f||_p$, $1

(b) $\{x\in(-\pi,\pi)||Hf(x)|>\lambda\}\leq \frac{C||f||_1}{\lambda}$

By the fact in this post, $S_Nf\to f$ in $L^p(-\pi,\pi)$

$\text{Q.E.D}\hfill \square$

$\mathbf{Remark:}$

### Partial sum operator of fourier series and L^p boundedness

Suppose $f\in L^1(-\pi,\pi)$ , consider the fourier series of $f$

$\displaystyle f\sim \sum \limits_{n\in\mathbb{Z}}c_ne^{inx}$

and its partial sum $\displaystyle S_N(f)=\sum \limits_{-N}^Nc_ne^{inx}$

$\mathbf{Problem:}$ The following statements are equivalent

(a)   $||S_Nf||_p\leq c_p||f||_p$, $1

(b)  $||S_Nf-f||_p\to \infty$ as $N\to \infty$

(c)   $||S_Nf||_{p'}\leq ||f||_{p'}$, $\frac{1}{p}+\frac{1}{p'}=1$

$\mathbf{Proof:}$

(a)$\Rightarrow$(b). Since trigonometric polynomial are dense in $L^p$, then we can choose $g$ to be a trigonometric polynomial are sufficiently close to $f$ in $L^p$.

$\displaystyle ||S_Nf-f||_p\leq ||S_Nf-S_Ng||_p+||S_Ng-g||_p+||g-f||_p$

$\displaystyle = ||S_N(f-g)||_p+||S_Ng-g||_p+||g-f||_p$

$\displaystyle \leq c_p||f-g||_p+||S_Ng-g||_p+||g-f||_p\to \infty$

(b)$\Rightarrow$(a). Uniform boundedness principle.

(a)$\Rightarrow$(c) and (c)$\Rightarrow$(a) are quite easy.

$\text{Q.E.D}\hfill \square$

$\mathbf{Remark:}$