A Function That is Continuous on 0 1 but is Not of Bounded Variation on 0 1

Bounded Variation

While bounded variation may serve efficient management for human purposes, the built-in rigidity will eventually lead to the systematic failure of hydrological, ecological, and biogeochemical functions that are maintained by self-organization and adaptation through the connectivity of wetlands.

From: Climate Vulnerability , 2013

Convex Functions, Partial Orderings, and Statistical Applications

In Mathematics in Science and Engineering, 1992

9.33 Theorem

Let f be of bounded variation on [a, b] = I and h, g be bounded functions such that h ∈ L(g) and fh ∈ L(g) on I. Let m = inf{f(x): a ≤ x ≤ b}, then

(9.44) ${\displaystyle \underset{a}{\overset{b}{\int }}f(s)h(s)dg(s)}\le m{\displaystyle \underset{a}{\overset{b}{\int }}h(s)dg(s)}+V(f,I)\underset{a\le u\le v\le b}{\sup ╡}{\displaystyle \underset{u}{\overset{v}{\int }}h(s)dg(s)},$

(9.45) ${\displaystyle \underset{a}{\overset{b}{\int }}f(s)h(s)dg(s)}\ge m{\displaystyle \underset{a}{\overset{b}{\int }}h(s)dg(s)}+V(f,I)\underset{a\le u\le v\le b}{\inf ╡}{\displaystyle \underset{u}{\overset{v}{\int }}h(s)dg(s)}.$

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Selected Topics of Real Analysis

Alexander S. Poznyak , in Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, Volume 1, 2008

Theorem 16.15

Let α : $ℝ$ → $ℝ$ be of bounded variation on [a, b] and f ∈ R _[a,b] (α). For any x ∈ [a, b] define

(16.81) $F(x):={\displaystyle \underset{s=a}{\overset{x}{\int }}f(s)d\alpha (s)}$

Then

(a)

F is of bounded variation on [a, b];

(b)

Every point of continuity of α is also a point of continuity of F;

(c)

If f ↑ on [a, b] then the derivative F′ (x) exists at each point x ∈ (a, b) where α′ (x) exists and where f is continuous. For such x

(16.82) $F^{\prime }(x)=f(x){\alpha }^{\prime }(x)$

Proof

It is sufficient to assume that α ↑ on [a, b]. If x ≠ y by (16.77) it follows that

$F(y)-F(x)={\displaystyle \underset{s=x}{\overset{y}{\int }}f(s)d\alpha (s)=c[\alpha (y)-\alpha (x)]}$

where c ∈ [m, M]. So, statements (a) and (b) follow at once from this equation. To prove (c) it is sufficient to divide both sides by (y − x) and observe that c → f (x) when y → x. Theorem is proven.

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Convex Functions, Partial Orderings, and Statistical Applications

In Mathematics in Science and Engineering, 1992

3.9 Theorem

Inequality (3.5) holds if λ is of bounded variation and satisfies

$\begin{array}{c}\lambda (y_{m-1})\le \lambda (x_{m-1})\le \lambda (y_{m-2})\le \cdots \le \lambda (x_1)\le \lambda (a)<\lambda (b)\\ \le \lambda (x_n)\le \cdots \le \lambda (x_{m+1})\le \lambda (y_m)\end{array}$

for every x _k in (y _k-1, y _k ) (k ≠ m, m ∈ {1, …, n}, y ₀ = a, y _n = b), and there exists a c ∈[y _m-1, y _m ] such that λ(x _m ) ≤ λ(y _m−1) for every x _m ∈ [y _m−1, c] and λ(x _m ) ≥ λ(y _m ) for every x _m ∈ (c, y _m ], provided that f is continuous and monotonic (in either direction) in each of the n − 1 intervals (y _k-1, y _k ).

The proof of Theorem 3.9 is similar to Pečarić's proof of the Jensen–Boas Inequality. In the limit as n → ∞, we obtain the inverse of Jensen inequality, i.e., Theorem 3.5.

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Compression

Stéphane Mallat , in A Wavelet Tour of Signal Processing (Third Edition), 2009

More Irregular Images

The mandrill and GoldHill are examples of images that do not have a bounded variation. This appears in the fact that their sorted wavelet coefficients satisfy $\left|f_B^r\left[k\right]\right|\sim C{k}^{-s}\text{for}s<1.$ Since $PSNR=10{\log }_{10}d(R,f)+K,$ it results from (10.53) that it increases with a slope of (2s – 1) 10 log₁₀ 2 as a function of ${\text{log}}_2\overline{R}$ . For the GoldHill image, s ≈ 0.8, so the PSNR increases by 1.8 db/bit. Mandrill is even more irregular, with s ≈ 2/3, so at low bit rates $\overline{R}<1/4$ the PSNR increases by only 1 db/bit. Such images can be modeled as the discretization of functions in Besov spaces with a regularity index s/2 + 1/2 smaller than 1.

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Mathematical preliminaries

Liansheng Tan , in A Generalized Framework of Linear Multivariable Control, 2017

2.7.3 Region of convergence

If f is a locally integrable function (or more generally a Borel measure locally bounded variation), then the Laplace transform F(s) of f(t) converges, provided that the limit

$\begin{array}{l}\hfill \underset{R\to \infty }{lim}{\int }_0^{R}f(t)e^{-st}dt\end{array}$

exists. The Laplace transform converges absolutely if the integral

$\begin{array}{l}\hfill {\int }_0^{\infty }\left|f(t)e^{-st}\right|dt\end{array}$

exists (as a proper Lebesgue integral). The Laplace transform is usually understood as conditionally convergent, meaning that it converges in the former instead of the latter sense.

Following the dominated convergence theorem, one notes that the set of values, for which F(s) converges absolutely satisfies Re(s) ≥ a, where a is an extended real constant. The constant a is known as the abscissa of absolute convergence, and depends on the growth behavior of f(t). Analogously, the two-sided transform converges absolutely in a strip of the form a ≤ Re(s) ≤ b. The subset of values of s for which the Laplace transform converges absolutely is called the region of absolute convergence or the domain of absolute convergence. In the two-sided case, it is sometimes called the strip of absolute convergence. The Laplace transform is analytic in the region of absolute convergence.

Similarly, the set of values for which F(s) converges (conditionally or absolutely) is known as the region of conditional convergence, or simply the region of convergence (ROC). If the Laplace transform converges (conditionally) at s = s ₀, then it automatically converges for all s with Re(s) > Re(s ₀). Therefore, the region of convergence is a half-plane of the form Re(s) > a, possibly including some points of the boundary line Re(s) = a.

In the region of convergence Re(s) > Re(s ₀), the Laplace transform of f can be expressed by integrating by parts as the integral

$\begin{array}{l}\hfill F(s)=(s-s_0){\int }_0^{\infty }{e}^{-(s-s_0)t}\beta (t)dt,\beta (u)={\int }_0^u{e}^{-s_{0}t}f(t)dt.\end{array}$

That is, in the region of convergence F(s) can effectively be expressed as the absolutely convergent Laplace transform of some other function.

There are several Paley-Wiener theorems concerning the relationship between the decay properties of f and the properties of the Laplace transform within the region of convergence.

In engineering applications, a function corresponding to a linear time-invariant (LTI) system is stable if every bounded original produces a bounded output. This is equivalent to the absolute convergence of the Laplace transform of the impulse response function in the region Re(s) ≥ 0. As a result, LTI systems are stable provided the poles of the Laplace transform of the impulse response function have a negative real part. This ROC is used in determining the causality and stability of a system.

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Signal and Image Representation in Combined Spaces

Victor E. Katsnelson , in Wavelet Analysis and Its Applications, 1998

§5 Mean-periodic continuation and Riesz bases

The following operator is one of the main objects in our considerations.

Definition 10

Let σ be a bounded variation nonconstant function on [a, b], [a, b] ⊂ ℝ, and let E, E ⊂ ℝ be a compact, (mes E > 0). By definition,

(5.1) ${\mathit{T}}_{\sigma ,E}:L^2\left(\left[a,b\right]\right)\to L^2\left(E\right),{\mathcal{D}}_{T_{\sigma ,E}}={\mathcal{D}}_{C_{\sigma }},T_{\sigma ,E}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_E{C}_{\sigma }\text{.}$

The operator T _σ,e is said to be a σ-mean-periodic transfer operator (from [a, b] onto E).

As direct consequence of Theorem 7 , we obtain:

Theorem 8

If the function σ satisfies the conditions (3.6), then ${\mathcal{D}}_{T_{\sigma ,E}}=L_{\left[a,b\right]}^2\text{,}$ and the operator T _{σ, e} acts continuously from L²([a, b]) into L ²(E).

Remark. If t ∈ (S_σ ) ((S_σ ) is a zero set of the function S_σ ), then

(5.2) ${\mathit{T}}_{\sigma ,E}\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_{\left[a,b\right]}\left(e^{it\omega }\right)=\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_E\left(e^{it\omega }\right)\text{.}$

The following result is (formally) the main result of this paper.

Main Theorem.

Let σ be a bounded variation function on [a, b], satisfying the conditions (3.6), and let the zero set (S_σ ) of the function S_σ (4.5) be separable (i.e. the condition (2.2) is satisfied).

i)

For the system of exponentials ${\left\{\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_E\left(e^{it\omega }\right)\right\}}_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}$ to be a Riesz basis in the Hilbert space L ²(E), it is necessary and sufficient that the transfer operator T _σ,e : L²([a, b}) → L ²(E) be invertible.

ii)

Let the operator T _σ,e be invertible, and let ${\left\{{\phi }_{\left[a,b\right];t_k}\right\}}_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}$ be a Riesz basis biorthogonal to the Riesz basis ${\left\{\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_{\left[a,b\right]}\left(e^{it\omega }\right)\right\}}_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}\text{.}$ Then the system ${\left\{{\phi }_{E;t_k}\right\}}_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}$ with

${\phi }_{E;t_k}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\left({{\mathit{T}}_{\sigma ,E}}^{*}\right)}^{-1}{\phi }_{\left[a,b\right];t_k}$

is a Riesz basis biorthogonal to the Riesz basis $\left\{\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_E\right(e^{it\omega }\}{}_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}\text{.}$

Proof. First of all, we remark that, according to (5.2), for each finite linear combination ${\displaystyle \sum c_t{e}^{it\omega }}$ with t ∈ $\mathcal{T}$ (S_σ ), the equality

(5.3) $T_{\sigma ,E}\left(\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_{\left[a,b\right]}\left({\displaystyle {\sum }_t{c}_t{e}^{it\omega }}\right)\right)=\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_E\left({\displaystyle {\sum }_t{c}_t{e}^{it\omega }}\right)$

holds.

a)

Assume that the operator T _σ,e is invertible. Since, from Theorem 4, the system ${\left\{\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_{\left[a,b\right]}\left(e^{it\omega }\right)\right\}}_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}\text{.}$ forms a Riesz basis in the Hilbert space L² ([a, b]), its image – the system ${\left\{\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_E\left(e^{it\omega }\right)\right\}}_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}-$ forms a Riesz basis in the Hilbert space L ²(E).

b)

Assume that the system ${\left\{\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_E\left(e^{it\omega }\right)\right\}}_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}$ forms a Riesz basis in L ²(E). Then this system is complete in L ²(E), and frame inequalities

$\begin{array}{l}{m}_E^2{\displaystyle \sum _{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}\left|c_t\right|{}^2}\le {‖{\displaystyle \sum _{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_E({\displaystyle \sum _{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}{c}_t{e}^{it\omega })}}‖}_{L^2\left(E\right)}^2\\ \le M_E^2{\displaystyle \sum _{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}\left|c_t\right|{}^2}\end{array}$

hold for each finite linear combination ${\displaystyle \sum c_t{e}^{it\omega }}$ with some frame constants m_e > 0, M_e < ∞. An analogous inequality holds also in L² ([a, b]) (with some frame constants m _[a,b]) > 0, M _[a,b] < ∞. Combining these frame inequalities, we obtain a double inequality

(5.4) $\begin{array}{l}{\left(m_{\left[a,b\right]}/M_E\right)}^2{‖{\displaystyle {\sum }_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_E\left({\displaystyle {\sum }_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}{c}_t{e}^{it\omega }}\right)}‖}_{L^2\left(E\right)}^2\\ \le {‖{\displaystyle {\sum }_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_{\left[a,b\right]}\left({\displaystyle {\sum }_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}{c}_t{e}^{it\omega }}\right)}‖}_{L^2\left(\left[a,b\right]\right)}^2\\ \le {\left(m_E/M_{\left[a,b\right]}\right)}^2{‖{\displaystyle {\sum }_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_E\left({\displaystyle {\sum }_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}{c}_t{e}^{it\omega }}\right)}‖}_{L^2\left(E\right)}^2\end{array}$

which holds for any finite linear combination ${\displaystyle \sum c_t}{e}^{it\omega }\text{.}$ According to (5.2), the inequality (5.4) means, that

(5.5) $\begin{array}{l}{\left(m_E/M_{\left[a,b\right]}\right)}^2{‖{\displaystyle {\sum }_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_{\left[a,b\right]}\left({\displaystyle {\sum }_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}{c}_t{e}^{it\omega }}\right)}‖}_{L^2\left(\left[a,b\right]\right)}^2\\ \le {‖{\mathit{T}}_{\sigma ,E}\left({\displaystyle {\sum }_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_{\left[a,b\right]}\left({\displaystyle {\sum }_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}{c}_t{e}^{it\omega }}\right)}\right)‖}_{L^2\left(E\right)}^2\\ \le {\left(M_E/m_{\left[a,b\right]}\right)}^2{‖{\displaystyle {\sum }_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_{\left[a,b\right]}\left({\displaystyle {\sum }_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}{c}_t{e}^{it\omega }}\right)}‖}_{L^2\left(\left[a,b\right]\right)}^2\end{array}$

Linear manifolds $\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_{\left[a,b\right]}\left({\displaystyle {\sum }_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}{c}_t{e}^{it\omega }}\right)$ and $\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_E\left({\displaystyle {\sum }_{t\in \mathcal{T}\left({\mathcal{S}}_{\sigma }\right)}{c}_t{e}^{it\omega }}\right)$ are dense in L² ([a, b]) and L ²(E) respectively. Hence, the operator T _σ,e is invertible, and

(5.6) $\begin{array}{cc}\hfill ‖{\left({\mathit{T}}_{\sigma ,E}\right)}^{-1}‖\le {\left(M_{\left[a,b\right]}/m_E\right)}^2,\hfill & \hfill ‖\left({\mathit{T}}_{\sigma ,E}\right)‖\le {\left(M_E/m_{\left[a,b\right]}\right)}^2\text{.}\hfill \end{array}$

The theorem is proved.

Remark. Let E be a finite union of intervals. If the sequence ${\left\{\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_E\left(e^{it\omega }\right)\right\}}_{t\in \mathcal{T}}$ is a Riesz basis in L ²(E), then, according to Landau [25, 26], d( $\mathcal{T}$ ) = (mes E)/2π (see (2.15)). If the sequence ${\left\{\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_{\left[a,b\right]}\left(e^{it\omega }\right)\right\}}_{t\in \mathcal{T}}$ (with the same $\mathcal{T}$ ) is also a Riesz basis in L ²([a, b]), then d( $\mathcal{T}$ ) = (b − a) / 2π. (Also according to Landau's Theorem. Of course, for the case E is an interval, this result was obtained by means of entire functional methods much earlier [38].) Comparing, we obtain:

Proposition

If σ is such that the system ${\left\{\mathbf{R}\mathbf{s}\mathbf{t}{\mathbf{r}}_{\left[a,b\right]}\left(e^{it\omega }\right)\right\}}_{t\in \mathcal{T}}$ be a Riesz basis in L ²([a, b}) (for example, if the conditions (3.6) are satisfied), and if the transfer operator T _σ,e is invertible, then

(5.7) $\begin{array}{cc}\hfill \mathrm{m}\mathrm{e}\mathrm{s}E=b-a\hfill & \hfill (=\hfill \end{array}\mathrm{m}\mathrm{e}\mathrm{s}\left[a,b\right])\text{.}$

However, this reasoning fails to work for sets E of more general structure than finite union of intervals.

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Convex Functions, Partial Orderings, and Statistical Applications

In Mathematics in Science and Engineering, 1992

5.43 Corollary

Let t ₀, f and ϕ satisfy the conditions of Corollary 5.39. Let λ be a function of bounded variation such that

$\begin{array}{llllll}{\displaystyle \underset{a}{\overset{t}{\int }}d\text{λ(t)}\ge \text{0}}\hfill & for\hfill & a\le t\le t_0,\hfill & {\displaystyle \underset{t}{\overset{b}{\int }}d\text{λ(t)}\ge 0}\hfill & for\hfill & t_0<t\le b,\hfill \\ \hfill & \hfill & \hfill & \hfill & and\hfill & {\displaystyle \underset{a}{\overset{b}{\int }}|d\text{λ(t)|>0}\text{.}}\hfill \end{array}$

Then the following inequality holds:

$\phi \left\{\left({\displaystyle \underset{a}{\overset{b}{\int }}f(x)d\text{λ(}x\text{)}}\right)/\left({\displaystyle \underset{a}{\overset{b}{\int }}|d\text{λ}(x)|}\right)\right\}\le \left({\displaystyle \underset{a}{\overset{b}{\int }}\phi (f(x))d\text{λ(}x\text{)}}\right)/\left({\displaystyle \underset{a}{\overset{b}{\int }}|d\text{λ}(x)|}\right).$

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Integration

Alexander S. Poznyak , in Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, Volume 1, 2008

15.2.4.4 Integrators of bounded variation

Definition 15.4

If P_n is a partition of a compact interval [a, b], Δα _k := α (x_k ) − α (x _k−1) and there exists a positive number M such that

for all partitions P_n of the interval [a, b], then α is said to be of bounded variation on [a, b].

Lemma 15.1

If α is monotonic on [a, b] then it is of bounded variation on [a, b].

Proof

Let α be nondecreasing. Then Δα _k ≥ 0 for all k = 1,…, n and, hence,

${\displaystyle \sum _{k=1}^n\left|\text{Δ}{\alpha }_k\right|}={\displaystyle \sum _{k=1}^n\text{Δ}{\alpha }_k}=\alpha (x_n)-\alpha (x_0)=\alpha (b)-\alpha (a)$

If f is nonincreasing then Δα _k ≤ 0 and $\text{Δ}{\alpha }_k\le 0\text{ and}\left|\text{Δ}{\alpha }_k\right|=-\text{Δ}{\alpha }_k$ which gives

Lemma is proven.

Lemma 15.2

If α is continuous on [a, b] and if α′ exists and is bounded (say, $\underset{x\in [a,b]}{\text{sup}}\left|{\alpha }^{\prime }(x)\right|\le M<\infty$ ) then α is of bounded variation on [a, b].

Proof

Since $\text{Δ}{\alpha }_k=\alpha (x_k)-\alpha (x_{k-1})={\alpha }^{\prime }(t_k)(x_k-x_{k-1})$ where $t_k\in (x_{k-1},x_k)$ it follows that

$\begin{array}{c}{\displaystyle \sum _{k=1}^n\left|\text{Δ}{\alpha }_k\right|}={\displaystyle \sum _{k=1}^n\left|{\alpha }^{\prime }(t_k)(x_k-x_{k-1})\right|}={\displaystyle \sum _{k=1}^n\left|{\alpha }^{\prime }(t_k)\right|(x_k-x_{k-1})}\\ \le M{\displaystyle \sum _{k=1}^n(x_k-x_{k-1})}=M(b-a)<\infty \end{array}$

which completes the proof.

Lemma 15.3

If α is of bounded variation on [a, b], say ${\displaystyle {\sum }_{k=1}^n\left|\text{Δ}{\alpha }_k\right|}\le M$ for all partitions of[a, b], then α is bounded on [a, b], namely,

Proof

For any x ∈ (a, b), using the special partition P := {a, x, b}, we find

$\left|\alpha (x)-\alpha (a)\right|+\left|\alpha (b)-\alpha (x)\right|\le M$

which implies $\left|\alpha (x)-\alpha (a)\right|\le M$ , or, equivalently, $\alpha (x)\le \alpha (a)+M$ . The same inequality is valid if x = a or x = b. Lemma is proven.

To work more exactly with functions of bounded variations we need the following definition.

Definition 15.5

For a function α of bounded variation on [a, b] the number

(15.49) $V_{\alpha }[a,b]:=\underset{P}{\text{sup}}\left\{{\displaystyle \sum _{k=1}^n\left|\text{Δ}{\alpha }_k\right|}\right\}$

(where sup is taken over all possible partitions of [a, b]) is called the total variation of α on the interval [a, b].

The following properties of V _α [a, b] are evident:

1.

Since α is of bounded variation the number V _α [a, b] is finite;

2.

(15.50) $V_{\alpha }[a,b]\ge 0$

3.

$V_{\alpha }[a,b]=0$

if and only if α (x) = const on [a, b];

4.

(15.51) $V_{\alpha +\beta }[a,b]\le V_{\alpha }[a,b]+V_{\beta }[a,b]$

5.

(15.52) $\begin{array}{c}{V}_{\alpha \cdot \beta }[a,b]\le A{V}_{\alpha }[a,b]+B{V}_{\beta }[a,b]\\ \begin{array}{cc}\begin{array}{cc}A:=\underset{x\in [a,b]}{\text{sup}}& \left|\beta (x)\right|\end{array},& \begin{array}{cc}B:=\underset{x\in [a,b]}{\text{sup}}& \left|\alpha (x)\right|\end{array}\end{array}\end{array}$

6.

If c ∈ (a, b) then

(15.53) $V_{\alpha }[a,b]=V_{\alpha }[a,c]+V_{\alpha }[c,b]$

7.

If x ∈ (a, b) then the function

(15.54) $V(x):=V_{\alpha }[a,x]$

possesses the following properties:

(a)

$V(a)=0$

(b)

V (x) is a nondecreasing function on [a, b];

(c)

[V (x) − α (x)] is a nondecreasing function on [a, b];

(d)

Any point of continuity of α (x) is a point of continuity of V (x) and inversely.

The following theorem gives the simple and elegant characterization of functions of bounded variations.

Theorem 15.8. (on a difference of increasing functions)

Let α be defined on [a, b]. Then α is of bounded variation on [a, b] if and only if α can be represented as the difference of two nondecreasing functions, namely, if and only if

where α⁺ ↑ on [a, b] and α⁻ ↑ on [a, b].

Proof

Define α⁺ (x) = V (x), where V (x) is the function (15.54), and α⁻ (x) := V (x) − α (x). By the statement 7(b−c) of the previous claim it follows that both α⁺ (x) and α⁻ (x) are nondecreasing which proves the theorem.

Corollary 15.2

If α (x) is continuous at the point x, then α⁺ (x) and α⁻ (x) are also continuous at x.

Example 15.3

Consider the function (see Fig. 15.3)

Fig. 15.3. The function of bounded variation.

$\alpha (x):=\{\begin{array}{lll}0\hfill & if\hfill & 0\le x<1\hfill \\ 3\hfill & if\hfill & 1\le x<2\hfill \\ 1\hfill & if\hfill & 2\le x<3\hfill \\ 2\hfill & if\hfill & 3\le x\le 4\hfill \end{array}$

Define (see Fig. 15.4)

Fig. 15.4. The first nondecreasing function.

${\alpha }^{+}(x):=\{\begin{array}{lll}0\hfill & if\hfill & 0\le x<1\hfill \\ 3\hfill & if\hfill & 1\le x<2\hfill \\ 4\hfill & if\hfill & 2\le x<3\hfill \\ 6\hfill & if\hfill & 3\le x\le 4\hfill \end{array}$

and (see Fig. 15.5)

Fig. 15.5. The second nondecreasing function.

${\alpha }^{-}(x):=\{\begin{array}{lll}0\hfill & if\hfill & 0\le x<2\hfill \\ 3\hfill & if\hfill & 3\le x<3\hfill \\ 4\hfill & if\hfill & 3\le x\le 4\hfill \end{array}$

Then, it is clear that $\alpha (x)={\alpha }^{+}(x)-{\alpha }^{-}(x)$ .

Corollary 15.3

(Royden 1968) For any function α (x) of bounded variation on [a, b] and for each point c ∈ (a, b) there exist $\begin{array}{lll}\underset{x\to c-0}{\text{lim}}\alpha (x)\hfill & and\hfill & \underset{x\to c+0}{\text{lim}}\alpha (x)\hfill \end{array}$ .

Corollary 15.4

(Royden 1968) Any monotone function and, hence, any function of bounded variation on [a, b] can have only a countable number of discontinuities.

Proof

It follows from the fact that for any monotone function α (x) the number of points where

$\begin{array}{l}\left|\alpha (x+0)-\alpha (x-0)\right|>1/s_n\hfill \\ s_n:={\displaystyle \sum _{k=1}^n{[\text{Δ}{\alpha }_k]}_{-,}\text{undefined}{[x]}_{-}:=\text{min}\left\{0;x\right\}}\hfill \end{array}$

for any partition P_n is finite.

Corollary 15.5

(Royden 1968) If α (x) is a function of bounded variation on [a, b], then α′ (x) exists for almost all x ∈ [a, b], that is, α (x) is differentiable almost everywhere on [a, b].

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Measure and Integration

G. de Barra , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

V Differentiation and Integration

Differentiation and integration are closely connected; since we have extended the elementary notion of integral we must deal carefully with differentiation so that this relation continues to hold. The first point to note is that continuous functions are not very relevant here; indeed continuous functions which are nowhere differentiable are easily constructed. For example, on the interval (0, 1) let f _n (x) denote the distance from x to the nearest number of the form m/10 ⁿ where m and n are nonnegative integers. Then f _n has a "sawtooth" graph with 10 ⁿ "teeth" and max f _n   = $\frac{1}{2}$ · 10⁻ⁿ . So f _n certainly continuous, and ∑ f _n (x) is uniformly convergent with sum f(x), say. Then f is continuous but for each x  ∈   (0, 1) by considering its decimal expansion we can show that the graph of y  = f(x) has no tangent at this point.

We now consider an important class of functions which although not necessarily continuous are well behaved as regards differentiability, namely, the functions of bounded variation. We suppose f is defined and finitevalued on the finite interval [a, b] and take a partition a  = x ₀  < x ₁  <   …   < x _k   = b. of [a, b]. Then we form the sums

$\begin{array}{c}\hfill \text{p}={\displaystyle \stackrel{\text{k}}{{\displaystyle \sum _{\text{i}=1}}}}{\left(\text{f}\left({\text{x}}_{\text{i}}\right)-\text{f}\left({\text{x}}_{\text{i}-1}\right)\right)}^{+}\\ \hfill \text{n}={\displaystyle \stackrel{\text{k}}{{\displaystyle \sum _{\text{i}=1}}}}{\left(\text{f}\left({\text{x}}_{\text{i}}\right)-\text{f}\left({\text{x}}_{\text{i}-1}\right)\right)}^{-}\\ \hfill \text{t}=\text{p}+\text{n}={\displaystyle \stackrel{\text{k}}{{\displaystyle \sum _{\text{i}=1}}}}\left|\text{f}\left({\text{x}}_{\text{i}}\right)-\text{f}\left({\text{x}}_{\text{i}-1}\right)\right|\end{array}$

where we are using the notation a ⁺  =   max(a, 0) and a ⁻  =   max(−a, 0). So t, p, n,   ≧   0 and f(b)   − f(a)   = p  − n. Taking upper bounds over all partitions of [a, b], and keeping to the same function f we let P  =   sup p, N  =   sup n, T  =   sup t and call these quantities the positive, negative, and total variations of f over [a, b]. If T is finite, f is said to be of bounded variation over [a, b] or to belong to the class BV[a, b]. Taking suprema over partitions the relations for t, p, n give T  = p  + N and f(b)   − f(a)   = P  − N. Also, each of these variations is additive over intervals, so if a  < c  < b then T[a, b]   = T[a, c]   + T[c, b] and similarly for P, N. This follows immediately from the corresponding identities for t, p, n, and leads to the important result that a function f is of bounded variation over [a, b] if, and only if, it may be written as the difference of two finite-valued monotone increasing functions g and h, say. These are obtained by defining g(x)   = P[a, x]   + f(a) and h(x)   = N[a, x]. The converse follows from the fact that any finite-valued monotone function is of bounded variation, and so therefore is the difference of two such functions. Indeed, the class of functions BV[a, b] forms a vector space; linear combinations of functions of bounded variations are again functions of bounded variation.

Functions of bounded variation share the "good" properties of monotone functions. Since any finite-valued monotone increasing function is continuous except possibly on a countable set the same is true for functions f  ∈   BV[a, b], so in particular these functions are measurable. An example of a monotone-increasing function with a countable number of discontinuities is provided by letting {r _i } be an enumeration of the rational numbers in [0, 1] and defining $\text{f}\left(\text{x}\right)={\sum }_{{\text{r}}_{\text{n}}\leqq \text{x}}{2}^{-\text{n}}$ . Then f is discontinuous at each rational in the interval. For an example of a function f not of bounded variation on [0, 1] define f arbitrarily at x  =   0 and let f(x)   =   sin(1/x) otherwise; another example is given by f(x)   = x sin(1/x), x  ≠   0, f(0)   =   0. This example shows directly that f may be continuous but not of bounded variation.

Lebesgue proved that if f  ∈   BV[a, b], then f is differentiable a.e. and its derivative is finite a.e. This is a significantly more difficult result to prove than the earlier results of this section. The two examples of the previous paragraph show that the converse of this theorem is not true.

We consider now indefinite integrals and write for any integrable function f, F(x)   =   ∫_a ^x fdt, so that F is the indefinite integral of f over the interval [a, b], say, on which f is integrable. Then Lebesgue's dominated convergence theorem, applied to the family of functions χ_{[a. x]} f where x  → x ₀ shows that F is continuous. It also follows easily from the definitions that F  ∈   BV[a, b], its total variation being bounded by ∫ ^b _a ∣f∣ dt. So F′ exists a.e. and it can easily be shown that F′   = f a.e. in [a, b]. We cannot expect F to be everywhere differentiable (e.g., let f be a step function, then F′ does not exist at the points of discontinuity). Nor can we assume that whenever F′ exists it equals f, for we may take a continuous function as f (so that F′ exists at all points) and then change f at a single point x ₀. Then F is unchanged but F′(x ₀) cannot equal f(x ₀).

Indefinite integrals, however, have the important property of being absolutely continuous. A function f is said to be absolutely continuous on [a, b] if given ε   >   0 there exists δ   >   0 such that

${\displaystyle \stackrel{\text{n}}{{\displaystyle \sum _{\text{i}=1}}}}\left|\text{f}\left({\text{x}}_{\text{i}}\right)-\text{f}\left({\text{y}}_{\text{i}}\right)\right|<ϵ$

whenever ∑ ⁿ _i=1 ∣x _i   − y _i ∣   <   δ for any finite set of disjoint intervals (x _i , y _i ) in [a, b]. Taking the special case n  =   1   we see that absolutely continuous functions are continuous. Considering any partition of [a, b] and introducing new partition points at a distance at most δ apart we can show that every absolutely continuous function is of bounded variation.

Now for any integrable function f we have that ∫ _E ∣f∣ dt tends to zero as m(E)   →   0. This is obvious for a bounded function f, and follows for an unbounded function since the integral of f over any set is the limit of integrals of the functions f _n where f _n   = f provided ∣f∣   ≦ n, f _n   =   ± n otherwise. From this it follows (with E  =   ∪ ⁿ _i=1 (x _i , y _i )) that if f is integrable over [a, b] its indefinite integral is absolutely continuous there. It is a little more difficult to prove the converse: every absolutely continuous function is an indefinite integral, indeed it is the indefinite integral of its derivative, a function which we know to exist a.e. That the derivative of any function of bounded variation f is measurable can be seen from the fact that it may be obtained as the limit of a sequence of ratios g _n (x)   = n(f(x  +   1/n)   − f(x)) which are themselves measurable. So, on finite intervals, a function is an indefinite integral if, and only if, it is absolutely continuous.

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Fourier Series

James S. Walker , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

III Convergence of Fourier Series

There are many ways to interpret the meaning ofEq. (13). Investigations into the types of functions allowed on the left side ofEq. (13), and the kinds of convergence considered for its right side, have fueled mathematical investigations by such luminaries as Dirichlet, Riemann, Weierstrass, Lipschitz, Lebesgue, Fejér, Gelfand, and Schwartz. In short, convergence questions for Fourier series have helped lay the foundations and much of the superstructure of mathematical analysis.

The three types of convergence that we shall describe here are pointwise, uniform, and norm convergence. We shall discuss the first two types in this section and take up the third type in the next section.

All convergence theorems are concerned with how the partial sums

${\text{S}}_{\text{N}}\left(\text{x}\right):=\stackrel{\text{N}}{\sum _{\text{n}=-\text{N}}}{\text{c}}_{\text{n}}{\text{e}}^{\text{<mi mathvariant="italic" is="true">inx</mi>}}$

converge to f(x). That is, does lim _{N  →   ∞} S _N   =   f hold in some sense?

The question of pointwise convergence, for example, concerns whether lim _{N  →   ∞} S _N (x ₀)   =   f(x ₀) for each fixed x-value x ₀. If lim _{N  →   ∞} S _N (x ₀) does equal f(x ₀), then we say that the Fourier series for f converges to f(x ₀) at x ₀.

We shall now state the simplest pointwise convergence theorem for which an elementary proof can be given. This theorem assumes that a function is Lipschitz at each point where convergence occurs. A function is said to be Lipschitz at a point x ₀ if, for some positive constant A,

(19) $\left|\text{f}\left(\text{x}\right)-\text{f}\left({\text{x}}_0\right)\right|\le \text{A}\left|\text{x}-{\text{x}}_0\right|$

holds for all x near x ₀ (i.e., ∣x  − x ₀∣   <   δ for some δ   >   0). It is easy to see, for instance, that the square wave function f₁ is Lipschitz at all of its continuity points.

The inequality inEq. (19) has a simple geometric interpretation. Since both sides are 0 when x  = x ₀, this inequality is equivalent to

(20) $|\frac{\text{f}(\text{x})-\text{f}({\text{x}}_0)}{\text{x}-{\text{x}}_0}|\le \text{A}$

for all x near x ₀ (and x  ≠ x ₀). Inequality (20) simply says that the difference quotients of f (i.e., the slopes of its secants) near x ₀ are bounded. With this interpretation, it is easy to see that the parabolic wave f₂ is Lipschitz at all points. More generally, if f has a derivative at x ₀ (or even just left- and right-hand derivatives), then f is Lipschitz at x ₀.

We can now state and prove a simple convergence theorem.

Theorem 3

Suppose f has period 2π, that ∫_−π ^π |f(x)| dx is finite, and that f is Lipschitz at x ₀. Then the Fourier series for f converges to f(x ₀) at x ₀.

To prove this theorem, we assume that f(x ₀)   =   0. There is no loss of generality in doing so, since we can always subtract the constant f(x ₀) from f(x). Define the function $\text{g}\left(\text{x}\right)=\text{f}\left(\text{x}\right)/\left({\text{e}}^{\text{ix}}-{\text{e}}^{{\text{ix}}_0}\right)$ . This function g has period 2π. Furthermore, ∫_−π ^π ∣g(x)∣ dx is finite, because the quotient $\text{f}\left(\text{x}\right)/\left({\text{e}}^{\text{ix}}-{\text{e}}^{{\text{ix}}_0}\right)$ is bounded in magnitude for x near x ₀. In fact, for such x,

$\begin{array}{ll}\left|\frac{\text{f}(\text{x})}{{\text{e}}^{\text{<mi mathvariant="italic" is="true">ix</mi>}}-{\text{e}}^{\text{x}0}}\right|\hfill & =\left|\frac{\text{f}(\text{x})-\text{f}({\text{x}}_0)}{{\text{e}}^{\text{<mi mathvariant="italic" is="true">ix</mi>}}-{\text{e}}^{\text{x}0}}\right|\hfill \\ \hfill & \le \text{A}\left|\frac{\text{x}-{\text{x}}_0}{{\text{e}}^{\text{<mi mathvariant="italic" is="true">ix</mi>}}-{\text{e}}^{\text{x}0}}\right|\hfill \end{array}$

and $\left(\text{x}-{\text{x}}_0\right)/\left({\text{e}}^{\text{<mi mathvariant="italic" is="true">ix</mi>}}-{\text{e}}^{\text{i}{\text{x}}_0}\right)$ is bounded in magnitude, because it tends to the reciprocal of the derivative of e ^ix at x ₀.

If we let d _n denote the n th Fourier coefficient for g(x), then we have ${\text{c}}_{\text{n}}={\text{d}}_{\text{n}-1}-{\text{d}}_{\text{n}}{\text{e}}^{\text{i}{\text{x}}_0}$ because $\text{f}\left(\text{x}\right)=\text{g}\left(\text{x}\right)\left({\text{e}}^{\text{<mi mathvariant="italic" is="true">ix</mi>}}-{\text{e}}^{{\text{<mi mathvariant="italic" is="true">ix</mi>}}_0}\right)$ . The partial sum S _N (x ₀) then telescopes:

$\begin{array}{ll}{\text{S}}_{\text{N}}({\text{x}}_0)\hfill & ={\displaystyle \sum _{\text{n}=-\text{N}}^{\text{N}}}{\text{c}}_{\text{n}}{\text{e}}^{\text{i}\text{n}{\text{x}}_0}\hfill \\ \hfill & ={\text{d}}_{-\text{N}-1}{\text{e}}^{-\text{<mi mathvariant="italic" is="true">iNx</mi>}0}-{\text{d}}_{\text{N}}{\text{e}}^{\text{i}(\text{N}+1)\text{x}0}.\hfill \end{array}$

Since d _n   →   0 as ∣n∣   →   ∞, by the Riemann-Lebesgue lemma, we conclude that S _N (x ₀)   →   0. This completes the proof.

It should be noted that for the square wave f₁ and the parabolic wave f₂, it is not necessary to use the general Riemann-Lebesgue lemma stated above. That is because for those functions it is easy to see that ∫_−π ^π∣g(x)∣² dx is finite for the function g defined in the proof ofTheorem 3. Consequently, d _n   →   0 as ∣n∣   →   ∞ follows from Bessel's inequality for g.

In any case,Theorem 3 implies that the Fourier series for the square wave f₁ converges to f₁ at all of its points of continuity. It also implies that the Fourier series for the parabolic wave f₂ converges to f₂ at all points. While this may settle matters (more or less) in a pure mathematical sense for these two waves, it is still important to examine specific partial sums in order to learn more about the nature of their convergence to these waves.

For example, inFig. 3 we show a graph of the partial sum S ₁₀₀ superimposed on the square wave. Although Theorem 3 guarantees that S _N   →   f₁ as N  →   ∞ at each continuity point,Fig. 3 indicates that this convergence is at a rather slow rate. The partial sum S ₁₀₀ differs significantly from f₁. Near the square wave's jump discontinuities, for example, there is a severe spiking behavior called Gibbs' phenomenon (seeFig. 4). This spiking behavior does not go away as N  →   ∞, although the width of the spike does tend to zero. In fact, the peaks of the spikes overshoot the square wave's value of 1, tending to a limit of about 1.09. The partial sum also oscillates quite noticeably about the constant value of the square wave at points away from the discontinuities. This is known as ringing.

FIGURE 3. Fourier series partial sum S ₁₀₀ superimposed on square wave.

FIGURE 4. Gibbs' phenomenon and ringing for square wave.

These defects do have practical implications. For instance, oscilloscopes—which generate wave forms as combinations of sinusoidal waves over a limited range of frequencies—cannot use S ₁₀₀, or any partial sum S _N , to produce a square wave. We shall see, however, inSection V that a clever modification of a partial sum does produce an acceptable version of a square wave.

The cause of ringing and Gibbs' phenomenon for the square wave is a rather slow convergence to zero of its Fourier coefficients (at a rate comparable to ∣n∣⁻¹). In the next section, we shall interpret this in terms of energy and show that a partial sum like S ₁₀₀ does not capture a high enough percentage of the energy of the square wave f₁.

In contrast, the Fourier coefficients of the parabolic wave f₂ tend to zero more rapidly (at a rate comparable to n ⁻²). Because of this, the partial sum S ₁₀₀ for f₂ is a much better approximation to the parabolic wave (seeFig. 5). In fact, its partial sums S _N exhibit the phenomenon of uniform convergence.

FIGURE 5. Fourier series partial sum S ₁₀₀ for parabolic wave.

We say that the Fourier series for a function f converges uniformly to f if

(21) $\underset{\text{N}\to \infty }{lim}\left\{\underset{\text{x}\in [-\pi ,\pi ]}{max}|\text{f}(\text{x})-{\text{S}}_{\text{N}}(\text{x})|\right\}=0.$

This equation says that, for large enough N, we can have the maximum distance between the graphs of f and S _N as small as we wish.Figure 5 is a good illustration of this for the parabolic wave.

We can verifyEq. (21) for the parabolic wave as follows. ByEq. (21) we have

$\begin{array}{ll}|{\text{f}}_2(\text{x})-{\text{S}}_{\text{N}}(\text{x})|=\hfill & \left|{\displaystyle \sum _{\text{n}=\text{N}+1}^{\infty }}\frac{4{(-1)}^{\text{n}}}{{\text{n}}^2}cos\text{<mi mathvariant="italic" is="true">nx</mi>}\right|\hfill \\ \hfill & \le {\displaystyle \sum _{\text{n}=\text{N}+1}^{\infty }}|\frac{4{(-1)}^{\text{n}}}{{\text{n}}^2}cos\text{<mi mathvariant="italic" is="true">nx</mi>}|\hfill \\ \hfill & \le {\displaystyle \sum _{\text{n}=\text{N}+1}^{\infty }}\frac{4}{{\text{n}}^2}.\hfill \end{array}$

Consequently

$\begin{array}{ll}\underset{\text{x}\in [-\pi ,\pi ]}{max}|{\text{f}}_2(\text{x})-{\text{S}}_{\text{N}}(\text{x})|\hfill & \le {\displaystyle \sum _{\text{n}=\text{N}+1}^{\infty }}\frac{4}{{\text{n}}^2}\hfill \\ \hfill & \to 0\text{as}\text{N}\to \infty \hfill \end{array}$

and thusEq. (21) holds for the parabolic wave f₂.

Uniform convergence for the parabolic wave is a special case of a more general theorem. We shall say that f is uniformly Lipschitz ifEq. (19) holds for all points using the same constant A. For instance, it is not hard to show that a continuously differentiable, periodic function is uniformly Lipschitz.

Theorem 4

Suppose that f has period 2π and is uniformly Lipschitz at all points, then the Fourier series for f converges uniformly to f.

A remarkably simple proof of this theorem is described in Jackson1941. More general uniform convergence theorems are discussed in Walter1994.

Theorem 4 applies to the parabolic wave f₂, but it does not apply to the square wave f₁. In fact, the Fourier series for f₁ cannot converge uniformly to f₁. That is because a famous theorem of Weierstrass says that a uniform limit of continuous functions (like the partial sums S _N ) must be a continuous function (which f₁ is certainly not). The Gibbs' phenomenon for the square wave is a conspicuous failure of uniform convergence for its Fourier series.

Gibbs' phenomenon and ringing, as well as many other aspects of Fourier series, can be understood via an integral form for partial sums discovered by Dirichlet. This integral form is

(22) ${\text{S}}_{\text{N}}(\text{x})=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\text{f}(\text{x}-\text{t}){\text{D}}_{\text{N}}(\text{t})\text{<mi mathvariant="italic" is="true">dt</mi>}$

with kernel D _N defined by

(23) ${\text{D}}_{\text{N}}(\text{t})=\frac{sin(\text{N}+1/2)\text{t}}{sin(\text{t}/2)}.$

This formula is proved in almost all books on Fourier series (see, for instance, Krantz1999), Walker1988, or Zygmund1968. The kernel D _N is called Dirichlet's kernel. InFig. 6 we have graphed D ₂₀.

FIGURE 6. Dirichlet's kernel D ₂₀.

The most important property of Dirichlet's kernel is that, for all N,

$\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{\text{D}}_{\text{N}}(\text{t})\text{<mi mathvariant="italic" is="true">dt</mi>}=1.$

FromEq. (23) we can see that the value of 1 follows from cancellation of signed areas, and also that the contribution of the main lobe centered at 0 (seeFig. 6) is significantly greater than 1 (about 1.09 in value).

From the facts just cited, we can explain the origin of ringing and Gibbs' phenomenon for the square wave. For the square wave function f₁,Eq. (22) becomes

(24) ${\text{S}}_{\text{N}}(\text{x})=\frac{1}{2\pi }{\int }_{\text{x}-\pi /2}^{\text{x}+\pi /2}{\text{D}}_{\text{N}}(\text{t})\text{<mi mathvariant="italic" is="true">dt</mi>}.$

As x ranges from −π to π, this formula shows that S _N (x) is proportional to the signed area of D _N over an interval of length π centered at x. By examiningFig. 6, which is a typical graph for D _N , it is then easy to see why there is ringing in the partial sums S _N for the square wave. Gibbs' phenomenon is a bit more subtle, but also results fromEq. (24). When x nears a jump discontinuity, the central lobe of D _N is the dominant contributor to the integral inEq. (24), resulting in a spike which overshoots the value of 1 for f₁ by about 9%.

Our final pointwise convergence theorem was, in essence, the first to be proved. It was established by Dirichlet using the integral form for partial sums inEq. (22). We shall state this theorem in a stronger form first proved by Jordan.

Theorem 5

If f has period 2π and has bounded variation on [0,2π], then the Fourier series for f converges at all points. In fact, for all x-values,

$\underset{\text{N}\to \infty }{lim}{\text{S}}_{\text{N}}(\text{x})=\frac{1}{2}[\text{f}(\text{x}+)+\text{f}(\text{x}-)].$

This theorem is too difficult to prove in the limited space we have here (see Zygmund,1968. A simple consequence ofTheorem 5 is that the Fourier series for the square wave f₁ converges at its discontinuity points to 1/2 (although this can also be shown directly by substitution of x  =   ± π/2 into the series in (Eq. (11)).

We close by mentioning that the conditions for convergence, such as Lipschitz or bounded variation, cited in the theorems above cannot be dispensed with entirely. For instance, Kolmogorov gave an example of a period 2π function (for which ∫_−π ^π ∣f(x)∣ dx is finite) that has a Fourier series which fails to converge at every point.

More discussion of pointwise convergence can be found in Walker (1998), Walter 1994, or Zygmund 1968.

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