**2**

votes

**1**answer

289 views

### Connection between the Fourier transform of f and |f|

If $f\in L^p(R)$ with $1\leq p\leq 2$, then Hausdorff-Young inequality implies that the Fourier transform $\widehat{f}\in L^{p'}$, $p'$ is the dual exponent of $p$, and
$$
...

**0**

votes

**0**answers

110 views

### The functional $L(\varphi)=\int_0^{2\pi}\frac{\sqrt{1-\varphi^2-(\varphi')^2}}{1-\varphi^2}d\theta$

Consider the $2\pi$-periodic inner product space $L^2[0,2\pi]$.
Let $F\triangleq\{f\in L^2[0,2\pi]|f(\theta)>0,(f(\theta),\cos\theta)=(f(\theta),\sin\theta)=0\}$.
Let $G\triangleq\{\varphi\in ...

**1**

vote

**0**answers

34 views

### Multidimensional Filters

Say you want to design a LP FIR filter with low pass cutoff $fc$, transition band $fc$ to $fs$ and ripple factor $dp$ at passband and $ds$ at stop band. If one divides the frequencies by $\pi$, then ...

**2**

votes

**0**answers

127 views

### the (2,2,1) boundedness of a “product” operator

Let $\{E_j\}_{j\in\mathbb{Z}}$ and $\{F_k\}_{k\in\mathbb{Z}}$ be two collections of pairwise disjoint sets in $\mathbb{R}$. Let $C(j,k)$ be a bounded function (e.g. $|C(j,k)|<1$) defined on ...

**0**

votes

**1**answer

185 views

### How to solve this differential equation with an infinite sum?

I would like to find solutions of the following differential equation:
$ \sum_{1}^{\infty} a_n f(nx) + f''(x)+ x^2 f(x)=\lambda f(x)$
For example in space of function from $\mathbb R^*$ to $\mathbb ...

**2**

votes

**1**answer

205 views

### Fourier transform of $sin(\frac{1}{x})$ for $x > 0 (x > 1)$

Please, give me the cue: does exist analytical representation of Fourier Transform of $sin(\frac{1}{x})$ for$ x>0$ (or $x>1$). Maybe exist an approximation of $FT(sin(\frac{1}{x}))$ by Bessel ...

**0**

votes

**1**answer

137 views

### A kind of Discrete Fourier Transform

Given a $z\in \mathbb{C}^N$, the DFT of $z$ is given for every $k\in [0,N-1]_\mathbb{N}$ by
$$DFT_z(k)=\frac{1}{N} \sum_{j=0}^{N-1} z_j\, \omega^{-k j}$$ where I have denoted by $\omega$ the $N$-th ...

**4**

votes

**3**answers

259 views

### Is there a compactly supported function that its Fourier transfrom vanishes at given n real points?

My question is as follows: Given ${{\lambda }_{1}},\,{{\lambda }_{2}},...,{{\lambda }_{n}}\in \mathbb{R}$ where $\underset{1\le j\le n-1}{\mathop{\min }}\,\left| {{\lambda }_{j+1}}-{{\lambda }_{j}} ...

**2**

votes

**0**answers

100 views

### Do position and momentum measurements determine a wave function?

Suppose we have a function $f\in L^2(\mathbb R^n)$ and we know the functions $x\mapsto|f(x)|$ and $p\mapsto|\hat f(p)|$, where $\hat f$ is the Fourier transform of $f$.
Can we reconstruct the function ...

**1**

vote

**0**answers

63 views

### decomposition of tempered distributions by entire analytic functions

Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1~~\text{if}~|\xi|\leq 1\}$$
Let $j\in \mathbb N$ ...

**4**

votes

**0**answers

226 views

### What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?

I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it.
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$ \operatorname{supp} \phi ...

**5**

votes

**1**answer

278 views

### Jackson's theorem for partial sum of Fourier series

There is a classical theorem of Jackson stating that the $N$-th partial sum $S_N f$ of the Fourier series of a Lipschitz continuous function $f$ (which is periodic with period 1) satisfies
$$
|f(x) - ...

**2**

votes

**0**answers

70 views

### request for any expository works in pointwise convergence of double Fourier series and especially a paper by Hardy

Quart. J. Math. Volume 37, Issue 1, Pages 53-79
On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters.
Hardy, G.H.
I am not ...

**5**

votes

**1**answer

171 views

### Largest area of a compactly supported positive definite function

Consider a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$, supported on $[-1,1]$, of positive type. Assume $f(0) = 1$; what is the "largest area" $\int f\,dx$ that can be achieved?
To be ...

**0**

votes

**2**answers

197 views

### “Convolution” for Multiplying Random Variables

The following situation arises frequently in probability.
Suppose we have two independent continuous random variables $X$ and $Y$ and we consider their sum, $Z=X+Y$. Then the pdf of $Z$ is the ...

**0**

votes

**1**answer

45 views

### Degree of polynomial approximating characeristic function of finte set

Given $C \geq 1$ and $\epsilon > 0$, is there a number $N = N(C,\, \epsilon)$ such that the following holds:
For every set $S \subseteq S^1$ of cardinality $C$, there is a function $f: S^1 \to ...

**-3**

votes

**1**answer

595 views

### A question about pointwise convergence of Fourier transform in $N$-dimensions

I am retreating back on this statement, after some explorations and calculation
Bow to Willie and others who were skeptical on this. Main difficulty can be seen in this reference. But I must mention ...

**3**

votes

**0**answers

171 views

### An upper bound for a average of a function in $L_{p}([0,1))$

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j},~
1\leq j \leq 2^{n} \}
\right\} \right)_{n ...

**4**

votes

**3**answers

555 views

### Textbook request for class field theory [duplicate]

I am studying class field theory. I need good reference books, notes, or other materials which explain the following topics: ideles and ideals, Haar measure and integration on local fields, Fourier ...

**1**

vote

**3**answers

147 views

### On the existence of compactly supported functions whose its Fourier transform satisfies a given condition

My question is concerned with the existence of compactly supported functions whose its Fourier transform satisfies a given condition: For $\gamma\ge 1$, one can prove that there is no compactly ...

**5**

votes

**1**answer

474 views

### Is fourier analysis necessary to prove this?

I have a couple of inequalities that I want to prove. The proof is easy using fourier analysis but I am wondering whether there is a proof that does not use fourier analysis.
1) For any $c, s > ...

**0**

votes

**0**answers

165 views

### Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1)) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j} \stackrel{\text{df}}{=} \left[ ...

**6**

votes

**2**answers

553 views

### Reverse Hausdorff Young for nonnegative functions

The classical Hausdorff-Young inequality states that
$$
\Vert \widehat{f} \Vert_{p'} \leq \Vert f \Vert_p \text{ for } 1 \leq p \leq 2.
$$
For $p=2$, we even have equality due to Plancherel.
If we ...

**1**

vote

**1**answer

165 views

### Fourier approximation error in L^2 for piecewise continuous functions

Let $u:[0,2\pi)\to \mathbb{R}$ be the step function
$$u(x) = \begin{cases}
1 & \text{if } x \in [0,\pi), \\
0 & \text{if } x \in [\pi,2\pi)
\end{cases}$$
By a direct computation, one ...

**2**

votes

**1**answer

124 views

### Can we count isospectral graphs?

On n-vertices, how many isospectral graphs exist?
[..I saw this previous "historic" discussion between two of the stalwarts in this field - Operation on Isospectral graphs ]
Given a graph are ...

**6**

votes

**1**answer

224 views

### Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$:
$$
\begin{cases}
\partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\
u(0,x)=u_0(x).
\end{cases}
$$
...

**0**

votes

**1**answer

148 views

### Convergence of complex series that are not absolutely convergent?

Does anyone know of a convergence test for a complex series of the form
$$\sum_n a_n \cdot \exp(i \cdot b_n)$$
?
The particular series I need to understand has a_n going to zero as n goes ...

**1**

vote

**0**answers

188 views

### Davenport's proof that almost all integers are the sum of 4 cubes

Where can I find a pdf that describes Davenport's proof that almost all integers are the sum of $4$ cubes?

**5**

votes

**2**answers

264 views

### Eigenstates of Fourier transformation

Let $\gamma$ be defined on $\mathbb R^n$ by $\gamma (x)=e^{-π x^2}$. With $\mathcal F$ standing for the Fourier transformation defined on the Schwartz space by
$$
(\mathcal F u)(\xi)=\int e^{-2iπ ...

**2**

votes

**1**answer

271 views

### Is a polynomial decay sufficient for a smooth function to be in $\mathcal{F}(L^1)$?

Background: I have a function $g(\omega)\in C^{\infty}(\mathbb{R})$, which vanishes like $O(|\omega|^{-\beta})$ at infinity for some $\beta>0$.
This answer states that functions that decays "too ...

**0**

votes

**2**answers

276 views

### A book about almost periodic functions [closed]

Can anyone give me suggestions for new books about Besicovitch's almost periodic functions? Thanks a lot.

**0**

votes

**0**answers

76 views

### Where the following interpolation method converges?

In this question about discrete-analytic functions (that is functions, who equal to their Newton series) I asked for a solution for the following problem:
Is there a method to extend the notion ...

**5**

votes

**1**answer

398 views

### Are Besov spaces $B^{s}_{p,q}$ invariant under Fourier transform?

(This may be very easy question for MO; as I am just trying to understand Besov spaces)
Let $\phi \in C^{\infty}(\mathbb R^{n})$ with
$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: ...

**2**

votes

**0**answers

109 views

### “Direct” proof (without hypercontractivity) of equivalence of moments?

Let $(x_i)_{i \in \mathbb{N}}$ be a family of independent $\pm 1$ centered Bernoulli random variables, and let $p, q > 1$. There exists a constant C such that for every (finite) linear combination ...

**3**

votes

**2**answers

433 views

### Exponential Sum Bound

In
http://131.220.77.52/files/preprints/diophantine/bruedern/wpminicubefinal.pdf, Bruedern and Wooley mention the following fact on the bottom of page 6:
Let
...

**5**

votes

**0**answers

373 views

### Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definitions
Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...

**1**

vote

**1**answer

335 views

### Prove that these two definitions of “natural” integration constant coincide when both converge

These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details).
The first one is based on ...

**-2**

votes

**1**answer

550 views

### Why calculus textbooks do not include the natural integration constants in the tables of integrals? [closed]

The formulas for integrals in the textbooks usually define indefinite integral up to a constant term. Yet the natural integration constant for antiderivative can be fixed from the following formula ...

**4**

votes

**2**answers

137 views

### Inverse Fourier of $\omega^{-1+{\rm i}\alpha} u(\omega-1)$

Let $\alpha$ be an arbitrary real number and define
\begin{align}
\widehat{f}(\omega)=\left\{\begin{array}{ll}
\omega^{-1+{\rm i}\alpha}, & \omega>1,\\
0, & \textrm{otherwise}.
\end{array}
...

**3**

votes

**2**answers

353 views

### How do functions operate in a Sobolev space $H^{s}$?

Let $s>\frac{1}{2};$ and define a Sobolev space as follows:
$$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$
Fact: Let $m$ ...

**1**

vote

**0**answers

197 views

### Is there an asymptotic bound for this oscillatory integral?

I have an oscillatory integral:
$$ \int u(x,y) e^{i\lambda f(x,y)} dx $$
with $f(x,y)\in \mathbb{C}^{\infty}$ a complex-valued function in a neighborhood of $(0,0)$ satisfying:
$$ \text{Im} f \geq ...

**1**

vote

**2**answers

489 views

### decomposition of Hilbert space into tensor product $L^2([0,\tfrac{1}{2}]) \otimes L^2([\tfrac{1}{2},1]) \simeq L^2([0,1])$

The definition of entanglement entropy in Quantum Field Theory involves decompositing a Hilbert space into a tensor product $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$.
As an example, is it ...

**1**

vote

**0**answers

287 views

### Non-Abelian Fourier Analysis

I'm currently thinking to generalize a known result on abelian groups to non-abelian groups. This is the problem. Fix an abelian group $G$. We know that:
$\mathbb{E}_{\chi\in ...

**0**

votes

**1**answer

202 views

### $\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$? [closed]

(This may be very simple question for MO; I had post it to math stack exchange few days back but I could not get any answer(or comment) to it)
It is well-known that, for $f,g \in L^{1}(\mathbb R).$ ...

**11**

votes

**2**answers

272 views

### Heuristic interpretation of the 'third index' for Besov and Triebel-Lizorkin spaces

For $p,q \in (0,\infty)$ and $s \in \mathbb{R}$, one can define certain function spaces, $B_s^{p,q}(\mathbb{R}^n)$ and $F_s^{p,q}(\mathbb{R}^n)$, the Besov and Triebel-Lizorkin spaces respectively. ...

**4**

votes

**2**answers

125 views

### Approximate Moment Conditions

It is known in classical probability that if two random variables $X$ and $Y$ obeys
$$\mathbb{E} X^k = \mathbb{E}Y^k, \ \forall \ k \geq 1$$
with additional condition that $\mathbb{E}X^k$ does not ...

**0**

votes

**0**answers

151 views

### $L^{p}(\mathbb R)\subset L^{1}(\mathbb R) \ast L^{p}(\mathbb R), (1< p< \infty)$?

Let $\mathbb T$ be a circle group.
In 1939, Salem, has shown that, every member of $L^{1}(\mathbb T)$ can written as a product(convolution) some other two members of $L^{1}(\mathbb T),$ that is, ...

**4**

votes

**0**answers

236 views

### Why is the density function of a sum of many iid random variables (i.e., the Gaussian) self-dual?

In the usual proof of the central limit theorem via characteristic functions, we note that if $X_1, \dots, X_n$ are independent and identically distributed, with probability density function $f(x)$, ...

**2**

votes

**0**answers

126 views

### seek another proof of a result in Fourier analysis

It was proved on page 26 of this note the following result:
Let $\xi$ be an algebraic number that is not a root of unity, then there exists an $n_0\geq 0$ with the property that ...

**0**

votes

**1**answer

249 views

### Paley-Wiener type theorem for integral functions with compact support

If $f\in L^{1}(\mathbb{R}^n)$, and $f$ has compact support, then how can I show that $\hat{f}$ cannot satisfy $\hat{f}(x)=O(e^{-\epsilon|x|})$ for any $\epsilon>0$?
This is similar in the spirit ...