The fourier-analysis tag has no usage guidance.

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56 views

### Integrating a series expansion of $\mbox{frac}(x)\lfloor x\rfloor$ coming from Fourier series of sawtooth function

Let me preface this question by saying that I am not exactly sure it counts as research level. It is crossposted on mathstackexchange: ...

**2**

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**1**answer

28 views

### What are the known conditions for the log of the Fourier transform of a 2D real discrete signal to have no branch cuts?

Suppose we sample a real 2D signal, f(x,y), at N by N evenly spaced points in x and y. Then we compute the Fourier transform of the sampled signal, F(u,v), and then take the log of F. There will be a ...

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230 views

### Completeness of nonharmonic Fourier Series

I have the following question:
The Exponential System $(\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}}$ constitutes an orthonormal basis of $L^2([-1/2,1/2])$.
Thus, certainly the oversampled system ...

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vote

**1**answer

99 views

### A metric on the set of BV functions, is it mentioned/studied in literature?

I'd like to propose the following metric which operates on the set $M$ of all square integrable functions that are also of bounded variation, of the form $f : (0,1) \to \mathbb{R}$.
Given any $x,y \in ...

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122 views

### Proof without distributions

I was wondering whether there is a way to show this identity
$$\pi \int_{\mathbb{R}^3} \frac{f(x)}{|x|} dx = \int_{\mathbb{R}^3} \frac{\widehat{f(x)}}{|x|^2} dx $$ without using distributions for $f ...

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votes

**1**answer

71 views

### Lower bounds from Fourier dimension?

According to Mattila, Geometry of sets and measures in Euclidean spaces, p. 168, the Fourier dimension $\text{dim}_F(A)$ of $A\subseteq \mathbb R^n$ is the unique number in $[0,n]$ such that for any ...

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62 views

### Combining oscillatory integrals of the first and second kind

Consider an oscillatory integral of the first kind
$$
I_\lambda(x)=\intop_{\mathbb{R}^{n}}e^{i\lambda\Phi(x,y)}a(x,y)\,d y,\quad \lambda\geq 0,\; a\in C_c^\infty(\mathbb{R}^{k+n}),\; \Phi\in ...

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**1**answer

213 views

### Maximal $L_1$ norm of Fourier Transform of a Subset

Let $A_n$ be the following $n\times n$ matrix: $(A_n)_{i,j}= \frac{1}{\sqrt{n}}\omega_n^{i\cdot j}$ for all $0 \le i,j <n$, where $\omega_n=e^{\frac{2\pi i}{n}}$.
I want to understand how $A_n$ ...

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81 views

### Error term for a Fourier integral

There is a well-known theorem that states that for $f$ continuous and $f,\hat f$ integrable,
$$f(0)=\frac{1}{\pi}\lim_{T\to\infty}\int_{-\infty}^\infty f(x)\frac{\sin(Tx)}{x}dx.$$
So it should be that
...

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235 views

### Bounding exponential sum with square roots

It is well known that for each $m\in\mathbb{N}$
$$\lim_{N\to\infty}\frac1N\sum_{n=1}^Ne^{2\pi i\sqrt{nm}}=0$$
My question is whether there is some uniformity in the variable $m$.
More precisely, is it ...

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votes

**1**answer

74 views

### Fractional Sobolev spaces on the circle with a Littlewood-Paley characterisation

Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L_p(\mathbb R)$.
Here, $F$ denotes the ...

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**1**answer

358 views

### Can exponential sums be small on a whole interval?

This is almost certainly routine to an analyst, so forgive me in advance.
Let $\alpha_i\in \mathbb{R}$. Consider the functional $$\varphi: L^1[0.9A,A]\to \mathbb{C}$$ via $$f\mapsto \sum_i ...

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162 views

### Name for an orthogonal decomposition of $L^2 (\mathbb{R}^2; \mathbb{C})$

The space $L^2 (\mathbb{R}^2; \mathbb{C})$ can be decomposed as
$$
L^2 (\mathbb{R}^2; \mathbb{C}) = \bigoplus_{k \in \mathbb{Z}} L^2_k (\mathbb{R}^2; \mathbb{C}),
$$
where
$$
L^2_k (\mathbb{R}^2; ...

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61 views

### Spaces $C^\infty(\mathbb T^n\times \mathbb R^n)$, $C^\infty_0(\mathbb T^n\times \mathbb R^n)$ and $\mathscr{S}(\mathbb T^n\times \mathbb R^n)$? [closed]

Is there any characterization of the space $C^\infty(\mathbb T^n\times \mathbb R^n)$ that I can take as a definition of it?
I assume it would be something like this:
$$C^\infty(\mathbb T^n\times ...

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52 views

### Looking for some “nontrivial” examples of pseudodifferential operators/symbols

I'm reading up on $\Psi DO$'s and trying to find some examples of symbols that are not quite so trivial.
Obviously, the first example of a symbol that most people talk about is just a polynomial in ...

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**1**answer

65 views

### Help with notations from 2D to 3D FFT representations as 1D FFT

I have this question on mathematics forum too, Notations, I thought of posting here, which ever place I get an answer, I will try to close it in the other.
I need some help and clarifications for my ...

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**1**answer

47 views

### Locality of homogeneous pseudo-differential operator

Let $P$ be a polynomial in several variables, and let $P(D)$ be the corresponding differential operator. Obviously, $P(D)$ is a local operator, in the sense that I need only to know the function $u$ ...

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101 views

### Uniqueness of the “Gubinelli” Derivative in the Theory of Paracontrolled Distributions

From the theory of Rough Paths it is well known that if we have a truly rough path $X$ and two controlled rough paths $(Y,Y'),(Y,\tilde{Y}')\in\mathcal{D}_X^{2\alpha}$, then we have already $Y' = ...

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33 views

### Wiener amalgam space $W(\mathcal{F}L^{2}, L^{1}) \subset L^{1}$?

(I have asked this question on SE but could not get any answer and hope this is o.k for MO)
Let $X=\mathcal{F}L^{p}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{p}(\mathbb R)\},$ and $\|f\|_{X}= ...

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87 views

### Swapping sums and integration for a kernel in Fourier space (the non absolutely convergent case)

Under what conditions on $c_{r}^{m}$
does
$$\int_0^{2\pi} k(p,q)\exp(-inq)dq=\sum_{r=0}^{\infty}c_{r}^{m}\exp\left(-imp\right) \text{ in } L^2_{per}$$
hold for
...

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70 views

### Is there a space in which the $\vec a$ in $\sin(a_1\cdot x)+\sin(a_2\cdot x)$ is linear?

Suppose one has equations of the form $\sin(a_1\cdot x_i)+\sin(a_2\cdot x_i)=y_i$ for $i = 1, \dots, n$ (there are also amplitudes and phase shifts, but let's ignore these for now). I want to solve ...

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346 views

### Proof of a Fourier pair with Bessel functions?

How can we prove that the Fourier transform of the function
$$
f(x)
=
\begin{cases}
(a^2-x^2)^{c/2} BesselJ[c,b\sqrt{a^2-x^2}] & \text{for }x^2 < a^2\\
0 & \text{otherwise}
\end{cases}
$$
...

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**1**answer

72 views

### Elaboration of a certain section of a paper by Thanigasalam

In section 11 of this paper by Thanigasalam, it says "... we get $G(10)\le 105$, and this implies that $H(10) \le 107$". However, it is very unclear how this follows. Why is it the case that $G(10)\le ...

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215 views

### Derivative of Band-limited functions [closed]

I'm trying to answer this problem: Consider a real function f, bandlimited by frequency $\omega$, which satisfy
$$\int_{-\infty}^\infty f(x)^2dx=c.$$
(For pure mathematicians: "bandlimited" means ...

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**1**answer

116 views

### Simplifying an expression using tools from Fourier transform

Can anyone simplify the following expression? I guess something from Fourier transform can help:
$f(\omega) = \lim_\limits{R \to \infty} \frac{1}{R^2} \int_{r=0}^{R}{re^{ \omega r^{-\gamma}}} ...

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200 views

### Localization arguments in the paper 'the proof of $l^2$ decoupling conjecture'

I am currently reading Jean Bourgain and Ciprian Demeter's 2015 paper The proof of the $l^2$ decoupling conjecture and would appreciate some help in understanding localization argument used in that ...

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223 views

### What is the importance of convergence of variation of Fourier reconstruction to that of variation of the function?

Let $f$ be a periodic function of bounded variation which jumps at a point $x_0\in\mathbb{R}$. Let $S_{N}[f]$ denote the partial Fourier sum of $f$ and let $C_{N}[f]$ denote the Cesaro partial sum. It ...

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152 views

### Composition operators on fractional-order (periodic) Sobolev spaces

(The question was originally posted on MSE.)
Preliminaries: We know that the fractional-order Sobolev spaces $\mathrm{H}^s(\mathbb{R})$ and $\mathrm{H}^s(\mathbb{T})$ are closed under multiplication ...

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152 views

### Fourier Transform of Eisenstein Series - Sum of Divisors or Ramanujan Sums?

I am stuck on this computation of the Fourier coefficients of Eisenstein series. For $\Gamma = SL(2, \mathbb{Z})$ and $\Gamma_\infty = \left\{ \left( \begin{array}{cc} 1 & m \\ 0 & 1 ...

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66 views

### A bound on the $\mathrm{L}^p$ norm in terms of the $\mathrm{L}^2$ norm in periodic Sobolev spaces

(The question was originally posted on Math StackExchange.)
Preliminaries: Given ${s \geq 0}$, let ${\mathrm{H}_P^s}$ denote the ${\mathrm{L}^2}$-based fractional-order Sobolev space of ...

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**1**answer

220 views

### The $L^2\times L^2\to L^2$ norm of the bilinear multiplier operator

Consider a general bilinear multiplier operator:
$$
T(f,g)(n)=\int_{\Pi}\int_{\Pi}\hat{f}(\xi)\hat{g}(\eta)e^{2\pi i(\xi+\eta)n}m(\xi,\eta)d\xi d\eta,
$$
where $\Pi$ is the torus, $n\in\mathbb{Z}$, ...

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358 views

### Can phase significantly concentrate a function's spectrum?

Let $F$ denote the Fourier transform over some group. What is known about the following quantity?
$$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$
Here, $|x|$ denotes the pointwise absolute ...

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60 views

### Is this similarity to the Fourier transform of the von Mangoldt function real? [duplicate]

This question proposed in SEM but no answer and it's interesting to know connection
between Fourier analysis and number theory .
Mathematica knows that the logarithm of $n$ is:
$$\log(n) = ...

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58 views

### Fourier coefficients of positive polynomials

Let $p(x) \geq 0$ be a positive polynomial on the hypersphere ($x \in S^{n-1}$) satisfying $\int_{S^{n-1}} p(x) = 1$. Writing $p(x) = \sum_{j=0}^s p_j(x)$ where $p_j(x) = \sum_m p_{jm} s_{jm}(x)$ with ...

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136 views

### Fourier series and transform related to Epicycles

Let $\gamma:\mathbb{R}\to\mathbb{C}$ be a continuous periodic curve having a bounded variation.
1) Is it true that one can find a sequence of numbers $(r_n)_{n\in\mathbb{N}}\subset (0,\infty)$ and ...

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711 views

### Is $f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}$ essentially bounded?

Let $$f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}
$$
Is it true that $\|f\|_{L^{\infty}(\mathbb{R}^2)}<\infty$? i.e. is $f$ essentially bounded?

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389 views

### Best known bounds on certain exponential sums

What are the best bounds currently known for the following exponential sum:
$$\sum_{x < p \le 2x} e(\alpha p^k)$$
for values of $\alpha$ far from a rational with small denominator. ($p$ refers ...

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70 views

### Fourier transform of the realization of a stationary process in the space of tempered distributions?

A path of a stationary sequence of random variables $y_t$ does not have a discrete-time Fourier transform in the classical sense because it is not summable. This leads to considering the spectral ...

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293 views

### Fourier expansion of Eisenstein Series

I have been reading a bit about the Fourier expansion of Eisenstein series (weight 1/2). I came across the fact that the coefficients contain Modified Bessel functions.
Further reading I found ...

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**1**answer

2k views

### Eliminating Gibbs phenomenon, and approximating with jumping functions in Fourier Analysis : An attempt and a question in this regard

Physical Motivation : Hear to these audio files S_f and P_f. S_f is Fourier partial
sum and P_f is the new reconstruction, both use spectrum only in the
region (0,4KHz) for reconstructing the ...

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97 views

### Inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$

Consider the inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$. My question is, how can we conclude about the decay properties, support and smoothness of the inverse Fourier transform? I ...

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69 views

### Does the Total variation of the Fourier partial sum of a bv function with jumps converge to TV of the function as $N\to\infty$

Does the total variation of the Fourier partial sum of a piecewise continuous bv function converge to the total variation of the function as $N\to\infty$. To explain briefly,
Let $f$ be a periodic ...

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153 views

### an analogue of Littlewood-Paley-Rubio de Francia theory

For any function $f$ defined on the set of integer $\mathbb{Z}$, we define its Fourier transform as the following periodic function: $$
\mathbb{F}f(\xi)=\sum_{n\in\mathbb{Z}}f(n)e^{-2\pi i n\xi}
$$
...

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**1**answer

135 views

### Ask the validity of Tauberian lemma in Sogge's book

In C.D.Sogge's Fourier Integrals in Classical Analysis pp.128-129, he proved Lemma4.2.3(Tauberian Lemma):
Lemma. Let$g(\lambda)$ be a piece-wise continuous tempered function of $\mathbb{R}$. Assume ...

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46 views

### Non interacting complex unit

How to work with two non interacting complex units say i and j. These two imaginary complex unit represent different quantities. For example i is for periodicity in theta and j is for frequency or ...

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**1**answer

308 views

### $L^1$ norm of exponential sum of $n^2 x$

What is the asymptotic order of
$$
\int_0^1 \left| \sum_{n=1}^N e^{2 \pi i n^2 x} \right| ~dx
$$
as $N \to \infty$. This should be known, but I cannot find it in the literature.

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**1**answer

204 views

### Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved).
Then the Fourier transform of this function is given by
...

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228 views

### The ring of modular forms for $\Gamma_0(11)$

Let $\mathcal M(11) = \oplus \mathcal M_k(11)$ be a graded algebra of modular forms for congruence group $\Gamma_0(11)$. I want to find generators and relations between them. I proved that $\dim ...

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49 views

### Value of prolate speroidal wave function at 0

I have a very basic question about prolate speroidal wave functions that can be defined as eigenfunctions to the following integral equation:
$$
\lambda\cdot \psi(x) = \int_{-1}^{1}\frac{\sin ...

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121 views

### Problem with operator and Fourier transform

I am currently dealing with a problem in functional analysis where I want to show the following.
Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$
if ...