What is Fourier series: Definition and 750 Discussions

In mathematics, a Fourier series () is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle (or period) of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.

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  1. T

    Complex Fourier Series: Uncovering the Mystery of Different Results for x^2

    Hello, First post. I will attempt to use latex, something that involves me jabbing my keyboard with a pen since my \ key is missing. We have an assignment question which I have solved, but there is a deeper issue I don't understand. We are asked to find the complex Fourier series...
  2. C

    Fourier Series and Energy Density

    dealing with absolute functions that are limited always throws me off so let's consider this f(x)=|x| for -∏ ≤ x < ∏ f(t)= f(t+2∏) it's not too bad however finding the energy density is throwing me off a little.. the questions tend to be generally phrased as below: Find the energy...
  3. B

    How Do Symmetry Properties Affect the Terms in a Fourier Series?

    Homework Statement Suppose, in turn, that the periodic function is symmetric or antisymmetric about the point x=a. Show that the Fourier series contains, respectively, only cos(k_{n}(x-a)) (including the a_0) or sin(k_{n}(x-a)) terms. 2. Homework Equations The Fourier expansion for the...
  4. D

    MHB Understanding theorem in Fourier series

    Suppose $f$ is continuous and periodic with period $2\pi$ on $(-\infty,\infty)$, and $f'$ exist and is in $\mathcal{P}\mathcal{C}[-\pi,\pi]$. Then $\sum\limits_{k = -\infty}^{\infty}\lvert A_k\rvert < \infty$.$f'$ has a Fourier series so let's call the coefficients $A_n'$. Then $f' =...
  5. D

    MHB Mean square convergence of Fourier series

    What is the statement of the mean square convergence of Fourier series?
  6. S

    Possible Error in Calculating Fourier Series for sin2x?

    Homework Statement Fx= sin2x for -pi<x<0 and 0 for 0<x<pi. Compute the coefficients of the Fourier series. Homework Equations The Attempt at a Solution I found Ao=0 even though I came up with a Fourier cosine series. There must be something wrong. And is that possible that I...
  7. E

    How Do You Calculate the Sum of a Fourier Series at Specific Points?

    Homework Statement h(x)=\left\{\begin{matrix} 9+2x , 0<x<\pi\\ -9+2x , -pi<x<0 \end{matrix}\right. \\ Find \ the \ sum \ of \ the \ Fourier \ series \ for \ x=\frac{3\pi}{2} and\ x=\pi \\ The \ Fourier \ series \ is: \\ h(x)=9+\pi + \sum_{n=1}^{inf}...
  8. Z

    Fourier Series of: 2sin(4000*pi*t)*sin(46000*pi*t)

    Homework Statement How can i find the magnitude spectra of 2sin(4000*pi*t)*sin(46000*pi*t) The Attempt at a Solution im not sure how to go about this question, can someone please give me some help on what i should do. I know that for a square wave i can find the Fourier series...
  9. D

    MHB Fourier series without integration

    Let $$ h(\theta) = \begin{cases} \frac{1}{2}(\theta + \pi), & 0 < \theta < \pi\\ 0, & \theta = 0, \pm\pi\\ \frac{1}{2}(\theta - \pi), & -\pi < \theta < 0 \end{cases} $$ How can I find the Fourier series without doing any integration?
  10. D

    MHB Differentiating a fourier series

    What are rules for differentiating a Fourier series? For example, given $$ f = \frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\sin(2n-1)\theta}{2n-1} = \begin{cases} 1, & 0 < \theta < \pi\\ 0, & \theta = 0, \pm\pi\\ -1, & -\pi < \theta < 0 \end{cases} $$ Can this be differentiating term wise? If so...
  11. K

    When is a function equal to its Fourier series?

    When is a function "equal to" its Fourier series? First of all - a bit unsure where this post fits in, there seems to be no immediately appropriate subforum. So I'm a physics student and currently looking at what it takes for a Fourier series to converge. I've looked at wiki...
  12. B

    Another question about Fourier series convergence

    I am trying to prove a theorem related to the convergence of Fourier series. I will post my proof below, so first check it and then my question will make sense. Is there any flaw in my proof? Also, here I proved it for integrable functions monotonic on an interval on the left of 0. But what if...
  13. B

    Pointwise Convergence of Fourier Series for a continuous function

    Where is the fallacy in this "proof" that the Fourier series of f(x) converges to f(x) if f is continuous at x and has period 2π? (I read in Wikipedia that a counterexample had been provided). Start with the Dirichlet integral for the N-th partial sum of the (trigonometric) Fourier series...
  14. D

    MHB How to Solve the Heat Problem in the Disk using Fourier Series?

    Suppose $f(\theta) = |\theta|$ for $-\pi < \theta < \pi$. Find the formal series solution of the corresponding heat problem in the disk. How many terms of the series will give $u(r,\theta)$ with an error $< 0.1$ throughout the disk? Evaluate $u\left(\frac{1}{2},\pi\right)$ to two decimals. Show...
  15. P

    Pressure field decompose into Fourier series

    Hi, please can I ask you, how would you decompose some field (eg. pressure field in a given space) into a Fourier serie with a linear term? This is my problem: In my equations, the pressure acts in the form (the pressure was transformed only in the x axis, y-axis stays in real space, the k...
  16. D

    MHB Understanding a question Fourier series

    Apply Theorem 1.4 to evaluate various series of constants. Theorem 1.4: Let $f$ be periodic and piecewise differentiable. Then at each point $\theta$ the symmetric partial sums $$ S_N(\theta) = \sum_{n=-N}^Na_ne^{in\theta} $$ converge to $\frac{1}{2}\left[f(\theta)+f(-\theta)\right]$; if $f$ is...
  17. Z

    Signals: Fourier series and frequency response

    Homework Statement A recursive DT system with input x[n] and output y[n] is given by y[n] = -0.25y[n-2] + x[n] a) Determine and plot the impulse response h[n] such that y[n] = x[n]*h[n] b) how would you classify the system and why? c) What modification, if any, should be made so...
  18. D

    Calculate the Fourier series of the function

    Calculate the Fourier series of the function $f$ defined on the interval [\pi, -\pi] by $$ f(\theta) = \begin{cases} 1 & \text{if} \ 0\leq\theta\leq\pi\\ -1 & \text{if} \ -\pi < \theta < 0 \end{cases}. $$ f is periodic with period 2\pi and odd since f is symmetric about the origin. So f(-\theta)...
  19. D

    MHB Calculating Fourier Series of $f(\theta)$ on $[\pi, -\pi]$

    Calculate the Fourier series of the function $f$ defined on the interval $[\pi, -\pi]$ by $$ f(\theta) = \begin{cases} 1 & \text{if} \ 0\leq\theta\leq\pi\\ -1 & \text{if} \ -\pi < \theta < 0 \end{cases}. $$ $f$ is periodic with period $2\pi$ and odd since $f$ is symmetric about the origin. So...
  20. S

    Fourier Series Help: Piecewise Smooth | x=-1 to 1

    Homework Statement Hello, Check each function to see whether it is piecewise smooth. If it is, state the value to which its Fourier series converges at each point x in the given interval and the end points (a.) f(x)=|x|+x, -1<x<1 (it would be very helpful to see if i did this right, as the...
  21. A

    Uses of eulers equation in fourier series

    greetings, why do we use Euler equation that is e ^(jωt)=cos(ωt)+i sin(ωt) in Fourier series and what does it represent? advanced thanks.
  22. D

    MHB Proving the Basic Identity of Fourier Series

    If you write $$ e^{ik\theta} = \cos k\theta + i\sin k\theta, $$ then $\sum\limits_{k = 0}^ne^{ik\theta} = \frac{1 - e^{i(n + 1)\theta}}{1 - e^{i\theta}}$ yields two real sums $$ \sum\limits_{k = 0}^n\cos k\theta = \text{Re}\left(\frac{1 - e^{i(n + 1)\theta}}{1 - e^{i\theta}}\right) $$ and $$...
  23. R

    Finding the Sum of Sin2(na)/n2 Using Fourier Series for f(x)

    Homework Statement Use the Fourier series of f(x) = { 1 |x|<a { 0 a<|x|<\pi for 0<a<\pi extended as a 2-Pi periodic function for x \inR to find \sum Sin2(na)/n2 [b]2. Homework Equations [/b I got that the Fourier series of f(x) was a/\pi+\sum (2/(m\pi) sin(ma) sin(mx)...
  24. P

    Finding the Fourier Series of a Cosine Function.

    Homework Statement Given the function f(x) = Acos(∏x/L), find its Fourier series Homework Equations Okay so, f(x) is even, so the Fourier series is given by: f(x) = a0 + \sumancos(nx) where a0 = 1/∏\int f(x).dx with bounds ∏ and -∏ and an = 1/∏\int f(x)cos(nx).dx with bounds ∏...
  25. M

    MATLAB Plotting the Fourier series in Matlab

    I am very confused on how to start this problem, would highly appreciate some help! Consider the function f(x)=   0, -1≤x≤0 2x, 0≤x≤1. The Fourier series coefficients for this function are given by [a][/0]=0.5, and for  k=1, 2, 3,...  ...
  26. J

    Convergence of Unique Fourier Series

    Hi, I'm new to this forum, so I apologize if my LaTeX looks messed up. 1. Find the Fourier Series for f(x) = \sqrt{|x|} and prove it converges to f(x) 3. So, I've thus far proved that \sqrt{|x|} is piecewise continuous by proving that the limit as x approaches 0 (from both the right and left)...
  27. ShayanJ

    A question about Fourier series

    We know that because \sin{nx} and \cos{nx} are degenerate eigenfunctions of a hermition operator(the SHO equation),and eachof them form a complete set so we for every f(x) ,we have: f(x)=\frac{a_0}{2}+\Sigma_1^{\infty} a_n \cos{nx} and f(x)=\Sigma_1^{\infty} b_n \sin{nx} But...
  28. R

    Fourier series, applications to sound

    Homework Statement The Attempt at a Solution I don't understand where that 2 comes from in the denominator in cos nπ/2
  29. R

    Fourier series odd and even functions

    Homework Statement The Attempt at a Solution I don't understand the step above. It has something to do with this equation I think. I'm supposed to expand it into an appropriate Fourier series.
  30. R

    Fourier series complex numbers

    Homework Statement The Attempt at a Solution I don't understand this equation. 2pi, 4pi, 6pi only = 0 when there is a sine function before it, so I don't see how the evens = 0. I don't see why the e vanishes. I also can't get the i's to vanish since one of them is in exponential...
  31. R

    Fourier Series Convergence at the Origin

    Homework Statement The Attempt at a Solution Obviously brackets mean something other than parentheses because .5[0 + 0] ≠ .5
  32. A

    Fourier Series to Fourier Integral

    1. Question Consider\ any\ periodic\ function\ f(x)\ of\ period\ 2L\ that\ can\ be\ represented\ by\ a\ Fourier\ series:\ {f(x)= a_0 + \sum_{n=1}^\infty\ a_ncos\ w_nx + b_nsin\ w_nx}\ ,\ w_n= {n\pi x\over \ L } How\ do\ I\ get\ this\ form\ :\ f(x)= \int_{0}^{\infty}\...
  33. R

    Understanding Fourier Series Convergence: Common Confusions Addressed

    Homework Statement This series is what dictates the graph above. The Attempt at a Solution I don't understand what's going on. If they're using the series that i pasted below then why aren't they multiply each value in the brackets by -2/pi? I also don't get why terms...
  34. R

    Understanding the Role of n in the Denominator of Fourier Series

    Homework Statement The Attempt at a Solution I don't see why the n is in the denominator
  35. R

    Fourier Series 2: Solving Homework Problems

    Homework Statement The Attempt at a Solution I don't see how if n = 0 then the answer is 1/2. By my reckoning 1/pi * sin nx/n = 1/pi * 0 = 0 I also don't see where the 2 comes from when sin(npi/2) first shows up.
  36. R

    Problem with the fourier series

    Homework Statement this comes from a problem with the Fourier series The Attempt at a Solution I don't get the above step.
  37. J

    Fourier Series in Complex Form

    Hi, Just wanted to verify my process to complete the Fourier series in complex form. With the Function f(t) = 8, 0≤t≤2 -8, 2≤t≤4 f(t) = \sum Cn e-int n=1 to ∞ Where Cn = \frac{1}{T} ∫ f(t) e-int\frac{2∏}{T} Cn = \frac{1}{T} 0∫T f(t) e-int2∏/T dt Cn = \frac{1}{4}...
  38. T

    Fourier series - question

    Homework Statement f(x) = |sin x|, -pi < x < pi, f(x) = f(x + 2pi) Determine the Fourier series of f(x) The attempt at a solution I am unsure how to evaluate an integral with absolute signs in it, however, I am wondering if I could reduce the bounds to 0<x<pi and and f(x) = sin x and assume...
  39. H

    Fourier Series of cos4t + sin8t

    Hi everyone, So I was trying to calcule the Fourier Series of x(t) = cos4t + sin8t, but I'm a little bit confused. What would be ω0 in this case since I have a combination of two functions with different frequencies? Thank you in advance.
  40. G

    Does Fourier series of x^2 converge?

    I'm trying to show that the Fourier series of f(x)=x^2 converges and I can't. Does anybody know if it actually does converge? (I'm assuming that f(x)=x^2 for x\in [-\pi,\pi]). The Fourier Series itself is \displaystyle\frac{\pi^2}{3}+4\sum_{n=1}^\infty \frac{(-1)^n}{n^2}\cos nx I tried...
  41. D

    Question about the start of a cosine fourier series

    Question about the "start" of a cosine Fourier series Hey. I was just looking through Paul's Online Notes http://tutorial.math.lamar.edu/Classes/DE/FourierCosineSeries.aspx to teach myself Fourier Series and I had a question about the a_{0} term of the cosine series. In the online lesson...
  42. P

    MHB Proof of Fourier Series: F(ax) = (1/a)f(k/a) with F(x) as Fourier Transform

    Let f(k) be the Fourier transform of F(x). Prove that the Fourier transorm of F(ax) is $\frac{1}{a}f(\frac{k}{a})$ where a>0 and the Fourier transform is defined to have a factor of 1/2pi.
  43. T

    Complex Fourier Series: n from -∞ to +∞?

    Hi, I don't understand why does n goes from -∞ to +∞ in the complex Fourier series, but it goes from n=1 to n=+∞ in the real Fourier series?
  44. J

    Fourier Series of an Odd Piecewise function

    Homework Statement Fourier Series of the following function f(x). f(x) is -1 for -.5<x<0 f(x) is 1 for 0<x<.5Homework Equations b_{n} = \frac{1}{L}\int^{L}_{-L}f(x)sin(nπx/L)dx Where L is half the period.The Attempt at a Solution Graphing the solution, I know that it is odd, which is why I...
  45. H

    How to Expand a Function in a Half-Range Fourier Series

    Half wave Fourier series help ! I am trying desperately to figure out a Fourier series question i have been given and have hit a big mental wall can someone please help and maybe point me in the right direction. The problems i have been given are:- a) f(t)=e[^2t] 0<t<1 (cosine series)...
  46. G

    Fourier series on a general interval [a, a + T]

    Are there general formulas for Fourier coefficients on an integral [a, a + T], where T is the period. There is a general formula for the coefficients of exponential Fourier series. Are there general formulas for the coefficients of the trigonometric Fourier series that would work on any interval...
  47. H

    Fourier series and Euclidean spaces

    A book I'm reading says that the set of continuous functions is an Euclidean space with scalar product defined as <f,g> = \int\limits_a^bfg and then defines Fourier series as \sum\limits_{i\in N}c_ie_i where c_i = <f, e_i> and e_i is some base of the vector space of continuous functions. What...
  48. Δ

    Where did I go wrong in computing Fourier Series for f(x)=x^2 on [-\pi,\pi]?

    Homework Statement Let f(x) = x^2 on [-\pi,\pi]. Computer the Fourier Coefficients of the 2π-periodic extension of f. Use Dirichlet's Theorem to determine where the Fourier Series of f converges. Use the previous two conclusions to show that \sum_{n=1}^\infty \frac{1}{n^2} =...
  49. I

    Fourier series (maybe) of e^x from 0 to 2pi

    Hey, I have to show: Should I try to find the Fourier series from -2pi to 2pi? I have tried this already but I can't seem to get rid of the cos(nx/2) and sin(nx/2) to turn them into just sin(nx) and cos(nx) and the denominator stays as (n^2+4 instead of n^2+1. Any suggestions would be...
  50. E

    Fourier Series: Small Waves & Equal Amplitude

    In Fourier series we have small waves on the top of big waves (the function seems like that), but the small waves do not have the same amplitude. Does somebody know how to get a function with waves and small waves on the top but with the same amplitude.
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