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Introduction to Linear Algebra with Mathematica

Glossary

Preface

There are know many sufficient conditions for a complex-valued, periodic function f to be equal to the sum of its Fourier series at a point of continuity. Moreover, the behavior of the Fourier series at points of discontinuity is determined as well (it is the midpoint of the values of the discontinuity).

This section prsents three famous tests of pointwise convergence of the Fourier series: the Dirichlet conditions, the bounded variation condition or the jJordan test, and Dini's test.

Pointwise convergence of Fourier Series

Example 3: We consider the function \[ f(x) = \begin{cases} \frac{\sin (1/x)}{x\,\ln (1/|x|)} , &\quad 0 < |x| < e^{-2} , \\ 0, &\quad e^{-2} \le |x| \le \pi \end{cases} \] that was considered previously in Example 2 of section. The given function is not absolutely integrable, but its sine Fourier coefficients exist: \[ b_n = \frac{2}{\pi} \int_0^{\pi} f(x)\,\sin (nx)\,{\text d}x = \frac{2}{\pi} \int_0^{1/e^2} \,\frac{\sin (1/x)}{x\,\ln (1/x)}\,\sin (nx)\,{\text d}x . \] By changing variable x = 1/t, we get \[ b_n = \frac{2}{\pi} \int_{e^2}^{\infty} a_n (t)\,\sin t\,{\text d}t, \qquad a_n (t) = \frac{\sin (n/t)}{t\,\ln t} . \] Tail of this integral is easy bounded. For t ≥ 2n, we have |n/t| ≤ ½, so |sin(n/t)| ≤ n/t. Thus, \[ | a_n (t) | \le \frac{n/t}{t\,\ln t} = \frac{n}{t^2 \ln t} . \] Hence, \[ \left\vert \int_{2n}^{\infty} a_n (t) \,\sin t \,{\text d}t \right\vert \le \int_{2n}^{\infty} \frac{n}{t^2 \ln t}\,{\text d}t \le \frac{1}{\ln n} . \] So the tail contributes O(1/lnn).

The main part of the integral e² ≤ t ≤ 2n has the Dirichlet-type estimate. On interval [e², 2n], we have \[ \left\vert a_n (t) \right\vert \le \frac{1}{t\,\ln t} , \] and 𝛂ₙ(t) is slowly varying and of size ∼ 1/(tlnt). Write \[ \int_{e^2}^{2n} a_n (t)\,\sin t\,{\text d}t \] as a Dirichelet-type integral: suint oscillates, while 𝛂ₙ(t) is monotone in t for fixed n and tends to zero as t → ∞. A standard Dirichlet argument gives \[ \left\vert \int_{e^2}^{2n} a_n (t)\,\sin t\,{\text d}t \right\vert \le \sup_{e^2 \le t \le 2n} \ |A_n (t) | , \] where derivative of Aₙ(t) is equal to 𝛂ₙ(t). A direct estimate of Aₙ(t) (integrating 𝛂ₙ(t) once and using |sin(n/t)| ≤ 1) yields \[ \left\vert A_n (t) \right\vert \le \int_{e^2}^{2n} \frac{{\text d}t}{t\,\ln t} \,\sim\, \ln\,\ln (2n) - \,\ln (e^2 ) \,\sim \,\ln\,\ln n . \] However, the dependence on n/t in sin(n/t) actually improves this estimate: for large n, the effective contribution comes from t of order n, and a refined estimate (using the same substitution u = n/t inside Aₙ(t)) shows \[ \left\vert \int_{e^2}^{2n} a_n (t)\,\sin t\,{\text d}t \right\vert \le \frac{1}{\ln n} . \] Heuristically, the extra oscillation in sin(n/t) kills the slow ln lnn growth and leaves a 1/lnn scale.

Combining both estimates, we obtain \[ | b_n | \le c_1 \frac{1}{\ln n} + c_2 \frac{1}{\ln n} \le c_3 \frac{1}{\ln n} . \]

All Fourier sine coefficients exist,
they decay very slowly, like )(1/lnn),
the Fourier series is not absolutely convergent---it is only conditionally convergent and converges very delicately,
partial sums exhibit strong oscillations and very slow convergence—this is convergence in a highly non‑uniform, fragile sense.

So we know that if x ≠ 0, the function is smooth in a neighborhood of x. Classical Dirichlet--Jordan theory applies:

f is piecewise ℭ¹ away from 0,
the Fourier series converges to f(x) at every such point.

So for every x ≠ 0, \[ S_N (x;f) \,\to\, f(x) \qquad \mbox{as } N\to\infty . \] At x = 0, every term in partial Fourier sum \[ S_N (x;f) = \sum_{n\ge 1} b_n \sin (nx) \] is zero; so the series converges to f(0) = 0. However, The Fourier series does not converges uniformely near x = 0 because coefficients decay too slowly. The uniform convergence requires bₙ = o(1/n). hus:

The partial sums oscillate violently near x = 0,
The Gibbs phenomenon becomes unbounded in a shrinking neighborhood of x = 0,
convergence is pointwise but not uniform, and not even locally uniform near 0.
classical Fejér or Lebesgue theorems do not apply,
function f does not belong to 𝔏¹,
but the Fourier series does converge in the sense of distributions.

■

End of Example 3

Before we start working with three advanced suffiocient tests, it is important to establish simple criteria that determine when a Fourier series converges.

A function f defined on [𝑎, b] is called piecewise continuous on this interval if

there is a subdivision 𝑎 = x₀ < x₁ < ⋯ < xₙ = b such that f is monomtone and continuous on each subinterval Iₖ = { x : xₖ_-₁ < x < xₖ } and
at each of the subdivision points x₀, x₁, … , xₙ both one-sided limits of f exist.

A function f is piecewise smooth on an intewrval [𝑎, b] if both f and its derivative f′ are piecewise continuous on the interval.

Theorem 1: Suppose that f is piecewise smooth and periodic with period T = 2ℓ. Then its Fourier series \begin{equation} \label{EqPoint.1} \frac{a_0}{2} + \sum_{n\ge 1} a_n\,\cos \left( n\,\frac{\pi}{\ell}\,x \right) + b_n\,\sin \left( n\,\frac{\pi}{\ell}\,x \right) \end{equation} converges to

f(x) if x is a point of continuity;

½(f(x + 0) + f(x − 0)) if x is a point of discontinuity.

Dirichlet conditions

Peter Dirichlet

The original test was established by Peter Gustav Lejeune Dirichlet in 1829, for piecewise monotone functions (functions with a finite number of sections per period each of which is monotonic). Dirichlet conditions are used for ensuring the exitence of convergent Fourier representations.

A function f : [𝑎, b] → ℝ is said to satisfy the Dirichlet conditions if
it is absolutely integrable;
it has a finite number of extrema,
f has a finite number of finite jump discontinuities over the interval, and these discontinuities must be finite.

Dirichlet conditions are used for ensuring the exitence of convergent Fourier representations. There are some key points:
,li> These conditions apply to most engineering and physical signals (e.g., square, triangular, sawtooth waves).
These are sufficient conditions for convergence, not necessary. A function might not satisfy them but still have a convergent Fourier series.
Gibbs Phenomenon: Near discontinuities, the Fourier series will exhibit high-frequency oscillations known as the Gibbs phenomenon, even though it converges in the limit.
,/ul>

Bounded variation

Camille Jordan

Marie Ennemond Camille Jordan (1838–1922) was a French mathematician, known both for his foundational work in group theory and for his textbook Cours d'analyse de l'École polytechnique. Camille Jordan is not to be confused with the geodesist Wilhelm Jordan (Gauss–Jordan elimination) or the physicist Pascual Jordan (Jordan algebras). Bounded variation (BV) functions of a single variable were first introduced by Camille Jordan, in the paper (1881) dealing with the convergence of Fourier series. The properties of functions of bounded variation became widely known because they were discussed by Jordan in a note appended to the third volume of his Course d’analyse (1887).

A function f : [𝑎, b] → ℝ is of bounded variation if the supremum of the sums \( \displaystyle \quad V_P (f) = \sum_{i=1}^n |f(x_i ) - f(x_{i-1})|, \ \) taken over all partitions P = { x₀, x₁, … , xₙ } of [𝑎, b], is finite. In other words, if the numbers V_P(f) form a bounded set, as P ranges over the set of all partitions of [𝑎, b]. We denote the supremum of the V_P(f) over all partitions P by \( \displaystyle \ V_a^b (f), \ \) the variation of f from 𝑎 to b.
The set BV[𝑎, b] of functions of bounded variation on a closed interval [𝑎, b] forms a vector space over ℝ (or ℂ). It is closed under function addition and scalar multiplication: if f and g are of bounded variation, then f + g and c f (for a scalar c) are also of bounded variation. The set BV[𝑎, b] is often endowed with the norm \( \displaystyle \quad \| f |_{BV}\, = |f(a)| + V_a^b (f) , \quad \) where \( \displaystyle \quad V_a^b (f) \ \) is the total variation of f. With this norm, BV[𝑎, b] is a complete normed space (a Banach space), but it is not separable.

We have the following chains of inclusions for continuous functions over a closed, bounded interval of the real line:
Continuously differentiable ⊆ Lipschitz continuous ⊆ absolutely continuous ⊆ continuous and bounded variation ⊆ differentiable almost everywhere

Dini's conditions

Ulisse Dini

Ulisse Dini

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
\[ \int_0^{\delta} \left\vert \frac{\psi (t)}{t} \right\vert {\text d} t < \infty \]
converges, then Fourier series of f(x) converges at x to f(x)

Theorem 2 (Dini’s convergence theorem): Suppose that for a periodic absolutely integrable function f(x) there exists a positive constant δ such that
\[ \int_{-\delta}^{\delta} \left\vert \frac{f(x-t) - f(x)}{t} \right\vert {\text d}t < + \infty . \]
Then the Fourier series S[f](x) converges to f(x) for every x from the whole interval.
This theorem was proved by the Italian mathematician Ulisse Dini (1845--1918).
Recall that if f has a continuous derivative, then
\[ \lim_{t\to 0} \left\vert \frac{f(x-t) - f(x)}{t} \right\vert \]
exists and is bounded for all x; in this case, taking δ = π,
\[ \int_{|t| < \pi} \left\vert \frac{f(x-t) - f(x)}{t} \right\vert {\text d}t \le 2\pi \| f' \|_{\infty} = 2\pi \max | f' (x) | < +\infty . \]

For arbitrary integrable function f, we compute
\begin{align*} \left\vert S_N (f; x) - f(x) \right\vert &= \frac{1}{2\pi} \left\vert \int f(x-y)\,D_N (y) \,{\text d}y - \int f(x)\, D_N (y)\,{\text d} y \right\vert \\ &= \frac{1}{2\pi} \left\vert \int \frac{f(x-y) - f(x)}{|y|} \,\sin \left( N + \frac{1}{2} \right) y \cdot \frac{|y|}{\sin (y/2)}\,{\text d} y \right\vert \\ &= \frac{1}{2\pi} \left\vert \frac{1}{2{\bf j}}\int_{-\pi}^{\pi} \frac{f(x-y) - f(x)}{|y|} \frac{|y|}{\sin (y/2)} \cdot \left( e^{{\bf j}y/2} e^{{\bf j} Ny} - e^{-{\bf j}y/2} e^{-{\bf j} Ny} \right) {\text d} y \right\vert \\ &= \frac{1}{2\pi} \cdot \frac{1}{2} \left\vert \int_{-\pi}^{\pi} \, \underbrace{\frac{f(x-y) - f(x)}{|y|} \frac{|y|}{\sin (y/2)} \cdot e^{{\bf j}y/2}}_{= g(y)} e^{{\bf j} Ny} {\text d} y - \int_{-\pi}^{\pi} \, \underbrace{\frac{f(x-y) - f(x)}{|y|} \frac{|y|}{\sin (y/2)} \cdot e^{-{\bf j}y/2}}_{= h(y)} e^{-{\bf j} Ny} {\text d} y \right\vert . \end{align*}
To complete the proof, it suffices to show that g, h ∈ 𝔏¹[−ℓ, ℓ]. For g, we need to show that
\[ \int_{-\pi}^{\pi} \frac{|f(x-y) - f(x)|}{|y|} \frac{|y|}{|\sin (y/2)|} \cdot \left\vert e^{{\bf j}y/2} \right\vert {\text d} y < + \infty . \]
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
\[ \int_{\delta < |y| < \pi} \frac{|f(x-y) - f(x)|}{|\sin (y/2)|} \left\vert e^{{\bf j}y/2} \right\vert {\text d}y \le M \int_{\delta < |y| < \pi} |f(x-y) - f(x)| \, {\text d}y \le 2M\, \| f \|_1 . \]
For domain |y| < δ, we observe that \( \frac{|y|}{|\sin (y/2)|} \le 2 , \) so that
\[ \int_{|y| < \delta} \frac{|f(x-y) - f(x)|}{|y|} \frac{|y|}{|\sin (y/2)|} \cdot \left\vert e^{{\bf j}y/2} \right\vert {\text d} y \le \int_{|y| < \delta} \frac{|f(x-y) - f(x)|}{|y|} \cdot 2 \,{\text d} y < + \infty \]
by assumprion.
h because |h| = |g|.

Ulisse Dini

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).

The smoothness of the function drastically affects the rate of decrease of the Fourier coefficients: the smoother the function the more rapidly its Fourier coefficient decreease. The study of pointwise convergence of Fourier series is largely the study of the interplay between assumprions of smoothness and conclusions about convergence.

Dirichlet, P., (1829), "Sur la convergence des series trigonometriques qui servent à represénter une fonction arbitraire entre des limites donnees", Journal für die reine und angewandte Mathematik (J. Reine Angew. Math.), 4: 157–169.

Jordan, Camille (1881), "Sur la série de Fourier" [On Fourier's series], Comptes rendus hebdomadaires des séances de l'Académie des sciences, 92: 228–230 (at Gallica).

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