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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 ω 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, δ ( t ) ) [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 ( x ) has bounded variation in the interval ( - π , π ) , the Fourier series corresponding to f ( x ) converges to the value f ( x ) at any point within the interval, at which the function is continuous; it converges tothe value 1 2 [ f ( x + 0 ) + f ( x - 0 ) ] at any such point at which the function is discontinuous. At the points π , - π it converges to the value 1 2 [ f ( - π + 0 ) + f ( π - 0 ) ] . [link]
  • If f ( x ) is of bounded variation in ( - π , π ) , the Fourier series converges to f ( x ) , uniformly in any interval ( a , b ) in which f ( x ) is continuous, the continuity at a and b being on both sides. [link]
  • If f ( x ) is of bounded variation in ( - π , π ) , the Fourier series converges to 1 2 [ f ( x + 0 ) + f ( x - 0 ) ] , bounded throughout the interval ( - π , π ) . [link]
  • If f ( x ) 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 ( x ) converges to f ( x ) at points of continuity and to 1 2 [ f ( x + 0 ) + f ( x - 0 ) ] at points of discontinuity. [link] , [link]
  • If f ( x ) is such that, when the arbitrarily small neighborhoods of a finite number of points in whose neighborhood | f ( x ) | has no upper bound have been excluded, f ( x ) becomes a function with bounded variation, then the Fourier series converges to the value 1 2 [ f ( x + 0 ) + f ( x - 0 ) ] , at every point in ( - π , π ) , except the points of infinite discontinuity of the function, provided theimproper integral - π π f ( x ) d x 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 [ f ( x + 0 ) + f ( x - 0 ) ] / 2 . If f is, in addition, continuous at every point of an interval I = ( a , b ) , its Fourier series is uniformly convergent in I . [link]
  • If a ( k ) and b ( k ) are absolutely summable, the Fourier series converges uniformly to f ( x ) which is continuous. [link]
  • If a ( k ) and b ( k ) are square summable, the Fourier series converges to f ( x ) where it is continuous, but not necessarily uniformly. [link]
  • Suppose that f ( x ) is periodic, of period X , is defined and bounded on [ 0 , X ] and that at least one of the following four conditions is satisfied: (i) f is piecewise monotonic on [ 0 , X ] , (ii) f has a finite number of maxima and minima on [ 0 , X ] and a finite number of discontinuities on [ 0 , X ] , (iii) f is of bounded variation on [ 0 , X ] , (iv) f is piecewise smooth on [ 0 , X ] : 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 ( x ) at a.a. x , to f ( x ) at each point of continuity of f , and to the value 1 2 [ f ( x - ) + f ( x + ) ] at all x . [link]
  • For any 1 p < and any f C p ( S 1 ) , the partial sums
    S n = S n ( f ) = | k | n f ^ ( k ) e k
    converge to f , uniformly as n ; in fact, | | S n - f | | is bounded by a constant multiple of n - p + 1 / 2 . [link]

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
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What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
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Source:  OpenStax, Brief notes on signals and systems. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10565/1.7
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