Fourier Transform 18 | Dirichlet Kernel [dark version]

TL;DR

Dirichlet kernel D_n is defined as the symmetric exponential sum ∑_{k=−n}^{n} e^{ikx}, and it can be rewritten using cosine sums or a sine-fraction formula.

Briefing Cornell Notes

Briefing

Dirichlet kernel D_n sits at the heart of Fourier series: it turns a partial Fourier sum into an integral (or convolution/inner product) built from D_n, making the kernel the main mechanism behind how Fourier series approximate functions. For each natural number n, D_n(x) is defined as a symmetric finite sum of complex exponentials from k = −n to n, and it can be rewritten in equivalent real forms using cosines or a closed expression involving sine terms. The kernel is 2π-periodic, continuous on ℝ, and its shape becomes increasingly oscillatory as n grows—especially near x = 0 where the central peak sharpens.

A key identity links Fourier partial sums directly to the kernel. If f is a fixed function and the Fourier series is truncated at n, the resulting continuous approximation at a point x can be written as (1/2π) ∫_{−π}^{π} f(y) D_n(x − y) dy. Because both f (in the periodic setting) and D_n are 2π-periodic, variable shifts don’t change the value, letting the same expression be recast as an L^2 inner product: the partial sum equals ⟨D_n, f(· + x)⟩ (equivalently, ⟨D_n(· − x), f⟩ depending on the chosen shift). This also yields a convolution form, D_n * f evaluated at x. These representations explain why the kernel is called a “kernel”: it sits inside the averaging/integration that produces the Fourier approximation.

The transcript then develops three properties that foreshadow both the power and the trouble of Fourier series. First, the zeros of D_n can be counted from the sine-based closed form: on (−π, π) the kernel has exactly 2n zeros, with the apparent zero at the origin canceled by the removable singularity in the sine-fraction representation. Second, D_n is normalized in the sense that integrating it against the constant function 1 returns 1 for every n; this follows because D_n is a sum of 2π-periodic exponentials whose integrals vanish except for the constant term. Third—and most consequential—the L^1 “mass” of the kernel grows without bound: ∫_{−π}^{π} |D_n(x)| dx → ∞ as n → ∞. The argument uses the sine-fraction form, symmetry to restrict to x ≥ 0, and estimates that compare |sin((2n+1)x/2)|/|sin(x/2)| to a simpler bound. The resulting lower bound reduces to a divergent harmonic-sum behavior, showing the kernel’s absolute area increases like a logarithmic divergence.

That unbounded growth is the warning label for pointwise convergence: even though D_n is normalized and the partial sums are continuous, the kernel becomes more extreme as n increases. The transcript closes by noting that these kernel properties are exactly what will be needed to prove pointwise convergence in the next installment.

Cornell Notes

The Dirichlet kernel D_n is the central object behind Fourier series. Defined as a symmetric sum of exponentials from k = −n to n, it has equivalent real forms (cosine sums and a sine-fraction expression) and is 2π-periodic and continuous. A truncated Fourier series at a point x can be written using D_n inside an integral: (1/2π)∫_{−π}^{π} f(y)D_n(x−y)dy, which can also be expressed as an inner product or convolution. D_n has three crucial properties: it has exactly 2n zeros in (−π, π), it is normalized so that integrating it against the constant function 1 gives 1, and its absolute integral ∫|D_n| grows unboundedly as n→∞. That last fact helps explain why pointwise convergence is subtle.

How is the Dirichlet kernel D_n defined, and why does it end up real-valued?

D_n(x) is defined for each natural number n as the symmetric finite sum of complex exponentials: D_n(x)=∑_{k=−n}^{n} e^{ikx}. The symmetry k↔−k pairs conjugate terms, so the imaginary parts cancel and the result is real. The transcript also notes equivalent representations: a cosine form “1 + 2 times a sum of cosines,” and a sine-fraction form involving sign/sine terms (with removable exceptions handled by continuity).

What identity turns a partial Fourier sum into an integral involving D_n?

For a fixed function f, the Fourier series truncated at n evaluated at x can be written as (1/2π)∫_{−π}^{π} f(y) D_n(x−y) dy. This comes from expressing the partial sum as a finite exponential sum with coefficients c_k computed by an integral, then regrouping the exponentials so the inner sum becomes D_n(x−y). Because D_n and the periodic setting use 2π-periodicity, shifting variables (e.g., replacing x−y by a new variable) leaves the value unchanged.

Why does the kernel’s normalization matter, and how is it shown?

Normalization means D_n acts like an averaging kernel for constants: in the L^2 inner product with the constant function 1, the result is 1 for every n. The reason is structural: D_n is a sum of 2π-periodic exponentials, and integrating each exponential over [−π, π] gives 0 except for the constant term. The constant term contributes exactly 2π, which cancels the (1/2π) factor in the inner-product/integral setup.

How many zeros does D_n have on (−π, π), and what happens at x=0?

Using the sine-fraction representation, the zeros correspond to zeros of the numerator sine term. In (−π, π) there are 2n+1 zeros from the numerator, but the origin is a removable exception: the numerator and denominator both vanish there, and the continuous extension removes the zero at x=0. The net result is exactly 2n zeros in (−π, π).

Why does ∫_{−π}^{π} |D_n(x)| dx blow up as n→∞?

Absolute value prevents cancellation of positive and negative oscillations. The transcript estimates |D_n(x)| using the sine-fraction form: compare the numerator |sin((2n+1)x/2)| to a simpler linear lower bound on [0, π], then bound the denominator by its maximum over each interval. Splitting the integral into pieces between consecutive zeros leads to a lower bound that behaves like a constant times ∑_{k=1}^{n} 1/k, the harmonic series. Since the harmonic sum diverges, the absolute integral tends to infinity.

Review Questions

What three equivalent forms of D_n are mentioned, and how do they relate to real-valuedness and 2π-periodicity?
How does the identity (1/2π)∫_{−π}^{π} f(y)D_n(x−y)dy connect Fourier partial sums to convolution/inner products?
Which property of D_n most directly signals why pointwise convergence of Fourier series is difficult, and what divergence does it produce?

Key Points

1
Dirichlet kernel D_n is defined as the symmetric exponential sum ∑_{k=−n}^{n} e^{ikx}, and it can be rewritten using cosine sums or a sine-fraction formula.
2
A truncated Fourier series at x can be written as (1/2π)∫_{−π}^{π} f(y)D_n(x−y)dy, showing D_n is the kernel inside the averaging integral.
3
Because of 2π-periodicity, variable shifts in the kernel-based integral do not change the value, enabling equivalent inner-product and convolution formulations.
4
D_n has exactly 2n zeros in (−π, π); the apparent zero at x=0 is removed by the continuous extension of the sine-fraction expression.
5
D_n is normalized: integrating D_n against the constant function 1 yields 1 for every n (in the appropriate inner-product/integral sense).
6
The absolute integral ∫_{−π}^{π} |D_n(x)| dx diverges as n→∞, with a lower bound tied to the divergent harmonic sum ∑_{k=1}^{n} 1/k.
7
The unbounded growth of the kernel’s L^1 mass helps explain why Fourier series convergence at individual points is subtle rather than automatic.

Highlights

D_n turns Fourier partial sums into an explicit kernel integral: (1/2π)∫_{−π}^{π} f(y)D_n(x−y)dy.

The kernel is normalized against constants, yet its absolute area grows without bound—an unusual combination that drives convergence difficulties.

On (−π, π), D_n has exactly 2n zeros, with the origin handled as a removable singularity in the sine-fraction form.

The divergence of ∫_{−π}^{π} |D_n(x)| dx is linked to a harmonic-series lower bound, showing how oscillations stop canceling once absolute values are taken.

Topics

Dirichlet Kernel
Fourier Series
Pointwise Convergence
Kernel Representation
Normalization