2022-11-13 02:11 pm

Type :Note Tags : Analysis

---

Good Kernels

Definition

Let $\{K_n\}$ be a sequence of functions defined on $\mathbb{T}$ satisfying:

• $\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{K}_n(t)dt=1\forall n\ge 1$
• $\exists M>0,s.t.\underset{-\pi }{\overset{\pi }{\int }}|K_n(t)|\le M$
• $\forall \delta >0,\underset{\delta \le |x|\le \pi }{\int }|K_n(x)|dx\to 0asn\to \infty$

Such a sequence of functions is called a family of good kernels or approximate identity.

---

Properties:

Proposition

If $f$ is integrable over $\mathbb{T}$, and $\{K_n\}$ be a family of good kernels, then: $li{m}_{n\to \infty }(f\ast K_n)(x)=f(x)\forall x$ where $f$ is continuous at x. If $f$ is continuous on $\mathbb{T}$, then the convergence is uniform.

Proof:

• Look at $|(f\ast K_n)(x)-f(x)|$, write it as $\left|\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}(f(x-y)-f(x))K_n(y)dy\right|$
• Split this up into $[-\pi ,-\delta ],[-\delta ,\delta ],[\delta ,\pi ]$
• Show that each of them is small.

---

Proposition

The Dirichlet Kernels are not good kernels.

Proof:

• They violate the second property of good kernels.
• Use $D_N(x)=\frac{sin(N+1/2)x}{sin(x/2)}$
• $A=\int |D_N(x)|\ge 2\int \frac{|sin(N+1/2)x|}{x}dx\ge 4{\int }_0^{\pi }\left|\frac{sin(N+1/2)x}{x}\right|dx$
• change variables, $y=(N+1/2)x$
• $A\ge 4{\int }_0^{(N+1/2)\pi }\frac{|sin(y)|}{|y|}dy$
• break this up into integrals from $k\pi$ to $(k+1)\pi$
• and get a sum of the form 1 + 1/2 + 1/3 … + 1/N
• ${\int }_{-\pi }^{\pi }|D_N(x)|dx\ge c\log (N)asN\to \infty$

---

Related Problems

Examples

Fejer Kernel

---

References

Convolutions Dirichlet Kernels