# Continuous-time signals  (Page 4/5)

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Note the derivative of a triangle wave is a square wave. Examine the series coefficients to see this. There are many books and web sites onthe Fourier series that give insight through examples and demos.

## Theorems on the fourier series

Four of the most important theorems in the theory of Fourier analysis are the inversion theorem, the convolution theorem, the differentiationtheorem, and Parseval's theorem [link] .

• The inversion theorem is the truth of the transform pair given in [link] , [link] , and [link] ..
• The convolution theorem is property 4 .
• The differentiation theorem says that the transform of the derivative of a function is $j\omega$ times the transform of the function.
• Parseval's theorem is given in property 6 .

All of these are based on the orthogonality of the basis function of the Fourier series and integral and all require knowledge of the convergenceof the sums and integrals. The practical and theoretical use of Fourier analysis is greatly expanded if use is made of distributions orgeneralized functions (e.g. Dirac delta functions, $\delta \left(t\right)$ ) [link] , [link] . Because energy is an important measure of a function in signal processing applications, the Hilbert space of ${L}^{2}$ functions is a proper setting for the basic theory and a geometric view can be especiallyuseful [link] , [link] .

The following theorems and results concern the existence and convergence of the Fourier series and the discrete-time Fourier transform [link] . Details, discussions and proofs can be found in the cited references.

• If $f\left(x\right)$ has bounded variation in the interval $\left(-\pi ,\pi \right)$ , the Fourier series corresponding to $f\left(x\right)$ converges to the value $f\left(x\right)$ at any point within the interval, at which the function is continuous; it converges tothe value $\frac{1}{2}\left[f\left(x+0\right)+f\left(x-0\right)\right]$ at any such point at which the function is discontinuous. At the points $\pi ,-\pi$ it converges to the value $\frac{1}{2}\left[f\left(-\pi +0\right)+f\left(\pi -0\right)\right]$ . [link]
• If $f\left(x\right)$ is of bounded variation in $\left(-\pi ,\pi \right)$ , the Fourier series converges to $f\left(x\right)$ , uniformly in any interval $\left(a,b\right)$ in which $f\left(x\right)$ is continuous, the continuity at $a$ and $b$ being on both sides. [link]
• If $f\left(x\right)$ is of bounded variation in $\left(-\pi ,\pi \right)$ , the Fourier series converges to $\frac{1}{2}\left[f\left(x+0\right)+f\left(x-0\right)\right]$ , bounded throughout the interval $\left(-\pi ,\pi \right)$ . [link]
• If $f\left(x\right)$ is bounded and if it is continuous in its domain at every point, with the exception of a finite number of points at which it mayhave ordinary discontinuities, and if the domain may be divided into a finite number of parts, such that in any one of them the function ismonotone; or, in other words, the function has only a finite number of maxima and minima in its domain, the Fourier series of $f\left(x\right)$ converges to $f\left(x\right)$ at points of continuity and to $\frac{1}{2}\left[f\left(x+0\right)+f\left(x-0\right)\right]$ at points of discontinuity. [link] , [link]
• If $f\left(x\right)$ is such that, when the arbitrarily small neighborhoods of a finite number of points in whose neighborhood $|f\left(x\right)|$ has no upper bound have been excluded, $f\left(x\right)$ becomes a function with bounded variation, then the Fourier series converges to the value $\frac{1}{2}\left[f\left(x+0\right)+f\left(x-0\right)\right]$ , at every point in $\left(-\pi ,\pi \right)$ , except the points of infinite discontinuity of the function, provided theimproper integral ${\int }_{-\pi }^{\pi }f\left(x\right)dx$ exist, and is absolutely convergent. [link]
• If f is of bounded variation, the Fourier series of f converges at every point $x$ to the value $\left[f\left(x+0\right)+f\left(x-0\right)\right]/2$ . If f is, in addition, continuous at every point of an interval $I=\left(a,b\right)$ , its Fourier series is uniformly convergent in $I$ . [link]
• If $a\left(k\right)$ and $b\left(k\right)$ are absolutely summable, the Fourier series converges uniformly to $f\left(x\right)$ which is continuous. [link]
• If $a\left(k\right)$ and $b\left(k\right)$ are square summable, the Fourier series converges to $f\left(x\right)$ where it is continuous, but not necessarily uniformly. [link]
• Suppose that $f\left(x\right)$ is periodic, of period $X$ , is defined and bounded on $\left[0,X\right]$ and that at least one of the following four conditions is satisfied: (i) $f$ is piecewise monotonic on $\left[0,X\right]$ , (ii) $f$ has a finite number of maxima and minima on $\left[0,X\right]$ and a finite number of discontinuities on $\left[0,X\right]$ , (iii) $f$ is of bounded variation on $\left[0,X\right]$ , (iv) $f$ is piecewise smooth on $\left[0,X\right]$ : then it will follow that the Fourier series coefficients may be defined through the defining integral,using proper Riemann integrals, and that the Fourier series converges to $f\left(x\right)$ at a.a. $x$ , to $f\left(x\right)$ at each point of continuity of $f$ , and to the value $\frac{1}{2}\left[f\left({x}^{-}\right)+f\left({x}^{+}\right)\right]$ at all $x$ . [link]
• For any $1\le p<\infty$ and any $f\in {C}^{p}\left({S}^{1}\right)$ , the partial sums
${S}_{n}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}{S}_{n}\left(f\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\sum _{|k|\le n}\stackrel{^}{f}\left(k\right){e}_{k}$
converge to $f$ , uniformly as $n\to \infty$ ; in fact, $||{S}_{n}-f{||}_{\infty }$ is bounded by a constant multiple of ${n}^{-p+1/2}$ . [link]

what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
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