We use the absolute value in equation [2] so that the norm is defined for complex functions, in case we felt like working with those. Can two integer polynomials touch in an irrational point? Using the Fourier Coefficients found on that page, we can plot the mean squared error between gn(t) and f(t): Figure 1. Please try the request again.

We could look at the distance (also called the L2 norm), which we write as: [Equation 1] For x and y above, the distance is the square root of 14. Generated Sun, 16 Oct 2016 00:56:27 GMT by s_ac15 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.6/ Connection Since you know $g$, you can find $N$ that will achieve the required precision. reference-request fourier-analysis fourier-transform share|cite|improve this question edited Oct 6 '14 at 9:55 asked Oct 6 '14 at 9:41 Paglia 390114 1 The Fourier series of an $L^2$ function on the

Related 0Deriving fourier series using complex numbers - introduction2Upper bound on truncation error of a fourier series approximation of a pdf?2Series evaluated to $m$ terms, approximating the error1The Fourier series of Generated Sun, 16 Oct 2016 00:56:27 GMT by s_ac15 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.8/ Connection The Mean Squared Error between gN(t) and f(t). In the next section, we'll look at deriving the optimal Fourier Coefficients (that is, the proof for equation [3] on the complex Fourier series coefficients page. Next: Derivation of Complex

Yes, I need an estimate –Paglia Oct 6 '14 at 9:50 1 Since continuous functions are dense in L^2, I don't think their rates of L^2 convergence can be any Please try the request again. Instead, integrate by parts, turning the integrals into $$ \frac{1}{2\pi N}\int_{-\pi}^\pi \frac{d}{dt} \left(g(t)\cos \frac{t}{2}\right)\cos Nt\,dt - \frac{1}{2\pi N}\int_{-\pi}^\pi \frac{d}{dt}\left(g(t)\sin \frac{t}{2}\right)\sin Nt\,dt $$ plus boundary terms (coming from discontinuities of $g$), each A bullet shot into a door vs.

Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the up vote 2 down vote favorite Suppose I have a piecewise smooth $2 \pi$-periodic function $f$ on $\mathbf{R}$ with a Fourier series $\sum_{n \in \mathbf{Z}}a_n e^{inx}$, a number $x_0 \in \mathbf What is the distance between f and g? Ideally, I would like an answer in the spirit of estimating the error term for Taylor series.

I would like an upper bound for $|f(x_0)-\sum_{n=-N}^N a_n e^{inx_0}|$. a bullet shot into a suspended block Animal Shelter in Java How to handle a senior developer diva who seems unaware that his skills are obsolete? And.. Your cache administrator is webmaster.

This "distance" is also known as the Mean Squared Error (MSE). question feed about us tour help blog chat data legal privacy policy work here advertising info mobile contact us feedback Technology Life / Arts Culture / Recreation Science Other Stack Overflow Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the For example, if $f$ is a periodic function so that $f(x)=x$ on $(-\pi,\pi)$, and I require a partial Fourier series which is within $\frac{1}{2}$ of $f(x_0)$ at, say, $x_0=1$, I want

Proof: Define $$g(t)=\frac{f(x-t)-f(x)}{\sin (t/2)}$$ so that $$s_N(f;x)-f(x)= \frac{1}{2\pi}\int_{-\pi}^\pi \left[g(t)\cos \frac{t}{2}\right]\sin Nt\,dt + \frac{1}{2\pi}\int_{-\pi}^\pi \left[g(t)\sin \frac{t}{2}\right]\cos Nt\,dt $$ For Rudin, an application of the Riemann-Lebesgue lemma ends the proof here. NO jumps) whose Fourier series converge slower than any prescribed rate of convergence. Generated Sun, 16 Oct 2016 00:56:27 GMT by s_ac15 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.10/ Connection We are interested in the distance (MSE) between gN(t) and f(t).

I've seen the wiki page, is there a particular section that would answer this particular question? –Steven Spallone Oct 1 '13 at 8:52 add a comment| 1 Answer 1 active oldest Please try the request again. Generated Sun, 16 Oct 2016 00:56:27 GMT by s_ac15 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.7/ Connection Your cache administrator is webmaster.

more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed Please try the request again. How do investigators always know the logged flight time of the pilots? To give an idea of the convergence, let's look again at the square function from the complex coefficients page.

Specifically, we are interested in knowing about the convergence of the Fourier Series Sum, g(t) (equation [3]), with the original periodic function f(t): [Equation 3] To get an idea of the How do computers remember where they store things? current community blog chat Mathematics Mathematics Meta your communities Sign up or log in to customize your list. Continuous FT) in terms of Sobolev order-3A question about pointwise convergence of Fourier transform in $N$-dimensions2What is the Fourier transform of this function?4Eliminating Gibbs phenomenon, and approximating with jumping functions in

The system returned: (22) Invalid argument The remote host or network may be down. Your cache administrator is webmaster. In it, you'll get: The week's top questions and answers Important community announcements Questions that need answers see an example newsletter By subscribing, you agree to the privacy policy and terms asked 3 years ago viewed 1832 times active 3 years ago Get the weekly newsletter!

asked 2 years ago viewed 180 times active 2 years ago Related 3Bandwidth approximation for a nonlinear problem2Behavior of the Fourier transform (FT) of a function and FT of its absolute For example, take Theorem 8.14 in Rudin's Principles of Mathematical Analysis (3rd edition): Fix $x$ and suppose there are constants $\delta>0$ and $M<\infty$ such that $$|f(x+t)-f(x)| \le M|t|$$ whenever $|t|<\delta$. So for vectors, it's pretty simple to define some sort of distance. The system returned: (22) Invalid argument The remote host or network may be down.

My (rough) estimate is $$|s_N(f;x)-f(x)|\le \frac{k}{\pi N}\sup |g|+ \frac{1}{N} \sup \left| \left(g(t)\cos \frac{t}{2}\right)' \right| + \frac{1}{N} \sup \left| \left(g(t)\sin \frac{t}{2}\right)' \right|$$ where $k$ is the number of discontinuities of $g$.