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_for some positive constants \(C\) and \(\delta\). Then \(s_{n}(x)\) converges to \(f(x)\) when \(n\) approaches \(\infty\)._
To motivate the definition of the Fejer kernel and a result on uniform convergence called the Fejer theorem, we begin with a simple observation that “averaging preserves convergence.” Suppose we are given a sequence of real numbers \(\{s_{n}\}\) and consider the sequence formed by their arithmetic means (or Cesaro sums)
\[\sigma_{n}=\frac{s_{1}+s_{2}+\cdots+s_{n}}{n}.\]
**Proposition 32**.: _If \(s_{n}\to s\), then \(\sigma_{n}\to s\)._
Proof
: The idea of the proof depends on the fact that if \(\{s_{n}\}\) is convergent, then given \(\epsilon>0\), we can choose \(n\) such that \(|s_{k}-s|<\epsilon\) for all \(k\geq n\). On the other hand if \(\{s_{n}\}\) is convergent, then it is bounded; thus there is an \(M>0\) such that \(|s_{n}|\leq M\) for all \(n\). Fixing this \(n\), one can rewrite \(\sigma_{N}\) for \(N>n\) as
\[\sigma_{N}=\frac{s_{1}+\cdots+s_{n}}{N}+\frac{s_{n+1}+\cdots+s_{N}}{N}\]
and prove that \(s-2\epsilon<\sigma_{N} Next, we take arithmetic means of the Dirichlet kernel to obtain the Fejer kernel. **Definition 84**.: The function \[K_{n}(x)=\frac{1}{n+1}\sum_{k=0}^{n}D_{k}(x)=\frac{1}{n+1}\frac{\sin^{2} \frac{1}{2}(n+1)x}{2\sin^{2}\frac{1}{2}x}\] is known as the _Fejer kernel_. Following figure shows the Fejer kernel when \(n=10\) (Figure 2.19). Note that \(K_{n}\) is an even, non-negative trigonometric polynomial and one of the property of the Dirichlet kernel (see Lemma 11, part b)) implies that \[\frac{1}{\pi}\int_{-\pi}^{\pi}K_{n}(x)dx=1,\quad\text{for}\quad n=1,2,\ldots.\] It is not difficult to see that \(\sigma_{n}(f)\) is a good approximation to \(f\) in the desirable situation of \(L_{2}\) norm. In other words, \(||f-\sigma_{n}(f)(x)||_{2}\to 0\) as \(n\to\infty\) (see Exercise 7 below). The point of the next theorem is that \(\sigma_{n}(f)(x)\) is actually a good _uniform_ approximation to \(f\). We state the Fejer theorem without proof. Its proof could be accessed in most books on Fourier Analysis. For example one can consult [49], p. 53. **Theorem 109** (Fejer theorem).: _Let \(f(x)\) be continuous on \(\mathbb{R}\) and periodic with period \(2\pi\). Let \(\sigma_{n}(x)=\sigma_{n}(f)(x)=\frac{1}{n+1}\sum_{k=0}^{n}s_{k}(x)\) be the arithmetic means of the partial sums of the Fourier series of \(f\). Then_ \[\sigma_{n}(x)\to f(x)\,\text{ uniformly on $\mathbb{R}$ as $n\to\infty$}.\] ### 2.3 Complex Fourier Series and Fourier Transform It is convenient to express the Fourier transform in the notation of complex exponentials \[e^{it}=\cos t+i\sin t.\] For a \(2\pi\) periodic function \(f:\mathbb{R}\to\mathbb{C}\) we define the complex form Fourier series of \(f\) by \[f(t)=\sum_{k=-\infty}^{\infty}c_{k}e^{ikt},\] where now we have a single formula for \(c_{k}\): \[c_{k}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)e^{-ikt}dt\qquad\quad n=0,\pm 1,\pm 2,\ldots.\] Figure 2.19: The Fejer kernel for \(n=10\) Note that \(c_{0}\) is real, i.e., \(c_{0}=c_{-0}=\overline{c_{0}}\), and \(c_{-k}=\overline{c_{k}}\) for \(k=0,1,\dots\), where \(\overline{c_{k}}\) we mean the complex conjugate of \(c_{k}\). The integral of a complex-valued function is defined in terms of the real and imaginary parts, thus we assume that both \(\mathrm{Re}f\) and \(\mathrm{Im}f\) are integrable on \([-\pi,\pi]\). Namely, \(\int f=\int\mathrm{Re}f+\int\mathrm{Im}f\). There are some advantages to this approach; one of which is that complex exponentials form a basis for \(L^{2}\). The exponentials \(e^{ikt}\) are now an orthonormal set. Specifically, we define \[\langle f,g\rangle=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)\overline{g(t)}\,dt,\] and then we have \[\langle e^{int},e^{imt}\rangle=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{int}e^{-imt} dt=\left\{\begin{array}{ll}0&\mathrm{for}\;\;m\neq n\\ 1&\mathrm{for}\;m=n.\end{array}\right.\] Furthermore, with this choice we can define the \(L^{2}\)-norm by \[||f||_{2}=\left(\frac{1}{2\pi}\int_{-\pi}^{\pi}|f(t)|^{2}\,dt\right)^{1/2}.\] The Dirichlet and Fejer kernels are essentially the same in this case, except now we take \(s_{n}(f)(x)=\sum_{k=-n}^{n}c_{k}e^{ikx}\). Loosely speaking, the Fourier transform and its inversion are the limiting case of Fourier series or the continuous analog of Fourier series. Even though some ideas carry over between Fourier series and Fourier transforms, some do not. Fourier transforms are considered to be a simpler approach and have other advantages. In what follows we will be dealing with a complex-valued function \(f(t)\) defined on the real line which is Riemann integrable over each bounded interval. For \(p>0\), we say \(f\in L^{p}\) when \(|f(t)|^{p}\) is integrable over \(\mathbb{R}\). **Definition 85**.: Let \(f\) be integrable, in other words \(\int_{-\infty}^{\infty}|f(x)|dx<\infty\). The Fourier transform of \(f(x)\) is denoted by \(\widehat{f}(x)\) and defined as: \[\widehat{f}(x)=\int_{-\infty}^{\infty}f(t)\,e^{-ixt}\,dt\qquad\quad-\infty The Fourier transform of an integrable function \(f\) plays the role of the sequence of Fourier coefficients and the function can be recovered from its Fourier transform by the formula \[f(t)=\int_{-\infty}^{\infty}\widehat{f}(t)\,e^{ixt}\,dt\qquad\quad-\infty which is a continuous analog of a Fourier series expansion. The following are some simple properties: **Lemma 12**.: _Suppose that \(f\) is integrable, then_ * \(|\widehat{f}(x)|<\infty\)_._
* _The Fourier transform is a linear operator._ : **Example 88**.: Consider the function \[f(t)=\left\{\begin{array}{ll}e^{-ct}&\mbox{for}\;\;t\geq 0\\ 0&\mbox{for}\;\;t<0\end{array}\right.\] for some constant \(c>0\). Then the Fourier transform of \(f\) is \[\widehat{f}(x)=\int_{0}^{\infty}e^{-ixt}\,e^{-ct}\,dt=\int_{0}^{\infty}e^{-ct} (\cos xt-i\sin xt)dt=\frac{c-ix}{c^{2}+x^{2}}.\] Observe that \(\widehat{f}\) is not an integrable function. **Definition 86**.: The _convolution_ of two functions \(f(x)\) and \(g(x)\) in \(L^{1}\) is defined by the integral \[(f*g)(x)=\int_{\mathbb{R}}f(y)g(x-y)dy.\] By changing variables \(y\mapsto x-y\), we may also write \[(f*g)(x)=\int_{\mathbb{R}}f(x-y)g(y)dy\]
Proof
* \(|\widehat{f}(x)|\leq\int_{-\infty}^{\infty}|f(t)|\,dt<\infty\) provided that \(f\) is integrable.
* If \(f\) and \(g\) are integrable functions and \(\alpha,\beta\in\mathbb{C}\) then \(\widehat{\alpha f+\beta}g=\alpha\widehat{f}+\beta\widehat{g}\) follows directly from the definition of Fourier transform.
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