A certain filter I'm writing uses two different kernels. The Fejer kernel (which is common) and the Jackson kernel:

$$ \Delta_T(x) = T \,\left( \frac{\sin \pi T x}{\pi T x}\right)^2 \quad\text{and}\quad J_T(x) = \frac{3T}{4} \left( \frac{\sin \pi T x/2}{\pi T x/2}\right)^4 $$

So I had to remind myself how these two kernels behaved and what their properties were:

For a fairly broad class of functions, convolving with the Fejer kernel is the same as integrating against a triangle:

$$ \int_{-T}^T \left( 1 - \frac{|\xi|}{T} \right) \hat{f}(\xi) \,e^{2\pi i \, x \xi} \,d\xi = (f * \Delta_T)(x) $$

Notably, the discrete Fejér kernel (over $S^1$) is a limit of the continuous Fejer kernel (over $\mathbb{R}$):

$$ F_N(x) = \frac{D_0(x) + \dots + D_{N-1}(x)}{N} = \frac{1}{N} \left(\frac{\sin \left(Nx/2\right)}{\sin (x/2)}\right)^2 = \Delta_{1/T}(x)$$

Here it emulates Cesaro summabilition on the Fourier series:

$$ \big(f*F_N\big)(x) = \frac{ S_0(f) (x) + \dots + S_{N-1}(f)(x)}{N} = \frac{1}{N} \sum_{n=0}^{N} \sum_{|x| \leq n} \widehat{f}(x)$$

The similarities continue that the $S^1$ Fejér kernel is the periodization of the $\mathbb{R}$ Fejér kernel:

$$ \sum_{n \in \mathbb{Z}} \Delta_{\frac{1}{N}}(x + n) = F_N(x) $$

These are taken from Fourier Analysis by Elias Stein. My question is what similar properties should occur for the Jackson kernel?

Googling the Jackson kernel doesn't return much. E.g. this note on trigonometric polynomials and modulus of continuity.


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