Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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A Fourier series exercise

Can anyone give me a hand with this exercise about Fourier series? Let $f(x)=-\log|2\sin(\frac{x}{2})|\,\,\,$ $0\lt|x|\leq\pi$ 1) Prove that f is integrable in $[-\pi,\pi]$. 2) Calculate the ...
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1answer
656 views

how to find absolute value for complex fraction

I have a Fourier transfer equation $H(jw) = \frac{jwL}{(jw)^2LC+jw\frac{L}{R}+1}$, and I need to find frequency to make $|H(jw)|$ is max. I know I should take the derivative of $|H(jw)|$ then find ...
5
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1answer
264 views

Highly Oscillating Integrals

I'd like to know the behavior of integrals of the form: $$ \int_0^1 f(x) \cos(k x) dx $$ as $ k \rightarrow \infty $ where f is a smooth function. It is easy to see, by expanding f in power series, ...
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425 views

Singular asymptotics of Gaussian integrals with periodic perturbations

At the bottom of page 5 of this paper by Giedrius Alkauskas it is claimed that, for a $1$-periodic continuous function $f$, $$ \int_{-\infty}^{\infty} f(x) e^{-Ax^2}\,dx = \sqrt{\frac{\pi}{A}} ...
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118 views

Intervals where the function is similar to the Fourier series

$$f(x)=\left\{\begin{array}{l l} 0,\quad x \in [-L,0[\\ 1,\quad x \in [0,L] \end{array}\right.$$ I need to know in which intervals the sum of the Fourier series is "equal to the function $f(x)$". ...
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47 views

How to prove this Fourier question?

How to prove this Fourier question? I hope for a procedure in detail.
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201 views

Fourier Analysis of Prime Counting Function

I was thinking about the following: Denote $\pi(x)$ as the prime counting function such that: $$ \pi(x) = \#\text{ of prime numbers}\leq x $$ It is well known from the prime number theorem that $$ ...
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181 views

Problem 25 pg 95, Stein and Shakarchi: $F(\xi) = 1/(1+|\xi|^2)^\epsilon$ is the Fourier transform of a function in $L^1(R^d,m)$.

Show that for any $\epsilon>0$, the function $F(\xi) = 1/(1+|\xi|^2)^\epsilon$ is the Fourier transform of a function in $L^1(R^d,m)$. [Hint: $K_{\delta}(X) = e^{-\pi|x|^{2/\delta}} ...
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150 views

Problem #23 pg-94, Stein and Shakarchi

As an application of the Fourier transform, show that there does not exist a function $I\in L^1(R^d,m)$ such that $f*I = f$ for all $f\in L^1(R^d,m)$.
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602 views

Fourier transform of a compactly supported function

In which space does the Fourier transform of a smooth compactly supported function $\phi$ lie? I would not say it lies in $\mathcal{S}$, heuristically as one can approximate the step function which is ...
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1answer
153 views

Are Trigonometric Functions Dense in $C^k(S^1)?$

Consider the functions $\{e^{2\pi i nx}\}_{n \in \mathbb{Z}}$ defined on the interval $[0,1].$ These are all smooth periodic functions (so functions on $S^1)$ and by the Stone-Weierstrass theorem ...
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41 views

Understanding the indices in a Fourier series

Sometimes the truncated Fourier series of a function with Fourier coefficients $\hat{u}_k$ is written $$\sum_{k=-N}^N\hat{u}_ke^{ikx}$$ which is a linear combination of $\cos(nx) +i\sin(nx)$ for ...
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1answer
226 views

Only when the function is odd its Fourier coefficient of cosnx is equal to 0?

Only when the function is odd its Fourier coefficient of cosnx is equal to 0?
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106 views

Inversion of a Fourier Transform

I am told that the functions $f(x), g(x)$ and $h(x)$ satisfy: $\hat{f}(k) = \dfrac{\hat{h}(k)}{A+\hat{g}(k)} $, where $\hat{f}(k)$ is the Fourier Transform of $f(x)$ (likewise for $h(x)$ and $g(x)$ ...
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117 views

What are the properties of this kind of Fourier transform when using it to solve Wave equqtion with Robin boudary data?

I am facinated by the following problem (Which is found in Budak & Samarskii's Book Collection of Problems on Mathematical Physics, about on Page 53.): Solve the following problem by using ...
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1answer
2k views

Is deconvolution simply division in frequency domain?

Is it correct to say that deconvolution simply division in frequency domain? And that convolution in time domain is multiplication in frequency domain. And is it a convention to notate a function in ...
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2answers
124 views

Fourier Transform calculation

I am trying to calculate the Fourier Transform of $$f(x)=\exp(-\frac{|x|^2}{2}). $$ Thus, I am looking at the integral $$ \hat{f}(u)=\int_{\mathbb{R}^n} \exp(-\frac{|x|^2}{2}) \cdot \exp(ix\cdot u) ...
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1answer
441 views

Discrete Time Fourier Transform example: $x = [1 \; 2 \; 3 \; 4]^T \; \rightarrow \; X=?$

How do I find the Discrete Fourier Transform of the sequence below? $$ x = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}$$ Show all steps.
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1answer
84 views

Why aren't these two question equal?

Firstly I doubt whether the 12 is right in Q1.If it is right,please give a proof. Secondly why (1) is not equal to (2) in Q2?
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1answer
73 views

I can't understand the last second step.

It is known that $f(x)=\sum_{n=-\infty}^{\infty}c_{n}e^{inx}$, with $c_{n}:=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\:dx$, for $n\in\mathbb{Z}$. To prove: ...
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1answer
159 views

Computing the Fourier transform (in the sense of tempered distributions) from a sequence of samples

Let $f\in L_{\operatorname{loc}}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity: $$\hat{f}=\sum_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$$ ...
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1answer
63 views

Fourier Transform is onto $L^2(\mathbb{R})$

In Introduction to Fourier Analysis on Euclidean Spaces, the authors explore the $L^2$ extension of the Fourier transform and argue that it is onto $L^2(\mathbb{R})$ but I can't follow their ...
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3answers
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Klein Bottle discrete harmonics?

Studying discrete representations of a function, on a $2$ Dimensional compact surface, brings to the use of spherical harmonics for the 2-sphere, and discrete Fourier transformations for the 2-torus. ...
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1answer
115 views

number of zeros of the superposition/interference of sine oscillations

There is a tricky problem to solve and we ask for your kind help. In a rather simple version of the problem, we have a set of oscillations of type $f(x)=\sin (\omega x + \phi)$ with different phases ...
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38 views

nasty exponentials

While trying to find the fourier transform of $\Large \frac{1}{1 + x^4} $, using the definition and the residue theorem has required me to evaluate nasty looking expressions like $$\large \rm ...
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92 views

Show existence of continuous functions $f$ with $f''=\delta_0-\delta_1$

Let $u$ be the distribution on $\mathbb{R}$ given by $$u=\delta_0-\delta_1 $$ (a) show there exists a continuous function $f$ such that $f''=u$ and indicate such one. I thought of doing this with ...
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1answer
55 views

Show that Fourier coefficients approach zero uniformly

Let $f(t)$, $g(t)$ be piecewise continuous functons on $[-\pi,\pi]$, periodically continued on $\mathbb R$. I want to show that $$ a_n(x) = \frac{1}{\pi} \int\limits_{-\pi}^{\pi} f(x+t)g(t) ...
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1answer
191 views

Problem evaluating an improper integral $\int_0^{\infty} \frac{(\sin{2x}-2x\cos{2x})^2}{x^6}$ using fourier transform

This is a question from one of the past papers of my university which I am unable to do. I am not being able to do question 2 from below. Let $f(x)= a^2-x^2 \,\,\,\,\, |x|<a ...
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Calculate the Fourier transform of ${\rm b}\left(x\right) = 1/\left(x^{2} +a^{2}\right)$

I need help to calculate the Fourier transform of this funcion $${\rm b}\left(x\right)=\frac{1}{x^{2} + a^{2}}\,,\qquad a > 0$$ Thanks.
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1answer
73 views

Extension of Fourier Transform

We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ ...
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Solving an initial value ODE problem using fourier transform

I am a physics undergrad and studying some transform methods. The question is as follows: $y^{\prime \prime} - 2 y^{\prime}+y=\cos{x}\,\,\,\,y(0)=y^{\prime}(0)=0\,\,\, x>0$ I am having some ...
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Fast Fourier Transform of 2D axisymmetric geometry

I need to compute the FFT of a signal in 2D axisymmetric geometry. The signal consists of a snapshot in time of a laser beam for which I have values in z (direction of propagation) and r (from 0 to ...
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Solving an integral equation using the Fourier transform

I have to solve the equation $\int_0^{\infty} f(x) \cos{(\alpha x)}\, dx=\frac{\sin{\alpha }}{\alpha}$ Using fourier transform. I know this is half of the usual fourier cosine transform, and so ...
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1answer
92 views

Wave-Function Series?

So I was basically exploring the function: $\displaystyle {\text{frac}(x)}$ which is the fractional part function and I noticed that it has a nice fourier series definition which is: ...
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1answer
76 views

Fourier transforms - don't understand this concept!!! Please help me on this

I have two Fourier transforms to solve, but the problem is that a I have a characteristic bijection or some etching that I don't know what it is and I don't know how to solve this... Please help ...
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1answer
71 views

Fourier transform parameter

I'm currently working on the problem: If $\hat{f}(k)$ is the complex Fourier Transform of the function $f(x)$ and $a$ is a real constant with $a>0$, show that the complex Fourier Transform of ...
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2answers
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Sum over cosines = dirac delta - how to get the coefficients?

Given this formula: $$\sum\limits_{n=0}^\infty a_n \cos(n \pi x / d) = \delta(x-x_0)$$ Where $0 \leq x \leq d$. How can one calculate the coeffciients $a_n$? I googled and searched all kinds of ...
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0answers
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Complex Fourier series of a function [duplicate]

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
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1answer
438 views

Complex Fourier series

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
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2answers
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Fourier Series of $f(x) = x$

I am having trouble finding the complex Fourier series of $f(x) = x$ and using that complex series to find 1)the real Fourier series of $f(x)$ and 2) the complex and real Fourier series of $h(x) = ...
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1answer
130 views

Unclear on relationship between different dimensionalities of Fourier transform

This is probably a silly question, but it's one that's directly relevant to a project of mine and I figured this was the place to go. I have some objects that contain a 1d and a 2d array of double ...
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2answers
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Is $\{g(n)\}_{n\in\mathbb{Z}}\in\ell_2$ if $g$ is a Sobolev function on the real line?

If we are given a function $g\in W_2^k(\mathbb{R})$ (even consider $k=1$ for simplicity), then is it true or not that $\{g(n)\}_{n\in\mathbb{Z}}\in\ell_2$? That is, do we have ...
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Tough Inverse Fourier Transform

In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
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1answer
51 views

A function whose derivatives always have a convergent fourier series

I am looking for a solid example that such a function that its derivatives can always be found by taking derivatives component-wisely in its Fourier series. A function with finitely many Fourier terms ...
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270 views

Fourier transform of $(1-\cos(tx))/x^2$

I am trying to compute the Fourier transform of $f(x) = \frac{1-\cos(tx)}{x^2}$, $(t > 0)$ directly. I tried contour integration, and could not seem to get it to work. So, I am wondering if it can ...
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3answers
714 views

How to reperesent $\sin^{4}(x)$ byFourier series? [closed]

how to represent $\sin^{4}(x)$ by Fourier series? Obviously,$\sin^{4}(x)$ is an even function, so $b_n=0$. How can i get $a_n$ ?
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2answers
132 views

How to solve this equation by Fourier series?

$$ y''+3y=\sin ^4 x ,\quad y=\frac{1}{8} +\frac{\cos2x}{2}-\frac{\cos4x}{104}.$$ Now the text book states the solution, but I don't know the process of solving this equation. I need your help!
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1answer
63 views

I need to find the Fourier transform of the following function

I need to find the Fourier transform of $e^{-2 \pi |x|}$. Normally, I can do something like this, but the absolute value is kind of confusing me.
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314 views

Show that f is a polynomial

Suppose $f$ is an entire function on $\mathbb{C}^n$ that satisfies for every $\epsilon>0$ a growth-condition $$|f(z)|\leq C_{\epsilon}(1+|z|)^{N_{\epsilon}}e^{\epsilon | \text{Im}\,z|}$$ ...
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299 views

Upper bound on truncation error of a fourier series approximation of a pdf?

Given a probability density function, $f\left(x\right)$, of a continuous random variable, $X$, and given an $N$-th order fourier series approximation: $$f_N\left(x\right)=\sum_{n=-N}^{N}c_n e^{inx}$$ ...