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Introduction to Linear Algebra with Mathematica

Glossary

Preface

Convergence in the sense of distributions extends classical analysis to encompass oscillatory and singular phenomena that would otherwise lie beyond its scope. In Fourier analysis, it provides a natural interpretation of series expansions for irregular functions, including those that are only conditionally integrable. The example of f(x) = sin(1/x)/x illustrates how oscillation, far from being an obstacle, can serve as the mechanism that ensures convergence—once the appropriate notion of limit is adopted.

Weak Convergence

We start with the general definition.
A sequence { xₙ } from a normed linear space is said to converge weakly to x if for every linear bounded functional T the sequence of scalars
\[ T\left( x_n \right) \,\to \, T(x) \qquad \mbox{as} \quad n \to \infty . \]
A weak convergence is denoted by xₙ ⇀ x.

This definition can be specified for our two Banach spaces, 𝔏¹[𝑎, b] and 𝔏²[𝑎, b].

A sequence of points { xₙ } in a Hilbert space ℌ is said to converge weakly to a point x in ℌ if
\[ \langle y\,\vert\, x_n \rangle \, \to \,\langle y\,\vert\, x \rangle \qquad \mbox{as} \quad n \to \infty \]
for all y in ℌ. Here ⟨ ⋅|⋅ ⟩ or ⟨ ⋅,⋅ ⟩ is understood to be the inner product in the Hilbert space.

A sequence of points { xₙ } in a Banach space 𝔏¹[−ℓ, ℓ] converges weakly to x ∈ 𝔏¹ if the sequence of scalars

\[ \int_{-\ell}^{\ell} f(t)\,x_n (t)\,{\text d} t \,\to \, \int_{-\ell}^{\ell} f(t)\,x (t)\,{\text d} t \qquad \mbox{as} \quad n \to \infty \]
for all bounded functions f ∈ 𝔏∞[−ℓ, ℓ]. When a sequence { xₙ } converges to x weakly, we abbreviate it as xₙ ⇀ x.
In particular, a sequence { xₙ }n∈ℕ of elements xₙ = [xn,0, xn,1, …] ∈ ℓ² converges weakly to x if and only if the following two conditions hold:
  1. there exists a positive constant C such that |xn,k| ≤ C;
  2. for each k, xn,k → xk.

We need weaker convergence than defined above weak convergence. This new definition requires usage of dual space X* that consists of all continuous linear functionals over the normed space X.

Let X be a normed linear space. A sequence of functionals { fₙ } ⊆ X* is weak* or weak-star convergent to f ∈ X* if sequence of scalars { fₙ(x) } converges to f(x) for all x ∈ X.
A weak-star convergence is denoted by xₙ ⇁ x.

The most famous application of the weak-star convergence provides the following lemma. It considers a trigonometric polynomial as a functional on 𝔏¹.

Riemann--Lebesgue Lemma: If f ∈ 𝔏¹[−ℓ, ℓ], then
\[ \lim_{\nu \to \infty} \int_{-\ell}^{+\ell} f(x)\, e^{{\bf j} \nu x} {\text d} x = 0 . \]
The Riemann--Lebesgue lemma was first published by Bernhard Riemann in 1867 and then was generalized for Lebesgue integrable functions in 1902 by Henri Lebesgue.
For a characteristic function f(x) = χ[𝑎, b], we have
\[ \int_a^b e^{{\bf j} \nu x} {\text d} x = \frac{1}{{\bf j}\nu} \left( e^{{\bf }b\nu} - e^{{\bf }a\nu} \right) \, \to \,0 \qquad \mbox{as} \quad \nu \to\infty . \]
The same conclusion holds for an arbitrary simple function that is a linear combination of characteristic functions.

Finally, let f ∈ 𝔏¹[−ℓ, ℓ] be arbitrary.

Let ϵ ∈ ℝ be fixed.

Since the simple functions are dense in 𝔏¹, there exists a simple function g such that

\[ \int_{-\ell}^{\ell} \left\vert f(x) - g(x) \right\vert {\text d} x < \epsilon . \]
By our previous argument and the definition of a limit of a complex-valued function, there exists N ∈ ℕ such that for all n > N
\[ \left\vert \int_{-\ell}^{\ell} g(x)\, e^{{\bf j} nx} {\text d} x \right\vert < \epsilon . \]
Then
\[ \int_{-\ell}^{\ell} f(x)\,e^{{\bf j} nx} {\text d} x = \int_{-\ell}^{\ell} \left[ f(x) - g(x) \right] e^{{\bf j} nx} {\text d} x + \int_{-\ell}^{\ell} g(x)\,e^{{\bf j} nx} {\text d} x . \]
By triangle inequality for complex numbers, the triangle inequality for integrals, multiplicativity of absolute value, and Euler formula,
\[ \left\vert \int_{-\ell}^{\ell} f(x)\,e^{{\bf j} nx} {\text d} x \right\vert \le \int_{-\ell}^{\ell} \left\vert f(x) - g(x) \right\vert {\text d} x + \left\vert \int_{-\ell}^{\ell} g(x)\,e^{{\bf j} nx} {\text d} x \right\vert \]
For all n > N, the right-hand side is bounded by 2ϵ. Since ϵ was arbitrary, this establishes
\[ \lim_{n\to\infty} \int_{-\ell}^{\ell} f(x)\,e^{{\bf j} nx} {\text d} x = 0 \]
for all f ∈ 𝔏¹[−ℓ, ℓ].
Lemma 1: If a sequence of points { xₙ } in a Banach space 𝔏¹[−ℓ, ℓ] converges strongly (in norm) to x, then { xₙ } is weakly convergent to x.
Example 10: We define a sequence { δₙ } ∈ ℓ² to be an indicator of position n:
\[ \delta_k = \begin{cases} 1, & \quad \mbox{if}\quad k=n , \\ 0, & \quad \mbox{if}\quad k\ne n . \end{cases} \]
This sequence does not converge in ℓ² because it is not Cauchy sequence: \( \displaystyle \| \delta_n - \delta_m \|_2 = \sqrt{2} \) for n ≠ m. However, the sequence converges weakly to 0. Indeed, for g ∈ ℓ², we have
\[ g(\delta_n ) = \sum_{k\ge 1} g(k)\, \delta_n = g(n)\,\delta_n = g(n) , \]
where we represent g as a vector [g(1), g(2), g(3), … ]. Since the general term of this vector tends to zero, we conclude that that the sequence {δn} converges weakly to zero.

However, this sequence { δₙ } of standard basis does not converge in ℓ¹. To prove this, we define the linear functional φ on ℓ¹ by
\[ \varphi (f) = \sum_{j\ge 1} f(j) \qquad (f \in \ell^{1}) . \]
Observe that |φ(f)| ≤ ∥ f ∥1 for each f ∈ ℓ¹. Then for each n ≠ m, we have
\[ |\varphi (\delta_n ) - \varphi (\delta_m )| = \varphi \left( \delta_n - \delta_m \right) = 2 . \]
So the sequence { φ(δₙ); } is not Cauchy in ℝ; hence, not convergent. Thus, the standard basis of ℓ¹ does not converge weakly.    ■
End of Example 10
Uniqueness Theorem: If {xn} converges weakly to both x and y, then x = y.
Suppose that {xn} converges weakly to both x and y, then for every functional T, the sequence of scalars {T xn} converges to both T x and T y. However, this is impossible because any sequence of scalars has at most only one limit value.
Radon--Riesz Theorem: Suppose { fₙ } converges weakly to f in 𝔏²[−ℓ, ℓ]. Then { fₙ } converges to f in 𝔏²[−ℓ, ℓ] if and only if \( \displaystyle \lim_{n\to\infty} \,\| f_n \| = \| f \|_2 . \)
The theorem was first proved in 1913 by the Austrian mathematician Johann Radon (1887--1956). A proof of the Radon--Riesz Theorem can be found in Royden and Fitzpatrick's book. The general proof for arbitrary p ≥ 1 is presented in Riesz and Sz.-Nagy’s Functional Analysis, London: Blackie & Son Limited (1956) (reprinted by Dover Publishing in 1990). It is also on the web: http://faculty.etsu.edu/gardnerr/5210/notes/Radon-Riesz.pdf.

Example 11: Let us consider the following sequence of points from ℓ²:
\[ x_n = [ 1, 0, 0, \ldots , 0 , 1, 0, \cdots ] , \]
where the second "1" is situated at the position n. This sequence convergese weakly to
\[ x = [ 1, 0, 0, \ldots , 0 , 0, 0, \cdots ] . \]
Indeed, all entries in xn,k are bounded by 1 and every its entry tends to zero as n → 0. However, this sequence does not converge to x uniformly because
\[ \| x_n - x \|_2 = 1 . \]

On the other hand, if we slightly modify the sequence above by placing on n-th position 1/n instead of 1, we get the sequence

\[ y_n = [ 1, 0, 0, \ldots , 0 , 1/n, 0, \cdots ] . \]
Then this sequence converges strongly to x because
\[ \| y_n - x \|_2 = \frac{1}{n} \to 0 \qquad \mbox{as}\quad n \to \infty . \]
   ■
End of Example 11

From the Radon-Riesz theorem, we immediately derive that every function f ∈ ℓ² has a weak convergent Fourier series S[f]. Since 𝔏²[−ℓ, ℓ] ⊂ 𝔏¹[−ℓ, ℓ], we have another result.

Schur’s Lemma: In ℓ¹, if a sequence { fₙ } converges weakly to f, fₙ ⇀ f, then this sequence strongly converges to f.
Proof of Schur's property can be found in Megginson's book, section 2.5.24.

Although weak and strong convergences are equivalent in ℓ¹, they are different in 𝔏¹[𝑎, b]. For example, the sequence of eigenfunctions \( \displaystyle \phi_n = e^{{\bf j} nx} \) converges to zero weakly, but does not converge in 𝔏¹ because their norms are constants:

\[ \| \phi_n \| = \int_{-\pi}^{\pi} \left\vert e^{{\bf j} nx} \right\vert {\text d} x = \int_{-\pi}^{\pi} {\text d} x = 2\pi . \]

Convergence in Ditributions of Fourier Series

Let 𝒟 or 𝒟 be a space of "good" test or probe functions.

The study of convergence lies at the heart of analysis, yet classical notions—pointwise, uniform, and norm convergence—are often inadequate for capturing the behavior of highly oscillatory or singular objects. The theory of distributions (or generalized functions), introduced by Laurent Schwartz, provides a robust framework in which convergence can be extended beyond classical limits. In this setting, oscillation itself becomes a mechanism of convergence, especially when interpreted through duality with smooth test functions. This perspective is particularly fruitful in Fourier analysis, where oscillatory expansions are central.

1. Distributional Convergence Let 𝒟(ℝ) denote the space of infinitely differentiable functions with compact support. A sequence (fₙ) is said to converge to a distribution T if, for every test function φ ∈ 𝒟(ℝ),

\[ \lim_{n\to\infty} \, \int_{\mathbb{R}} f_n (x)\,\varphi (x) \,{\text d}x = \langle T , \varphi \rangle . \]
This notion—often called weak convergence—replaces pointwise evaluation by averaging against smooth probes. It is strictly weaker than convergence in 𝔏p spaces but is stable under differentiation and Fourier transformation, making it especially suited for analysis involving irregular functions.

2. Oscillatory Functions and Conditional Integrability Certain oscillatory functions illustrate the limitations of classical integrability. Consider

\[ f(x) = x\,\sin (1/x), \qquad x \ne 0. \]
Near x = 0, the function oscillates with increasing frequency and amplitude, behaving roughly like 1/x in magnitude. Consequently, it is not absolutely integrable in any neighborhood of the origin. Nevertheless, the rapid oscillations produce cancellation, and improper integrals of the form
\[ \int_{|x| \ge \varepsilon} x\,\sin \left( \frac{1}{x} \right) {\text d}x \]
may converge as ε→0+, rendering the function conditionally integrable in the Riemann sense.

Such examples highlight a recurring theme: oscillation can compensate for growth. However, classical frameworks struggle to encode this compensation, whereas distribution theory accommodates it naturally.

3. Fourier Series and Weak Convergence Fourier series provide a canonical setting in which convergence issues arise. For a 2π-periodic function f, one formally writes

\[ f(x) \,\sim\, \mbox{V.P.} \sum_{n=-\infty}^{\infty} c_n e^{\mathbf{j}nx}, \quad c_n = \frac{1}{2π} \int_{-\pi}^{\pi} f(x)\, e^{-\mathbf{j}nx} \,{\text d}x . \]
Even when f is integrable, its Fourier series may fail to converge pointwise. Classical counterexamples are abundant, and the convergence theory is subtle (as developed extensively by Antoni Zygmund). However, if f is regarded as a distribution, then its Fourier coefficients can be defined via duality:
\[ c_n = \frac{1}{2\pi} \,\langle f, e^{−\mathbf{j}nx} \rangle . \]
In this framework, the partial sums
\[ S_N (x) = \sum_{n=−N}^N c_n e^{\mathbf{j}nx} \]
need not converge pointwise, but they converge in the sense of distributions. That is, for every smooth periodic test function φ,
\[ \lim_{N\to\infty} \,\int_{-\pi}^{\pi} S_N (x)\,\varphi (x)\,{\text d}x = \intt_{-\pi}^{\pi} f(x)\,\varphi (x)\,{\text d}x . \]
This result reflects the fact that Fourier series approximate functions after smoothing. The oscillations of the Dirichlet kernel, which obstruct pointwise convergence, are neutralized when integrated against smooth test functions.

4. Application to Highly Oscillatory Functions For functions like f(x) = sin(1/x)/x, classical Fourier analysis encounters serious difficulties. The singularity at the origin and the lack of absolute integrability prevent direct application of standard theorems. However, in the distributional framework: The function defines a distribution via its action on test functions. Oscillatory cancellation ensures that integrals against smooth functions remain finite. Fourier coefficients can be interpreted in a generalized sense. The resulting Fourier series converges weakly, even if it diverges pointwise or in norm. This phenomenon exemplifies a broader principle: highly oscillatory behavior can yield convergence when viewed through an averaging process. Techniques such as stationary phase and oscillatory integral estimates formalize this intuition.

5. Conceptual Significance The distributional approach to convergence has far-reaching implications:

From this viewpoint, Fourier series are not merely expansions of functions but representations of distributions, and their convergence must be understood accordingly.

Example 3: Let us consider the odd, 2π-periodic function \[ f(x) = \begin{cases} \frac{\sin (1/x)}{x\,\ln (1/|x|)} , &\quad 0 < |x| < e^{-2} , \\ 0, &\quad e^{-2} \le |x| \le \pi \end{cases} \] extended by f(0) = 0.

Near the origin, we observe its asymptotic behavior \[ f(x) \,\sim\, \frac{\sin (1/x)}{x\,\ln (1/|x|)} . \] Its absolute value satisfies \[ | f(x) | \,\sim\,\frac{1}{|x|\, \ln (1/|x|)} , \] and change of variable t = ln(1/|x|) shows \[ \int_0^{e^{-2}} \, |f(x)|\,{\text d}x \,\sim\, \int_2^{\infty} \frac{{\text d}t}{t} = \infty , \] so f ∉ 𝔏¹(−π, π). Thus is not a Lebesgue integrable function on one period, and in particular it does not belong to the usual framework of Fourier series.

Nevertheless, as we have already seen, the Fourier sine coefficients \[ b_n = \frac{2}{\pi} \int_0^{1/e^2}\,\frac{\sin (1/x)}{x\,\ln (1/|x|)} \,\sin (nx)\,{\text d}x \] exist as improper Riemann integrals and satisfy the decay estimate \[ b_n = O \left( \frac{1}{\ln n} \right) . \]

2. The associated distribution Define a distribution Tf ∈ 𝒟′(ℝ) by \[ \left\langle F_f , \varphi \right\rangle = \mbox{V.P.}\,\int_{-\pi}^{\pi} f(x)\,\varphi (x)\,{\text d}x , \qquad \varphi \in ℭ_c^{\infty} (\mathbb{R}) , \] where the integral is understood as an improper Riemann (or principal value) integral near x = 0, and the integrand is extended -periodically outside (−π , π).

Because φ is smooth and compactly supported, and has only a logarithmically moderated singularity at 0, the oscillatory factor sin(1/x) ensures that the pairing ⟨ Tf , φ ⟩ is well defined and finite for every test function φ. Thus, Tf is a well-defined tempered distribution representing the “generalized function” f.

3. Fourier coefficients as distributional moments On , the trigonometric system is a complete orthogonal system. In the distributional setting, the Fourier coefficients of are defined by \[ \hat{T}_f = \frac{1}{2\pi} \,\left\langle T_f , e^{-{\bf j}nx} \right\rangle , \qquad n \in \mathbb{Z} . \] Since f is odd, only the sine coefficients survive. Writing \[ e^{-{\bf j}nx} = \cos (nx) -{\bf j}\,\sin (nx) , \] and using the oddness of f, we obtain \[ \hat{T}_f (n) = - \frac{1}{2\pi}\,\left\langle T_f , \sin (nx) \right\rangle = - \frac{1}{2\pi}\,\int_{-\pi}^{\pi} f(x)\,\sin (nx)\,{\text d}x = = -\frac{1}{2}\, b_n . \] Thus, the distributional Fourier coefficients coincide (up to the factor −½) with the classical sine coefficients computed as improper integrals.

The estimate bₙ = O(1/lnn implies \[ \hat{T}_f (n) = ) \left( \frac{1}{\ln n} \right) , \] so the Fourier transform of Tf is a slowly decaying sequence, but still of at most logarithmic growth in the dual index.

4. Distributional Fourier series In the classical setting, the Fourier series of is Pointwise, this series converges to for every , and to at the origin, but the convergence is highly non-uniform and fails in any sense. In the distributional setting, we interpret the series \[ \sum_{n\ge 1} b_n \,\sin (nx) \] as a series of distributions: \[ \sum_{n\ge 1} b_n \,\sin (nx) \qquad \mbox{in } 𝒟,(\mathbb{R}) , \] For any test function \( \quad \varphi \in ℭ_c^{\infty}, \quad \) we consider \[ \left\langle \sum_{n\ge 1} b_n\,\sin (nx) , \varphi \right\rangle = \sum_{n\ge 1} b_n \int_{-\pi|^{\pi} \sin (nx) \,\varphi (x)\,{\text d}x . \] The integral \( \quad \int_{-\pi}^{\pi} \sin Onx)\,\varphi (x)\,{\textd}x \quad \) decays faster than 1/n any power of by repeated integration by parts, while bₙ = O(1/lnn) decays very slowly. The product is therefore absolutely summable in n, and the series converges in 𝒟′. One checks that \[ \left\langle \sum_{n\ge 1} b_n\,\sin (nx) , \varphi \right\rangle = \left\langle T_f , \varphi \right\rangle \] for all φ, so we obtain the identity \[ T_f = \sum_{n=1}^{\infty} b_n \,\sin (nx) \qquad \mbox{in } 𝒟;(\mathbb{R}) \] In other words, the Fourier series of f converges to f in the sense of distributions, even though f ∉ 𝔏¹ and the series is only conditionally convergent pointwise.

Interpretation: From the distributional viewpoint, the function is best regarded as a tempered distribution whose Fourier coefficients are given by the classical oscillatory integrals defining bₙ. The slow decay bₙ ∼ 1/lnn places this example at the boundary of classical Fourier analysis:

  • the coefficients exist but are barely summable in any sense;
  • the pointwise convergence is extremely delicate near the singularity;
  • yet the distributional Fourier series converges cleanly and unambiguously to the distribution Tf.
This illustrates the conceptual advantage of the distributional framework: it accommodates functions with strong, oscillatory singularities for which the usual 𝔏p theory is too rigid, while still retaining a precise and robust notion of Fourier expansion.    ■
End of Example 3

 

 

  1. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications.
  2. Kadets, M.I., About strong and weak convergence, Доклады АН СССР. – 1958. – Т. 122, No 1. – С. 13–16. http://testuvannya.com.ua/M.I.Kadets/PDF/W9-weak-and-strong.pdf
  3. Kadets, M.I., About connection between strong and weak convergence, Доклады АН СССР. – 1959. – Т. 122?, No 9. – С. 949–952.
  4. Kadets, V.M., A Course in Functional Analysis and Measure Theory, ‎ Springer, 2018 Edition
  5. Walter Rudin, Functional Analysis and Real and Complex Analysis.
  6. Laurent Schwartz, Théorie des distributions (1950–1951).
  7. Elias M. Stein and Rami Shakarchi, Fourier Analysis: An Introduction.
  8. Robert S. Strichartz, A Guide to Distribution Theory and Fourier Transforms.
  9. Antoni Zygmund, Trigonometric Series.

 

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