The Dirichlet kernel is the convolution kernel that represents the Fourier partial sums: $S_N f = f * D_N$. Every question about convergence of [Fourier series](/page/Fourier%20Series) — pointwise, uniform, or in $L^p$ — reduces to a question about $D_N$. The fundamental difficulty of the theory is that $D_N$ is *not* a positive kernel: it oscillates, its $L^1$ norm grows without bound, and it fails to be an approximate identity. This is why Fourier series can diverge even for [continuous](/page/Continuity) [functions](/page/Function), and why the [Fejér kernel](/page/Fejér%20Kernel) (which *is* positive) was introduced as a remedy. [motivation] ## Motivation ### From Partial Sums to Convolution The Fourier partial sum $S_N f(x) = \sum_{|n| \leq N} \hat{f}(n) e^{inx}$ is a finite linear combination of Fourier coefficients, each of which is an [integral](/page/Integral) against $f$. Exchanging sum and integral collapses this into a single integral: $S_N f(x) = \frac{1}{2\pi}\int_{-\pi}^\pi f(x - t) D_N(t) \, d\mathcal{L}^1(t)$, where $D_N$ absorbs the oscillatory sum $\sum e^{int}$. The convergence of $S_N f$ to $f$ is therefore entirely determined by the properties of $D_N$ as a [convolution](/page/Convolution) kernel. ### What Goes Wrong For a convolution $f * \varphi_\varepsilon$ to converge to $f$, the kernel $\varphi_\varepsilon$ should ideally be an [approximate identity](/page/Convolution): positive, unit mass, and concentrating at the origin. The Dirichlet kernel satisfies the second condition (unit mass) but violates the first (positivity) and the third (concentration in $L^1$). Its lobes of alternating sign allow the partial sums to overshoot the function's values — producing the Gibbs phenomenon — and its growing $L^1$ norm means the operator $f \mapsto S_N f$ is unbounded on $C(\mathbb{T})$, leading to the existence of continuous functions whose Fourier series diverge at a point. ### The Role of This Page This page develops the Dirichlet kernel as a standalone object: its definition, closed form, algebraic properties, and the critical $L^1$ norm growth. The consequences for convergence — [Dini's criterion](/theorems/583), Carleson's theorem, the Gibbs phenomenon — are developed on the [Fourier Series (Trigonometric)](/page/Fourier%20Series%20(Trigonometric)) page, which cites the results here. The positive counterpart is the [Fejér kernel](/page/Fejér%20Kernel), whose page explains how averaging cures the pathologies of $D_N$. [/motivation] ## Definition and Closed Form The Dirichlet kernel arises by summing the complex exponentials $e^{inx}$ over the symmetric range $|n| \leq N$. This sum can be evaluated in closed form by the geometric series formula, yielding a ratio of sines. [definition: Dirichlet Kernel] For $N \in \mathbb{N}_0$, the **Dirichlet kernel** of order $N$ is the function $D_N: \mathbb{T} \to \mathbb{R}$ defined by \begin{align*} D_N(x) := \sum_{n=-N}^{N} e^{inx}. \end{align*} [/definition] The sum has $2N + 1$ terms and is manifestly a trigonometric polynomial of degree $N$. The closed form and the convolution property are established by the following theorem. [quotetheorem:581] The closed form $D_N(x) = \sin((N+1/2)x)/\sin(x/2)$ makes the key features of $D_N$ visible: the peak at $x = 0$ has height $2N + 1$ (from L'Hôpital or direct evaluation), and the zeros occur at $x = 2k\pi/(2N+1)$ for $k = 1, \ldots, 2N$ (where $\sin((N+1/2)x) = 0$ but $\sin(x/2) \neq 0$). Between consecutive zeros, $D_N$ alternates in sign. [example: Small Values Of N] For $N = 0$: $D_0(x) = 1$ (the constant function — the "partial sum" of order $0$ is just $\hat{f}(0)$, the average of $f$). For $N = 1$: $D_1(x) = e^{-ix} + 1 + e^{ix} = 1 + 2\cos x$. This oscillates between $-1$ (at $x = \pi$) and $3$ (at $x = 0$), with zeros at $x = \pm 2\pi/3$. For $N = 2$: $D_2(x) = 1 + 2\cos x + 2\cos 2x$. At $x = 0$: $D_2(0) = 5$. At $x = \pi$: $D_2(\pi) = 1 - 2 + 2 = 1$. The function has more oscillations and a taller central peak. The pattern: as $N$ grows, the central peak sharpens (height $2N+1$, width $\sim 1/N$) while the side lobes persist with amplitude $\sim 1/\sin(x/2)$. The kernel concentrates more mass near $x = 0$ but never becomes non-negative. [/example] ## Properties The Dirichlet kernel has several algebraic properties that follow immediately from the definition and the closed form. **Periodicity.** $D_N$ is $2\pi$-periodic (each $e^{inx}$ is). **Evenness.** $D_N(-x) = \sum e^{-inx} = \overline{D_N(x)} = D_N(x)$, where the last equality holds because $D_N$ is real-valued (the closed form is a ratio of real functions for $x \in \mathbb{R}$). **Unit mass.** $\frac{1}{2\pi}\int_{-\pi}^\pi D_N(x) \, d\mathcal{L}^1(x) = 1$, since $\frac{1}{2\pi}\int e^{inx} \, d\mathcal{L}^1(x) = \delta_{n,0}$ and the $n = 0$ term contributes $1$. **Reproduction of trigonometric polynomials.** If $p(x) = \sum_{|n| \leq N} c_n e^{inx}$ is a trigonometric polynomial of degree $\leq N$, then $S_N p = p$ (the Fourier coefficients of $p$ are exactly the $c_n$, and the partial sum recovers them all). Equivalently, $p * D_N = p$: the Dirichlet kernel acts as the identity on the space of trigonometric polynomials of degree $\leq N$. **Non-positivity.** For every $N \geq 1$, there exist points $x$ where $D_N(x) < 0$. This is visible from the closed form: $\sin((N+1/2)x)$ changes sign at $x = k\pi/(N+1/2)$, while $\sin(x/2) > 0$ for $x \in (0, 2\pi)$, so $D_N$ takes negative values. ## The $L^1$ Norm and Its Growth The non-positivity of $D_N$ has a quantitative consequence: the $L^1$ norm $\|D_N\|_{L^1}$ grows without bound. If $D_N$ were non-negative, the unit mass property would force $\|D_N\|_{L^1} = 1$. The oscillations add "wasted" mass that the $L^1$ norm counts but the unit-mass integral cancels. [quotetheorem:589] The logarithmic growth of $\|D_N\|_{L^1}$ has two important consequences. **Unboundedness of the partial sum operator.** The operator norm of $f \mapsto S_N f$ on $C(\mathbb{T})$ equals $\|D_N\|_{L^1}$ (by duality: the supremum of $|S_N f(x)|$ over $\|f\|_\infty \leq 1$ and $x \in \mathbb{T}$ is achieved by choosing $f$ to have the same sign pattern as $D_N(x - \cdot)$). Since $\|D_N\|_{L^1} \to \infty$, the [uniform boundedness principle](/theorems/549) (Banach-Steinhaus) implies there exist continuous functions $f$ for which $\sup_N |S_N f(x_0)| = \infty$ at some point $x_0$. This is the du Bois-Reymond theorem (1876): *there exists $f \in C(\mathbb{T})$ whose Fourier series diverges at a point*. **The Gibbs phenomenon.** Near a jump discontinuity, the partial sums overshoot the function value by approximately $9\%$ of the jump height. The overshoot is caused by the negative lobes of $D_N$: when $f$ jumps, the convolution $f * D_N$ integrates the jump against both the positive and negative parts of $D_N$, producing a ringing artifact. The $9\%$ figure is $\frac{2}{\pi}\operatorname{Si}(\pi) - 1 \approx 0.0895$, where $\operatorname{Si}(\pi) = \int_0^\pi \frac{\sin t}{t} \, d\mathcal{L}^1(t)$. ## Comparison with the Fejér Kernel The Dirichlet kernel's pathologies are precisely what the [Fejér kernel](/page/Fejér%20Kernel) cures. The Fejér kernel $F_N = \frac{1}{N+1}\sum_{n=0}^N D_n$ is the Cesàro average of the first $N+1$ Dirichlet kernels. The averaging cancels the oscillations, producing a non-negative kernel with $\|F_N\|_{L^1} = 1$ (no growth). The price is slower convergence and loss of the reproduction property for trigonometric polynomials of degree exactly $N$ — but the gain is guaranteed [uniform convergence](/page/Uniform%20Convergence) for all continuous functions, which the Dirichlet kernel cannot provide. | Property | $D_N$ | $F_N$ | |---|---|---| | Positivity | No | Yes | | Unit mass | Yes | Yes | | $L^1$ norm | $\sim \frac{4}{\pi^2}\log N$ (growing) | $= 1$ (constant) | | Reproduces degree-$N$ trig. polys. | Yes | No | | $f * K_N \to f$ uniformly for $f \in C(\mathbb{T})$ | Not always | Always | ## References - Katznelson, Y. (2004). *An Introduction to Harmonic Analysis* (3rd ed.). Cambridge University Press. - Grafakos, L. (2014). *Classical Fourier Analysis* (3rd ed.). Springer. - Stein, E. M. and Shakarchi, R. (2003). *Fourier Analysis: An Introduction*. Princeton University Press.