# Preface

Since a Fourier series may fail to converge at individual points, we are led to try to overcome this failure by interpreting the limit of partial sums in a different sense. This section presents an introduction to Cesàro summation and its application to Fourier series.

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Introduction to Linear Algebra with *Mathematica*

## Glossary

# Cesàro summation

Fourier series allow us to represent a rather complicated function defined on a finite interval as a linear combination (which is usually infinite) of its projections onto a basis that consists of trigonometric functions. Such a compact representation has proven exceedingly useful in the analysis of many real-world systems involving periodic phenomena, such as waves propagating on a string, electrical circuits with oscillating current sources, and heat diffusion on a metal ring---an application we will later examine in detail. More generally, Fourier series usually arise in the ubiquitous context of boundary value problems, making them a fundamental tool among mathematicians, scientists, and engineers.

However, there lies a caveat. Except in degenerate cases, a Fourier series (more precisely, its finite sum truncation version,

*f*?

One notion of convergence between functions is 𝔏²-convergence, or convergence in the square mean. We consider only functions defined on a finite interval of length 2ℓ. Once we represent this function as a Fourier series, we get its periodic extension with period *T* = 2ℓ. Therefore, it is convenient to assume that the original function is a periodic function. Then its convergent Fourier series gives exact equality for the given function. For a 2ℓ-periodic function *f*, we have 𝔏²-convergence of the Fourier series *F*_{N} if

One of the first results regarding Fourier series convergence is that if *f* is square-integrable (that is, if \( \int_{-\ell}^{\ell} \left\vert f(x) \right\vert^2 {\text d} x < \infty \) ), then its Fourier series 𝔏²-converges to *f*. This is a nice result, but it leaves more to be desired. 𝔏²-convergence only says that over an interval of length 2ℓ, the average deviation between *f* and its Fourier approximation *F*_{N} must tend to zero. However, for a fixed *x* in \( (-\ell , \ell ) , \) there are no guarantees on the difference between *f*(*x*) and the series approximation at *x*.

A stronger---and quite natural---sense of convergence is pointwise convergence, in which
we demand that at each point \( x \in (-\ell ,\ell ), \) the series approximation converges to *f*(*x*). The Pointwise Convergence Theorem then states that if *f* is sectionally continuous and *x*_{0} is such that the one-sided derivatives \( f' (x_0 +0) = \lim_{\epsilon \,\mapsto 0, \epsilon >0} \, f(x_0 +\epsilon ) \) and \( f' (x_0 -0) = \lim_{\epsilon\, \mapsto 0, \epsilon >0} \, f(x_0 -\epsilon ) \) both exist, then the Fourier series converges to \( f (x_0 ) . \) However, in cases when the function experiences a finite jump of discontinuity pointwise convergence is not an appropriate notion of convergence as Gibbs phenomenon shows.

To avoid such problems, we desire the even stronger notion of uniform convergence, such that the rate at which the series converges is identical for all points in \( [-\ell , \ell ] , \) and consequently, everywhere (due to periodicity). By adopting the metric

To define uniform convergence of Fourier series, we need a more general
definition of convergence for infinite sum, that is known as
**Cesàro summation**, named after the Italian analyst
Ernesto Cesàro (1859--1906). Recall the definition
of Cesàro summability of infinite series (which is capable
of finding sums of divergent series).

*n*th partial sum of the series

*k*= 1 or

*k*= 0. In both cases, we will call

*s*

_{n}or

*S*

_{n}

*n*-th partial sum of the given infinite series despite that

*S*

_{N}is the sum of

*n*+ 1 terms. The series \( \sum_{k\ge 1} a_k \quad \mbox{or} \quad \sum_{k\ge 0} a_k \) is called

**Cesàro summable**, with Cesàro sum \( A \in \mathbb{R} \quad \mbox{or} \quad A \in \mathbb{C} , \) if the average value of its partial sums tends to

*A*:

*n*partial sums of the series, as

*n*goes to infinity.

If a series is convergent, then it is Cesàro summable and its Cesàro sum is the usual sum. For any convergent sequence, the corresponding series is Cesàro summable and the limit of the sequence coincides with the Cesàro sum. The Fourier series in complex form \( f(x) \,\sim \,\lim_{N\,\to \infty} \,\sum_{k=N}^N \alpha_k e^{k{\bf j} \pi x/\ell} \) is Cesàro summable to

*A*if and only if

*f(x)*has the Cesàro sum

*A*if the following limit exists:

*f(x)*has the Cesàro sum

*A*if the following limit exists:

**Example:**
Consider the infinite series \( 1-1+1-1+1- \cdots , \) also written

Indeed, its partial sums are

*n*. In any way, the limit is

**Example 2:**It is well-known that the geometric series has the sum value when its general term is less than 1:

*Mathematica*confirms:

*q*has absolute value less than 1. In the previous example of Grandi, we found the sum of geometric series for

*q*= −1. Now we are going to extend it for a general case. According to Cesàro, the sum of the geometric series is

*Mathematica*, we find

*q*| ≤ 1 and

*q*≠ 1 (to avoid zero in the denominator):

We are now primed to appreciate Fejér’s remarkable theorem that was proved in 1899 by a Hungarian mathematician Lipót Fejér (1880--1959). He was born into a Jewish family as Leopold Weiss. He was the chair of mathematics at the University of Budapest since 1911 and led a highly successful Hungarian school of analysis. He was the thesis adviser of mathematicians such as John von Neumann, Paul Erdős, George Pólya, Marcel Riesz, Gábor Szegő, and Pál Turán.

**Let \( f\,: \,[-\ell , \ell ]\, \mapsto \,\mathbb{R} \) be a continuous function with \( f(-\ell )= f(\ell ) . \) Then the Fourier series of**

**Theorem (Lipót Fejér, 1899)**:*f*(C,1)-converges to

*f*in \( C([-\ell , \ell ], \mathbb{R}) , \) where \( C([-\ell , \ell ], \mathbb{R}) \) is the metric space of continuous functions from \( [-\ell , \ell ] \) to the set of real numbers. ▣

**Corollary:**If

*f*is integrable on some finite interval, then the Fourier series of

*f*is Cesàro summable to

*f*at every point of continuity of

*f*. Moreover, if

*f*is continuous on the interval, then the Fourier series of

*f*is uniformly Cesàro summable to

*f*.

Without imposing any additional conditions on *f* aside from being continuous and periodic, Fejér’s theorem shows that Fourier series can still achieve uniform convergence, granted that we instead of regular convergence consider the arithmetic means of partial Fourier sums.

A more general form of the theorem applies to functions which are not necessarily continuous. Suppose that *f* is absolutely integrable on the finite interval \( [-\ell , \ell ] . \) If the left and right limits \( f(x_0 \pm 0) \mbox{ of } f(x) \) exist at *x*_{0}, or if both limits are infinite of the same sign, then

**If**

**Lemma:***f*is integrable on the interval [-ℓ,ℓ], then Fourier series in Cesàro sense

*K*

_{n}(0) = (

*n*+1)

^{-1}. ▣

**Example 3:**Let

*f*be the periodic function defined on the interval [-1,1]:

*f*(0) = 1, we have

*n = 20*terms for Fourier series and Cesàro one, we don't see Gibbs phenomenon in Cesàro partial sums:

alphak = (1/2)*Integrate[Exp[I*k*Pi*x], {x, -1/2, 1/2}]]

an = Integrate[Cos[k*Pi*x], {x, -1/2, 1/2}]

cos[m_] = 1/2 + (2/Pi)* Sum[(1/k)*Sin[k*Pi/2]*Cos[k*Pi*x], {k, 1, m}]

cecos[m_] = 1/2 + (2/Pi)* Sum[(1/k)*(1 - k/(m + 1))*Sin[k*Pi/2]*Cos[k*Pi*x], {k, 1, m}]

Plot[{cecos[20], f[x]}, {x, -2, 2}, PlotStyle -> {{Thick, Blue}, {Thick, Orange}}]

complex[m_] = Sum[alphak*Exp[k*Pi*I*x], {k, -m, m}]

cesaro[m_] = 1/2 + Sum[alphak*Exp[k*Pi*I*x]*(1 - Abs[k]/(m + 1)), {k, 1, m}] + Sum[alphak*Exp[k*Pi*I*x]*(1 - Abs[k]/(m + 1)), {k, -m, -1}]

Plot[{cesaro[10]}, {x, -3, 3}, PlotStyle -> {Thick, Red}]

**Example 4:**Let \( f(x) = \mbox{sign}(x) -x \) on the interval [-1,1]. Expanding this odd function into sine-Fourier series, we get

*n = 20*terms, we don't observe Gibbs phenomenon in the latter one.

bn=Integrate[f[x]*Sin[k*Pi*x], {x, -1, 1}]

sin[m_] = (2/Pi)*Sum[(1/k)*Sin[k*Pi*x], {k, 1, m}]

cesin[m_] = (2/Pi)*Sum[(1/k)*(1 - (k-1)/m)*Sin[k*Pi*x], {k, 1, m}]

Plot[{sin[20], f[x]}, {x, -2, 2}, PlotStyle -> {{Thick, Blue}, {Thick, Orange}}]

Plot[{cesin[20], f[x]}, {x, -2, 2}, PlotStyle -> {{Thick, Blue}, {Thick, Orange}}]

**Example 5:**Let \( f(x) = 4x^3 -3x^2 -6x \) on the interval [-2,2]. Expanding this function into Fourier series, we get

a0=Integrate[f[x], {x, -2, 2}]/2

ak=Integrate[f[x]*Cos[k*Pi*x/2]/2, {x, -2, 2}]

bk=Integrate[f[x]*Sin[k*Pi*x/2]/2, {x, -2, 2}]

fourier[m_]:=-4+Sum[ak*Cos[k*Pi*x/2] + bk*Sin[k*Pi*x/2], {k,1,m}]

*n = 20*terms:

pp = Plot[{f[x], fourier[20]}, {x, -3, 3}, PlotStyle -> {{Thick, Blue}, {Thick, Red}}, Epilog -> {Text[11.58, {-.66, 11.579}], Text[-35.58, {-.66, -35.58}]}]

p = Graphics[{Green, PointSize[Large], {Point[{0, 11.57959}], Point[{0, -35.58}]}}]

Show[pp, p]

*x = 2*and

*x = -2*. Since \( f(2+0) = -32 \quad\mbox{and}\quad f(2-0) =8 , \) the given function has finite jump of 40 at both points of discontinuity. Therefore, we expect overshoot/undershoot by the value \( (1.1789797444721675*40 - 40)/2 \approx 3.579594889 . \) As a result, we expect overshoot at points of discontinuity to be about 11.57959 and undershoot to be around -35.5796. Notably, we do not observe the Gibbs phenomenon in the Cesaro approximation.

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