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Introduction to Linear Algebra with Mathematica
One of the main issues of harmonic analysis is a possibility of
restoring a function from its Fourier coefficients. It turns out that this problem
is an
ill-posed problem. Therefore, practical applications of Fourier series may require a regularization, which is related to the scrutiny of
convergence of Fourier series. This topic is known as classical harmonic analysis, a branch of pure mathematics---it will be discussed further in next section and in section devoted to Cesàro summation.
Historically, Fourier series was the first example of expansion of a function
with respect to eigenfunctions corresponding to a Sturm--Liouville problem.
Before Fourier's discovery,
Taylor series were mostly in use; however, their expansions require an utilization of uniform and absolute convergence inside the disk, and pointwise convergence on its boundary. Taylor's series coefficients
are defined by derivatives evaluated at one point---the center of convergence, so only
infinitesimal information is used for their determination. In opposite,
eigenfunction expansions are based on the integration over whole interval.
This caused development of another methods: 𝔏² convergence, summability, and the the Cesàro mean.
This section is about some conditions that guarantee pointwise convergence of (formal) Fourtier series S[f] to f(x) either pointwise or uniformly.
There are known many sufficient conditions that guarantee convergence of Fourier series to the generating it function. However, we discuss only two important tests, known as Dirichlet and Dini, that provide pointwise convergence. If a function satisfies a Hölder condition, then its Fourier series converges uniformly.
where S[f] denotes a formal Fourier series, whose
Fourier coefficients \( \hat{f}(k) \) and 𝑎_{n}, b_{n} are determined through Euler--Fourier formulas
Here T = 2ℓ is the period and «P.V.» abbreviates the Cauchy principle value. As usual, j denotes the imaginary unit vector in the positive vertical direction on complex plane ℂ, so j² = −1. These coefficients are related via the Euler formula:
There are two standard notations to represent a complex conjugate: either with overline or with asterisk, so we utilize both.
Series \eqref{EqConverge.1}
is an example of an
expansion of f(x) with respect to eigenfunctions corresponding to a Sturm--Liouville problem (see section for detail). Since this problem contains periodic boundary conditions, this expansion reflects this property. Therefore, we will always assume that a function f(x) is periodically expanded to ℝ with the same period T = 2ℓ to match the boundary condition: f(x) = f(x + T).
Example 1:
We consider a 2π-periodic function whose Fourier series diverges. This function was constructed by Lipót Fejér in 1900.
Kolmogorov, A., Une serie de Fourier--Lebesgue divergente presque partout, Fundamenta Mathematicae, 1923, 4, 324--328.
■
Since Fourier series S[f] of a function f(x) is an example of infinite series, its convergence (or justification) depends on a rule how its partial sums
approach a limit. Therefore, we need a compact formula for finite partial sums S_{N}(f;, x); fortunately, such formula is known and it is based on Fourier secomposition of the Dirac delta function.
We expand the delta-function into Fourier series, Using Euler--Fourier formulas, we get
for a smooth function f(x). Actually there is a neat formula for the partial sum δ_{N}(x). It is called the Dirichlet kernel after the German scientist who first dicovered it in 1824.
Lemma 1:
The following trigonometric identity holds:
Before we get into the topic of convergence, we need to define some classes of functions that are sufficient for Fourier expansions and wide enough to include the majority of practical applications. First, we extend the set of continuous functions on the interval [−ℓ, ℓ], which is usually denoted as C[−ℓ, ℓ], that have an additional condition f(−ℓ) = f(ℓ). We need to consider piecewise continuous functions that consist of finite number of continuous pieces. We say that f(x) has a jump discontinuity at x = 𝑎 if the limit of the function from the left, \( \lim_{t\uparrow a} f(t) , \) denoted \( f(a-0) \ \mbox{ or }\ f(a^{-}) \) and the limit of the function from the right, \( \lim_{t\downarrow a} f(t) , \) denoted \( f(a+0) \ \mbox{ or }\ f(a^{+}) , \) both exist and \( f(a+0) \ne f(a-0) . \) A function
f(x) is called piecewise continuous on an interval [𝑎, b], if it is continuous on [𝑎, b] except for at most finitely many points \( x_1 , x_2 , \ldots , x_n \) at each of them the function has both the left-hand and right-hand limits: \( f(x_k -0) \ \mbox{ and }\ f(x_k +0) , \quad k=1,2,\ldots , n . \) Now we give the formal definition.
Let 𝑎 and b be real numbers such that 𝑎 < b. A function \( f\, : \, [a,b] \mapsto \,\mathbb{R} \) is said to be piecewise continuous on \( [a,b] \)
if the following conditions are satisfied:
there exists a finite set \( \{ x_1 , x_2 , \ldots , x_n \} \subset (a, b) \) such that \( x_1 < x_2 < \cdots < x_n \) and f is continuous and monotone on each subinterval
\[
\lim_{x\downarrow a} f \left( x \right) , \quad \lim_{x\uparrow x_k} f\left( x \right) , \quad \lim_{x\downarrow x_k} f\left( x \right) ,\quad k=1,2,\ldots , n-1, \quad \lim_{x \uparrow b} f\left( x \right) .
\]
A function \( f\, : \, \mathbb{R} \mapsto \,\mathbb{R} \) is piecewise continuous on \( \mathbb{R} , \) if it is piecewise continuous on every finite subinterval of \( \mathbb{R} . \)
Example 5:
The function
\[
f(x) = \begin{cases} 2x , & \ \mbox{ if } 0 < x < 1, \\
1, & \ \mbox{ if } 1 < x < 2, \end{cases}
\]
is piecewise continuous on [0,2], but not continuous on [0,2].
Next, we plot partial sums along with the given function.
Fourier approximation with 10 terms
Fourier approximation with 20 terms
Fourier approximation with 100 terms
■
The first sufficient conditions for the pointwise convergence of the Fourier series were discovered by a German mathematician Johann Peter Gustav Lejeune Dirichlet (1805--1859). A function that satisfies the Dirichlet conditions is also called piecewise monotone. Such functions must have right and left limits at each point of discontinuity. We prove the corresponding theorem in convergence section.
is the Fourier series for a piecewise continuous function f(x) over the interval [−ℓ, ℓ]. If its derivative f'(x) is piecewise continuous on the interval \( [- \ell , \ell ] \) and has both a left- and right-hand derivative at each point in this interval, then F(x) is pointwise convergent for all x ∈ [−ℓ, ℓ]. The relation \( F(x) = f(x) \) holds at all points x ∈ [−ℓ, ℓ], where f(x) is continuous. If x = 𝑎 is a point of discontinuity of f, then
\[
F(x) \,=\, \frac{f(a+0)+ f(a-0)}{2} ,
\]
where \( f(a-0) = f(a^{-}) = \lim_{t\uparrow a} f(t) \) and \( f(a+0) = f(a^{+}) = \lim_{t\downarrow a} f(t) \) denote the left- and right-hand limits, respectively. ⧫
Since the class of continuous functions C[−ℓ, ℓ] contains functions that are different from their corresponding sum-Fourier series, we consider its subclass containing so called piecewise smooth functions.
We say that f(x) is piecewise smooth on some interval if
the interval can be broken up into a finite number of pieces (or sections) such
that in each piece the function f(x) is continuous and its derivative df/dx is also continuous. The function f(x) may not be continuous, but the only kind of discontinuity allowed
is a finite number of jump discontinuities.
If derivatives in the condition above are replaced by weaker condition so that function f is piecewise monotonic, then such function is said to satisfy the Dirichlet conditions. A function \( f\, : \, \mathbb{R} \mapsto \,\mathbb{R} \) is piecewise smooth on \( \mathbb{R} , \) if it is piecewise smooth on every finite subinterval of \( \mathbb{R} . \) The tangent
function is not piecewise smooth because its discontinuity is of infinite jump. On the other hand, the function sin(1/x) is not piecewise continuous because it has infinite number of maxima and minima in the neighborhood of the origin.
A real-valued function f defined on an interval [𝑎, b] is called piecewise smooth on it if it is piecewise continuous and its derivative is piecewise continuous.
Example 6:
Let f(x) = |sin(x)| be the absolute value of the sine function on the interval |−π, π|. This function is continuous on any subinterval of the real axis ℝ. Its derivative is piecewise continuous:
\[
f'(x) = \begin{cases}
\phantom{-}\cos x , & \ \mbox{ if }\quad 0 < x < \pi , \\
-\cos x , & \ \mbox{ if }\quad -\pi < x < 0 .
\end{cases}
\]
However, \( f(x) = |x|^{1/2} = \sqrt{|x|} \) is continuous on [-1,1], but it is not piecewise smooth on [-1,1] because we can break the interval [-1,1] into two subintervals [-1, 0} and (0, 1] on each of these the given function has no a bounded derivative.
■
as N → ∞ for every fixed x∈[−ℓ, ℓ].
It turns out that not every integrable (in Riemann sense, 1855) function possesses a convergent Fourier series. in 1923, Andrew Kolmogorov gave an example of a function from 𝔏¹ with an almost everywhere divergent Fourier series.Soon after, in 1926, he proved another example of Fourier series that diverges at every point. Some function have convergent Fourier series that have limiting values different from generating Fourier series. We denote the relation between a (periodic) function f(x) and its Fourier series as
This relation indicates that the Fourier coefficients are evaluated according to the Euler--Fourier formulas.
The first convergence theorem was proved in 1829 by the German
mathematician Peter Gustav Lejeune Dirichlet (1805--1859). Its proof is based on application of the second mean value theorem. It provides sufficient conditions for Fourier series to comverge pointwise without requirement on existence of a derivative of function f(x).
Second mean value Theorem:
Let g(x) be a bounded, increasing real valued function defined on the interval [𝑎,b], that is continuous at 𝑎 and b; and f(x) a Riemann integrable function on [𝑎,b]. Then there exists a real number c∈[𝑎,b] such that
\[
\int_a^b g(x)\,f(x)\,{\text d} x = g(a+0) \int_a^c f(x)\,{\text d} x + g(b-0) \int_c^b f(x)\,{\text d} x .
\]
There are known other sufficient conditions that guarantee pointwise convergence of a Fourier series; however, we start with a classical statement credited to Dirichlet.
Theorem (Dirichlet):
Let f(x) be a periodic function with period T = 2ℓ
and satisfy the following conditions (that are usually referred to as the
Dirichlet conditions):
f(x) must be of bounded variation (which means that the function has finite number of local maximum and minimum points and it monotonically increases/decreases between them) in any given bounded interval;
f(x) must have a finite number of discontinuities in any given bounded interval, and the discontinuities cannot be infinite.
The Dirichlet theorem assures that the Fourier series
\( \frac{a_0}{2} + \sum_{k\ge 1} a_k \cos \frac{k\pi x}{\ell} + b_k \sin \frac{k\pi x}{\ell} \) converges to
f(x) at all points where f is continuous inside the interval (−ℓ, ℓ);
\( \displaystyle \frac{f(-\ell +0) + f(\ell -0)}{2} \)
at endponts.
The absolutely integrability condition is a sufficient condition to guarantee existence of Fourier coefficients. The last two conditions of Dichlet's theorem are imposed to apply the second mean value theorem:
In section on Fourier series, a class of functions,
called piecewise smooth, was introduced. Such functions satisfy the Dirichlet conditions.
Lemma 2:
If f is integrable on the interval [-ℓ,ℓ], then
The denominator in the Dirichlet kernel \eqref{EqFourier.4} can vanish and hence this expression holds at the points, where it is not well-defined, according to L'Hôpital's rule.
Lemma 3:
The Dirichlet kernel possesses the following properties.
Allowing n → ∞, we obtain the stated limit. A similar argument holds for the sine.
Suppose now that f(x) ∈ 𝔏[𝑎. b]. Given ε > 0, we can find a polynomial p(x) such that
\( \int_a^b \left\vert f(x) - p(x) \right\vert {\text d}x \le \varepsilon. \) Now,
Notice that in the integral over [δ, π], the integrand is an integrable function and hence the Riemann-Lebesgue lemma guarantees that it approaches zero as n → ∞.
Lemma 7:
If function ψ(x) is monotonic in an interval [0, δ], for some positive δ, then
Since the function \( \displaystyle \frac{\sin (mz)}{z} \) is integrable on the interval [0, δ] for any δ, we can apply the second mean value theorem and obtain
In order to prove the Dirichlet theorem, we need to show that the difference S_{n}(x) − f(x) approaches zero as n → ∞. Let us fix x and consider the function
has no singulariry in a neighborhood of the origin, it is integrable over [0, δ]. Hence, by the Riemann--Lebesgue lemma, the latter integral → 0 as n → ∞. Application of the second mean value theorem provides the required result because ψ(0) = 0.
Lemma 8:
Let f be a continuous periodic function. If its Fourier series converges absolutely, then the Fourier series converges to f uniformly.
; ⧫
Theorem 3:
For piecewise smooth f(x), the Fourier series of f(x) is continuous
and converges to f(x) for −ℓ ≤ x ≤ ℓ if and only if f(x) is
continuous and f(−ℓ) = f(ℓ).
⧫
Theorem 4:
A Fourier series for a 2ℓ-periodic function f(x) that is continuous can be differentiated term by
term if its derivative f'(x) satisfies the Dirichlet conditions.
⧫
Theorem 5 (Riemann principle of localization):
Let function f∈𝔏¹ be absolutely integrable on an interval of length T = 2ℓ and have a Fourier series \eqref{EqConverge.1}. The convergence of this series to f(x) at a fixed point x depends only upon the behavior of f(x) in an arbitrary small neighborhood of x.
From Lemma 5, it follows that the convergence to f(x) depends upon
where the supremum is taken over every partition of interval [𝑎, b] such that x_{i} ≤ x_{i+1}
If f is differentiable and its derivative is Riemann-integrable, its total variation is the vertical component of the arc-length of its graph, that is,
\[
V_a^b (f) = \int_a^b \left\vert f' (x) \right\vert {\text d} x .
\]
According to Boris Golubov, a bounded variation functions of a single variable were first introduced by Camille Jordan, in the paper (1881) dealing with the convergence of Fourier series. Using this definition, he improved the Dirichlet theorem.
Theorem 6 (Dirichlet--Jordan criterion):
Suppose that f(x) is of bounded variation over interval [−ℓ, ℓ]. Then
at every point x_{0}, the Fourier series S[f] converges to the value \( \displaystyle \tfrac{1}{2} \left[ f(x_0 +0) + f(x_) -0) \right] \) ; in particular, the Fourier series S[f] converges to f(x) at every point of continuity of f;
if further f is continuous at every point of a closed interval [𝑎, b], then its Fourier series S[f] converges uniformly in [𝑎, b].
The proof of this theorem follows from the Dirichlet theorem and the following lemma.
Lemma (Jordan decomposition):
Every function of bounded variation on a finite closed interval is a difference of two bounded monootonic (nondecreasing) functions.
Recall that thetotal variation of functionf is defined to be
Thus, f is of bounded variation on [𝑎, b] if and only if V^{b}_{𝑎}(f) is bounded on [𝑎, b]. Furthermore, given any function of bounded variation f on [𝑎, b], we may rewrite it as a difference of two monotonic functions:
Observe that \( f^{+} (x) = \tfrac{1}{2} \left[ V_a^b (f)(x) + f(x) \right] \) and \( f^{-} (x) = \tfrac{1}{2} \left[ V_a^b (f)(x) - f(x) \right] \)
are both monotone increasing functions. That is, a function is of bounded variation if and only if it is the difference of two monotonicaly increasing functions.
Since the Direchlet kernel is an even function (as sume of cosine functions), the partial Fourier sum can be written as
Note that a constant K is independenyt on C, N, and δ. This completes the proof
The following theorem was proved in 1880 by an Italian mathematician Ulisse Dini (1845--1918).
Theorem 6 (Dini's criterion):
Suppose that a function f(x) ∈ 𝔏[−ℓ, ℓ] is absolutely integrable, and let
ψ(t) = f(x + t) + f(x −t) −2f(x) for a fixed x. If the integral
We split the integral into the regions where |y| < δ and δ < |y| < 2π. For latter domain of integration, we have
\( \frac{1}{|\sin (y/2) |} \le M \) for some positive constant M. Then
Observe that for Dini’s theorem to hold, it is in fact enough to have that there existconstants H > 0 and α ∈ (0, 1] such that that whenever |y ≤ 2ℓ, we have
\[
| f(x-y) - f(y)| \le H \left\vert y \right\vert^{\alpha} .
\]
Such functions are called α-Hölder continuous. Note that if function f has a continuous derivative, then f is automatically 1-Hölder (also known as Lipschitz).
It is well known that the Fourier series of a
continuous function is not necessarily uniformly convergent in
an interval of continuity of the sum-function. However, if, in
addition to being continuous, the function is also of bounded
variation then it is known that the Fourier series is uniformly
convergent in the interval of periodicity. This is the so-called
Jordan criterion for uniform convergence,
A second criterion for uniform convergence is the Dini-
Lipschitz condition. If for 0 < |t| < 1
\[
| f(x+t) - f(x) | < \frac{K}{\left( -\log |t| \right)^{1+\alpha}} , \qquad K, \alpha > 0 ,
\]
then the Fourier series of f ( x ) converges uniformly.
Corollary:
Let f(x) be continuous and periodic in [−ℓ, ℓ] and have Fourier coefficients 𝑎_{k}, b_{k} according to Eq.\eqref{EqFourier.5T}. If \( \displaystyle \sum_{k\ge 1} \left( \left\vert a_k \right\vert + \left\vert b_k \right\vert \right) \) converges, then the Fourier series of f converges absolutely and uniformly to f(x), x∈[−ℓ, ℓ]. Similarly, if \( \displaystyle \sum_{\nu =-\infty}^{+\infty} \left\vert \hat{f}(\nu ) \right\vert < \infty , \) then the Fourier series S[f] converges absolutely and uniformly.
The smoothness of the function drastically affects the speed of convergence of the Fourier series: the smoother the function the more rapid is the decrease of the Fourier coefficients. The study of convergence of Fourier series is largely the study of the interplay between assumptions of smoothness and conditions about convergence.
Theorem 7:
Let f(x) ∈ C^{n}[−ℓ, ℓ], that it has n continuous derivatives, and have period 2ℓ, that is, f(−ℓ) = f(ℓ), f'(−ℓ) = f'f^{(n)}(−ℓ) = f^{(n)}(ℓ). Suppose further that its n-th derivative is a Lipschitz function of order α, 0 < α ≤ 1. The the Fourier coefficients of f satisfy the inequalities
This notation means that there exists a constant K such that |α_{n}| ≤ K/|n|^{m}.
Let f be a periodic function on [−ℓ, ℓ], let t be some point and let δ be a positive number. We define the local modulus of continuity at the point t by
Theorem 2 (Dini's test):
Suppose that f is continuous in a closed interval [𝑎, b] and let ω(δ) be its modulus of continuity there. If ω(δ)/δ is integrable near δ = 0, and if the integrals
P.G.L. Dirichlet, "Sur la convergence des séries trigonometriques qui servent à réprésenter une fonction arbitraire entre des limites données"
Journal für die reine und angewandte Mathematik, 1829,
Volume: 4, pages 157-169.
ISSN: 0075-4102; 1435-5345/e
Golubov, B.I., Dini criterion, Encyclopedia of Mathematics, 2001, EMS Press
Hewitt, E. and Hewitt, R.E., The Gibbs--Wilbraham phenomenon: An episode in Fourier analysis, Archive for History of Exact Science, 1979, Vol. 21, No. 2, pp. 129--160.
C. Jordan, "Sur la série de Fourier" C.R. Acad. Sci. Paris , 92 (1881) pp. 228–230.
Zygmund, A., Trigonometric Series, Volumes I and II combined, Third edition, Cambridge University Press, 2003. ISBN-13 : 978-0521890533
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