The fourier-analysis tag has no usage guidance.

**2**

votes

**2**answers

155 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 ...

**3**

votes

**1**answer

158 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 ...

**1**

vote

**1**answer

70 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 ...

**1**

vote

**1**answer

241 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, $n\in\mathbb{Z}$, ...

**7**

votes

**2**answers

360 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 ...

**2**

votes

**0**answers

61 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) = ...

**3**

votes

**0**answers

61 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 ...

**0**

votes

**1**answer

155 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 ...

**8**

votes

**2**answers

755 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?

**4**

votes

**0**answers

406 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 ...

**0**

votes

**0**answers

71 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 ...

**6**

votes

**1**answer

314 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 ...

**7**

votes

**1**answer

3k views

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

This problem seems like a nightmare to me. I tried to expand
$K_{\omega}^f(t)$, but I am clueless of getting some kind of a closed
form or some kernel like structure. If I try to take ...

**0**

votes

**0**answers

99 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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vote

**0**answers

73 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 ...

**2**

votes

**1**answer

163 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: $$
\mathbb{F}f(\xi)=\sum_{n\in\mathbb{Z}}f(n)e^{-2\pi i n\xi}
$$
...

**1**

vote

**1**answer

138 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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votes

**0**answers

47 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 ...

**9**

votes

**1**answer

313 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.

**5**

votes

**1**answer

259 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
...

**3**

votes

**0**answers

234 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 ...

**5**

votes

**0**answers

51 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 ...

**0**

votes

**0**answers

124 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 ...

**7**

votes

**2**answers

301 views

### Radial limit does not exist almost everywhere

Problem 4 in Chapter 4 of Stein's book "Real Analysis" says
$\sum_{n\geqslant 0}z^{2^n}$
doesn't have radial limit as $z$ approaches the unit circle from inside almost everywhere. It's fairly easy ...

**0**

votes

**0**answers

35 views

### Is Wiener amalgam spaces $W^{2,1}(\mathbb R)\subset C_0(\mathbb R)$?

I have been learning Wiener amalgam spaces.
In Wiener amalgam spaces $W(X, L^2)$, I am taking $X=\mathcal{F}L^{1}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}\},$ and $m(x)=1.$
Take $f(x)= ...

**1**

vote

**1**answer

177 views

### Finite trigonometric polynomial

I noticed by numerical and some explicit calculations for a few examples that for real-valued finitely supported functions $\phi \in L^2(\mathbb{R})$ we have that
$T(x):= \sum_{n \in \mathbb{Z}} ...

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**0**answers

49 views

### $A_n \not \rightharpoonup A$ in $L_1[-\pi; \pi] $ ( $A_n$ is partial fourier sum )

Let
\begin{equation*}
(A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k cos(kt) + b_k sin(kt), \\
a_k = \frac{1}{\sqrt{\pi}} \int_{-\pi}^{\pi} x(t) cos(kt) dt, \\
b_k = \frac{1}{\sqrt{\pi}} ...

**2**

votes

**2**answers

172 views

### Are there examples of functions in $L_1$ and $L_\infty$ whose Fourier series divergent (“weakly”)?

It is wellknown that there is a convergence in norm for Fourier series in $L_p$, if $1<p<\infty$, but are there some examples for pointwise divergence if $p=1,\infty$ in books, or somewhere? I ...

**1**

vote

**0**answers

104 views

### Transformation of kernel

I have the following problem at hand.
Define the kernel
$$K(x_1,x_2) = \int_{-1}^1\int_{-1}^1 \exp(-2\pi\jmath x_1 y_1)R(y_1,y_2)\exp(2\pi\jmath x_2 y_2)\mathrm{d}y_1\mathrm{d_2}.$$
Now, if ...

**3**

votes

**1**answer

231 views

### Variations on the Mellin and Dirichlet transforms

There are a number of variations on the Laplace transform that turn up all over math. Some examples:
$\int_{-\infty}^{\infty} f(t)e^{-st} dt$ - The Laplace transform
$\sum_{-\infty}^{\infty} ...

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**0**answers

83 views

### Proof sought that a certain function is approximatively 3-dimensional

Let $D_n(x)$ be a diagonal matrix of size $N\times N$ where the $k$th element is $\exp(2\pi\jmath x(n+(k-1)/N)$.
Let $P_n$ be a random diagonal $N\times N$ matrix where each diagonal element is a ...

**2**

votes

**0**answers

153 views

### Largest Fourier coefficient of sparse boolean function

Consider a Boolean function $f: GF(2)^n \rightarrow \{0, 1 \}$. I would like to show that if $f$ is sparse, i.e. $\sum f(i) \leq t$, then $f$ must have a large Fourier coefficient. (A Fourier ...

**2**

votes

**1**answer

852 views

### Is the integral always nonzero?

Let
$$I_{n,p,a}:=\int_0^{\infty-} \frac{g_{n,a}(t)}{t^{p+1}} \, dt,$$
where
$$(*)\qquad\qquad\qquad n\in\mathbb N,\quad -\infty<a<\infty,\quad p_{n,a} < p < ...

**0**

votes

**0**answers

35 views

### Gaussian gabor frame

It is widely known that $\phi(x)=e^{-\frac{x^2}{2}}$ does not define a Gabor frame if we consider translations by units of $1$ and multiplication by $e^{2 \pi inx}$for $n \in \mathbb{N}.$ A way to ...

**3**

votes

**0**answers

96 views

### Nonlinear Schrödinger blow-up for non radial solutions

I am studying a paper of Frank Merle and Pierre Raphaël,
http://math.unice.fr/~praphael/Publications/blow-up-norme-critique.pdf.
The equations are
$$
i\partial_tu+\Delta u=-|u|^{p-1}u
$$
on ...

**3**

votes

**1**answer

144 views

### Characterizations of Wiener algebra

The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that
$$
\mathcal W\subset ...

**1**

vote

**0**answers

88 views

### Is my particular finite dimension Toeplitz matrix always strictly positive?

Let $H(\omega), \; -\pi \leq \omega \leq \pi$ be a real-valued function with a continuous band of zeros, that is (for simplicity) $H(\omega)=0, \; |\omega|\geq \beta \pi$.
Define a sequence of banded ...

**1**

vote

**1**answer

196 views

### Fourier coefficients of real analytic functions on an n-dimension torus

Let $(\mathbf{R}^n,\langle\;,\; \rangle)$ be the n-dimensional euclidean space endowed with the standard inner product. For a lattice $L\subseteq \mathbf{R}^n$ we let $cov(L)$ denote the covolume of ...

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votes

**3**answers

157 views

### Reference request : Besov spaces on ubounded domains

As I am relatively new to these matters, I would like to know if you could provide me a reference for Besov spaces on unbounded domains, because when I checked the first tome of Triebel's Theory of ...

**1**

vote

**0**answers

52 views

### Can we choose $g$ so that $\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$?

[I have asked this question on S.E. M; but I have not got any answer; and hope this is o.k. for M.O]
Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and ...

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votes

**1**answer

174 views

### Hardy-Littlewood-Sobolev inequality using fractional sobolev norm on the RHS

Using Hardy-Littlewood-Sobolev inequality, we can prove that:
$$\left| \int_0^1\int_0^1 |x-y|^{-\frac{1}{2}} f(x)f(y) \mathrm{d}x \mathrm{d}y \right| \leq C \left\| f \right\|_{L^{4/3}(0,1)}^2 \leq C ...

**4**

votes

**1**answer

268 views

### Is there a name for this space?

I'm just asking if there is a name for the space of functions on $\mathbb R^n$ whose norm is defined by
$$ \|f\|=\|\hat f\|_{L^p} $$
for $p\in [1,\infty]$. I find it handy to give it a name when ...

**2**

votes

**0**answers

66 views

### How is the structure of spectrum in cap-sets with no strong increments unrealistic if density is too large?

I am reading this excellent paper by Bateman and Katz on improved bounds on the cap set problem.
Let A be a set in $\mathbb{F}_3^n$ containing no 3-term arithmetic progression and let $A(x)$ denote ...

**2**

votes

**0**answers

105 views

### On uniform or simple convergence of Poisson Summation formula

Under good conditions on an even function $f(x)$ we have the Poisson Summation formula ($x>0$):
$$f(0) + 2 \sum\limits_{n =1}^{\infty} f(nx)= \frac{1}{x} \left( \hat{f}(0) + 2 \sum\limits_{n ...

**3**

votes

**1**answer

153 views

### Determining the Fourier transform

Let $d>2$. Let $M$ be a 2-dimensional submanifold of $\mathbb{R}^d$. For instance (and this is the type of example I primarily care about) we could have $M$ being the set of scalar multiples of a ...

**4**

votes

**1**answer

80 views

### Estimate self crossings of a curve parameterized by a trigonometric polynomial

Given z on the unit circle, let $P(z)= \sum\limits_{k=-D}^D p_k z^k $.
Can one estimate the number of self crossings of the following curve with an analytic expression in terms of the coefficients ...

**4**

votes

**1**answer

287 views

### Frequency of a representation of SO(3)

When generalizing the basic tenets of Fourier Theory to the symmetric group $S_n$, we can define a notion of the frequency of a basis function (i.e. an irreducible representation of $S_n$). In ...

**1**

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**0**answers

156 views

### Coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$

I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is striclty greater than ...

**1**

vote

**0**answers

89 views

### Summing a function at integer points

For a real function $f$ defined on the reals, and real $y$, define the 2-way infinite sum
$$ F_f(y) = \sum_{i\in\mathbb Z} f(y+i). $$
If $F_f(y)$ is defined for all $y$, it is periodic of period 1.
...

**0**

votes

**0**answers

98 views

### Condition for boundedness in stationary phase theorem

I am trying to understand theorem 7.7.1 in Hormander's Analysis of linear partial differential operators, vol.1.
Let $K \subset \mathbb{R}^n$ be a compact set, $X$ an open neighborhood of $K$ and ...