Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials/sines and cosines are used.

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How to solve this Real Analysis?

Given that $f(x)= \sin(x + \pi/4)$ is periodic with period $2\pi$. Find the complex Fourier series. It's quite a moderate tough question. Can someone help me out. Thanks in advance
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151 views

Exponentials: Don't understand the simplification step

Can you explain how they go from the first line to the second? More specifically, what are the steps for making the negative exponential terms positive? i.e. how does the third term: $-e^{j(4\pi t + ...
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136 views

Order of partial sums in the derivatives of the Fourier series

Given periodic function $f\in C^{w}[0,1]$ with its Fourier series $f(x)=\sum\limits_{s=-\infty}^{\infty}f_{s}\exp(2\pi isx)$. What can one say about the asymptotic order of ...
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1answer
329 views

Power series for the sawtooth wave

This wikipedia article described a Fourier expansion of the sawtooth wave. Does this wave have a power series expansion (around any point)? If so, what is it? Does every function with a Fourier ...
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253 views

Bounding a function by its second derivative using Fourier series

I came across this Putname problem (2007, B2) the other day: Suppose that $f: [0,1] \to \mathbb{R}$ has a continuous derivative and that $\int_0^1 f(x)\,dx = 0$. Prove that for every $\alpha \in ...
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310 views

Finding the Fourier coefficients of this function

I need help finding the Fourier coefficients of: $f(x) =\begin{cases} \sum_{n=0}^\infty{\frac{e^{inx}}{1+n^2}} & \text{if } x\neq 2k\pi \\0& \text{if } x= 2k\pi ...
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2answers
260 views

Wave equation with initial and boundary conditions - is this function right?

If $y(x,t)$ satisfies the 1-dimensional wave equation $$\frac{\partial^2y}{\partial t^2}=c^2\frac{\partial^2y}{\partial x^2}\quad\text{for }0\leq x \leq l$$ with boundary conditions ...
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1k views

Relationship of Fourier series and Hilbert spaces?

I just read in a textbook that a Hilbert space can be defined or represented by an appropriate Fourier series. How might that be? Is it because a Fourier series is an infinite series that adequately ...
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3answers
557 views

Fourier Series (in reverse)

In electronics (and in other fields of engineering), we study that every periodic signal (whatever its shape) may be decomposed in a series of sine waves with frequency $f, 2f, 3f, \ldots, nf$ with ...
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2k views

Poisson's summation formula

It is said that the Fourier transform $\hat{f}(\omega)$ of a function $f(t)$ and the Fourier transform $\hat{b}(\omega)$ of its samples $b(k)=f(t)|_{t=k}$ are related by Poisson's summation formula ...
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290 views

Prove $\forall \delta >0, \, \exists C>0 \, \,s.t.\, |\int_{x}^{\pi}D_n(t)\, dt| \leq \frac{C}{n}$. (Dirichlet kernel)

I'm trying to prove that $\forall \delta >0, \, \exists C>0 \, \,s.t.\, |\int_{x}^{\pi}D_n(t)\, dt| \leq \frac{C}{n}$ for all $x \in [0,\delta]$, where $D_n$ denotes the Dirichlet kernel. I'm ...
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1answer
311 views

Convergence of: $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}\sin(nx)}{n}$

Need help with checking: $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}\sin(nx)}{n}$ for point-wise convergence and uniform convergence of: ${-\pi} \leq x \leq {\pi}$.
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111 views

finding the fourier coefficients of $f(x) = \sum_{n=1}^{\infty} \frac{\cos(nx)}{n^2}$

This is what i know so far: The given series uniformly converges by the M-test and that i can swap the integration and the sum when calculating the coefficients. Apparently i am supposed to use the ...
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2answers
64 views

Product in frequency domain of discrete data.

I have discrete data x[n]=[1 2 2] and h[n]=[3 4 1]. I can find their frequency counterparts using the fourier series x[iw]=[5 -1 -1] & h[iw]=[8 (1/2-3*sqrt(3)*i/2) (1/2+3*sqrt(3)*i/2)]. How can I ...
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1answer
517 views

What is the sum of only half the exponential terms that give the Dirac comb?

The following infinite sum of exponential terms gives a Dirac comb: $$ \sum_{n=-\infty}^\infty e^{i n x} = 2 \pi \sum_{n=-\infty}^\infty \delta(x - 2 \pi n) $$ Of course the sum doesn't strictly ...
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545 views

Is $\sin\left(\frac{\pi}{2}\sin\left(\frac{\pi}{2}\cdots\sin x\cdots\right)\right)=\frac4{\pi}\sum\limits_{k=0}^\infty\frac{\sin(2k+1)x}{2k+1}$?

We can see intuitively that $$ f(x)=\sin\left(\frac{\pi}{2}\sin\left(\frac{\pi}{2}\sin\left(\frac{\pi}{2}\cdots\sin{x}\cdots\right)\right)\right) $$ is the square wave with period $2\pi$ and has the ...
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1answer
165 views

Fourier series with trig parameters

Suppose I want to write the function $x \sin(t)$ as the series over the interval $x \in (0,\pi)$ $$x\sin(t) = \sum_{n=1}^{\infty}(a_n \cos(t) + b_n \sin(t) )\sin(nx)$$ Then would the coefficients ...
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1answer
163 views

Bounds on integral

I am calculating Fourier coefficients for certain functions and have come across an integral of the form $$I=\int_0^{2\pi} \int_0^1 r^2e^{2\pi i r(m\cos\theta+n\sin\theta)}drd\theta,$$ where ...
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77 views

Bounds on Fourier coefficients of Euclidean distance functions

I am interested in the bounding the Fourier coefficients $a_{m,n}$ of the function $f(x,y)=\sqrt{x^2+y^2}$ defined on the interval $[-1,1]^2$. I am specifically interested in understanding the ...
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1answer
367 views

Real part of an integral with complex argument

This is a paper about Fourier cosine series approximation to option pricing problem. The coefficient $A_k$ is defined as $$A_k=\frac{2}{b-a}\int_a^bf(x)\cos\left(k\pi\frac{x-a}{b-a}\right)dx$$ Then ...
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3answers
345 views

Easy Fourier series example: where is my mistake?

I'm doing exercise 15 on page 255 in Kreyszig: To illustrate that a Fourier series of a function $f$ may converge even at a point where $f$ is discontinuous, find the Fourier series of $$ f(x) = ...
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3answers
405 views

About completeness of the Fourier series.

The Fourier series of a function is given by $$ \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos n \theta + \sum_{n=1}^\infty b_n \sin n \theta . $$ Here what does the statement " $\sum_{n=1}^\infty b_n ...
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1answer
140 views

A trigonometric identity

If one sees the simplification done in equation $5.3$ (bottom of page 29) of this paper it seems that a trigonometric identity has been invoked of the kind, $$\ln(2) + \sum _ {n=1} ^{\infty} ...
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1answer
314 views

Fourier and half range series for $\sin x$

Expand in Fourier series of $f(x) = \sin x$ for $0<x<l$. Deduce the result \[ \frac1{1 \cdot 3} - \frac{1}{3\cdot5} +\frac{1}{5\cdot 7} - \cdots = \pi-\frac{2}{4}. \] Obtain half range ...
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1answer
123 views

Question about Fourier series

The Fourier series of a function $f: G \to \mathbb C$ where $G$ is a group is the representation of $f$ in terms of characters $\chi_g \in \mathrm{Hom}(G, S^1)$ of $G$. I understand the case where ...
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2answers
731 views

Fourier series expansion of $x^3$

Determine the value of $a_n$ in the Fourier series expansion of $$f(x) = x^3 , \quad - n \lt x \lt n.$$
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1answer
245 views

Can I apply Fourier Transform to Fourier Series?

Can I apply the Fourier Transform to a Fourier series?
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980 views

How can I apply a low pass filter to a Fourier Series?

I have a square wave signal and its Fourier Series. This signal pass through an ideal low pass filter,H(f), which has a cutoff frequency of 4KHz. What is the resulting baseband bandwidth of the ...
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1answer
205 views

Riemann sum estimate of Cauchy residue formula

Let $e(k) = \exp \left(\frac{2 \pi i k}{N}\right)$ be a root of unity. I wanted to numerically verify the Cauchy residue formula in Mathematica. $$ \lim_{N \to \infty}\frac{1}{N}\sum_{k=0}^{N-1} ...
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147 views

Cosine series as a Fourier series.

Theorem: Let $a_{n}\downarrow 0$ and suppose that $\left(a_{n}\right)$ is a quasi-convex. Then $\displaystyle \frac{a_{0}}{2}+\sum a_{n}\cos nx$ is the Fourier series of the $L^{1}$ function ...
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141 views

Function $f$ such that Fourier-series converges uniformly, but the series of the derivatives are divergent

I am studying Fourier-transformation right now, and I am asking if there exists a function $f$ such that is Fourier-series converges uniformly, the Fourier-series of $f'$ only in $L_2$ and that $f''$ ...
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1answer
61 views

norm of a variant of Fejer 's kernel

Let $K_N$ the Fejer's kernel on $\mathbb{T}$. Let $l$ be a positive integer. Let $Q$ the function defined by $$ Q(t)=K_N(lt). $$ In Hewitt/Ross "Abstract Harmonic Analysis 2" page 438, I can read that ...
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136 views

Check my solution - Modelling of a spring with Differential Equation

I am doing some work with differential equations. I have solved the following problem but am uncertain if I'm doing it correctly. Could someone look over it for me and check if I'm doing something ...
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1answer
779 views

Finding Fourier series with function not centered at the origin

I am trying to find both Fourier cosine and sine series which represent the function F(t) in the interval $(0, \pi)$ where $F(t)=\begin{cases} \frac{\pi}{2} & \ \ 0<t< \frac{\pi}{2}\\ 0 ...
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1answer
242 views

Does a closed form sum for this fourier series exist?

Continuing from an earlier question of mine: Fourier-Series of a part-wise defined function? I now got a fourier series which I believe is the correct one: $$\frac{\pi(b-a)}{2} + ...
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1answer
85 views

Squared Series Fourier [duplicate]

Possible Duplicate: Fourier 1st step? How to find fourier transform of a series of the such form: $$y_k=\left[f(x) \right]^{2},$$ but I am not sure of the step by step for going about this ...
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1answer
153 views

Fourier-Series of a part-wise defined function?

I have a function f given as $$ f(x) = \begin{cases} ax&\text{ if }\quad-\pi \leq x \leq 0\\ bx&\text{ if }\quad 0<x\leq\pi \end{cases} $$ I am supposed to develop the fourier series of ...
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157 views

Trigonometric series as a Fourier series of essentially bounded function.

A trigonometric series $ \displaystyle \frac{a_0}{2}+\sum_{n=1}^{\infty}(a_{n}\cos nx +b_{n}\sin nx)$ is a Fourier series of a essentially bounded function if and only if there exists a constant $K$ ...
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250 views

expansion for $1-|t|$

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $$ I am wondering if one can expand ...
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2answers
88 views

Fourier and integral

Given the below trigonometric series: $1 + \sum_{n=1}^{\infty} \frac{2}{1+n^{2}}\cos (nt)$ Where $f(t)$ is the value of the series. Can I then deduce that $\int_{-\pi}^{\pi} f(x) dx$ is ...
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1answer
179 views

Convergence in mean of differentiated Fourier series

Let a function $f: \mathbb{R} \rightarrow \mathbb{R}$ be integrable on $[-\pi,\pi]$ and $2\pi$-periodic. Let $$ \frac{a_0}{2}+\sum\limits_{n=1}^\infty (a_n \cos nx+b_n \sin nx) $$ be the Fourier ...
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1answer
352 views

Poisson summation formula and Schwartz functions

I am reading a proof of the Poisson summation formula which states that (with my version of the Fourier transform - I think they sometimes vary by a constant factor) for $f$ a Schwartz function on ...
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32 views

Asymptotic order of some sums with the Fourier coefficients

Given $f\in C^{w}[0,1]$ with periodic conditions $f(0)^{(j)}=f(1)^{(j)},\ j=0,\dots, w-1$ and its Fourier series are $f(x)=\sum_{l}f_{i}\exp(2\pi ix)$. I need to find the asymptotic order of errors ...
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2answers
306 views

Equating coefficients in a Fourier series

Suppose, for example, using Fourier series techniques to solve a differential equation leads to the following: $a_0 + \sum_{n=1}^{\infty}a_n\sin(nx)+b_n\cos(nx)=4\sin x$ At this point, why can you ...
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1answer
802 views

A function and its Fourier transform cannot both be compactly supported

I am stuck on the following problem from Stein and Shakarchi's third book. I can't figure out how to use the hint productively. Once I know $f$ is a trigonometric polynomial, I see how to finish the ...
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1answer
252 views

Fourier Coefficients of periodic function

Consider a Function $f\in L^2(\mathbb{T})$. Is there any lower bound for the decay of the Fourier coefficients $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt$$ known? There are a lot ...
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227 views

Polynomials in Fourier trigonometric series

I'm successively integrating $x^{n} \cos{k x}$ for increasing values of positive integer n. I'm finding: $\frac{\sin{kx}}{k}$, $\frac{\cos{kx}}{k^2}+\frac{x\sin{kx}}{k}$, $\frac{2 x ...
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1answer
521 views

Extending a function to become odd or even?

"Suppose we have a function defined on an interval [0,K], then we extend it as an even or odd function of period K so as to produce a Fourier cosine or sine series." (1): What exactly is extending a ...
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1answer
581 views

Dirichlet kernel.

I have a function $h\in L^1(\mathbb{T})$, and I want to show that: $$\int_{\pi\geq |t|>\delta>0} h(x+t)D_N(t) dt/2\pi \leq \xi_N(h,\delta)$$ where $\xi_N(h,\delta) \rightarrow 0$ as ...
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1answer
207 views

Trigonometric series as a Fourier series.

I want to show that if $f(x)\in L^{p}(p>1)$ and $\phi(x)\in L^{q}$, where $\displaystyle \frac{1}{p}+\frac{1}{q}=1$ then the trigonometric series $\displaystyle ...