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0
votes
3answers
62 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 ...
0
votes
0answers
94 views

A question on the Euclidean domain $\mathbb{Z}[\omega]$ [on hold]

Let $\omega=\frac{-1+i\sqrt{3}}{2}=e^{\frac{2 \pi i}{3}}$ be a complex cube root of unity, and $\mathbb{Z}[\omega]$ the Euclidean domain. In view of that $\int_0^\infty e^{ix} ...
0
votes
0answers
149 views

Mellin transform on $\mathbb{Z}[\omega]$ [on hold]

I'm eager to ensure some facts which are elementary for many experts here. Let $\omega=\frac{-1+i\sqrt{3}}{2}$ be a complex cube root of unity. The Eisenstein integers $\mathbb{Z}[\omega]$ (a unique ...
5
votes
1answer
390 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
0answers
142 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[ ...
-1
votes
1answer
92 views

Oddify an even function and vice versa: need a Fourier transform-based formula [closed]

I have constructed an operator that applied to an odd function will give its even counterpart and also an inverse operator that would transform an even function into an odd one. I need a ...
5
votes
2answers
121 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
1answer
111 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 ...
0
votes
0answers
29 views

computational question concerning singular integral theory

Let $m\in C^\infty(\mathbb{R}^d-\{0\})$ be homogeneous of degree zero and has mean zero on the sphere$(S^{d-1})$. Then $m$ defines a tempered distribution and $\partial_j^dm\in S'(\mathbb{R}^d)$ is ...
1
vote
1answer
86 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 ...
5
votes
0answers
89 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
1answer
108 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
0answers
160 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
2answers
184 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
1answer
146 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
2answers
177 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
0answers
54 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 ...
4
votes
1answer
207 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
0answers
86 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 ...
2
votes
2answers
379 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 ...
4
votes
0answers
171 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 ...
0
votes
1answer
225 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
1answer
424 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
2answers
99 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} ...
0
votes
0answers
68 views

what is the best estimation for the following

Suppose a continuous $2\pi$-periodic function $f:R\rightarrow R$ satisfies $$ |f(x+\delta)+f(x-\delta)-2f(x)|\leq \mathrm{const}\frac{\delta}{\Big(\log\frac{1}{\delta}\Big)^{\gamma}}, \,\,\, ...
3
votes
2answers
174 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
0answers
156 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
2answers
258 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 ...
0
votes
0answers
177 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
1answer
142 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
2answers
214 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
2answers
110 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
0answers
126 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
0answers
94 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
0answers
112 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
1answer
139 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 ...
1
vote
0answers
104 views

Uniform bound for an alternating series of functions

I have mainly two questions, the first one being motivated by the second one. 1) Is there a way to prove that $F(x) = \sum_{k=1}^\infty \frac{(-1)^{k+1} x^{2k}}{(2k)!}$ is bounded on $\mathbb{R}_+$ ...
0
votes
0answers
38 views

Godement-Jacquet and L-functions

Let $M_{r}(F)$ be the matrices with coefficients in a local nonarchimedean field $F$ and $q$ the cardinal of the residue field. We have a Fourier tansform on $M_{r}(F)$ with kernel ...
4
votes
1answer
147 views

A Fourier series that does not converge in $L^1$

I'm reading Katznelson's book "Harmonic Analysis" and there is an exercise that I can't solve : Show that if the sequence $\left\{ N_j \right\} $ tends to infinity fast enough, then the Fourier ...
9
votes
1answer
250 views

Is a Bessel function larger than all other Bessel functions when evaluated at its first maximum?

Let $\mathcal{J}_{n+1/2}$ be the Bessel function of order $n+1/2$. Let $x_n$ denote the first zero of its derivative, which is also the location of the first maximum of $\mathcal{J}_{n+1/2}$. My ...
2
votes
0answers
135 views

A decomposition of a representation via characters of a normal compact subgroup

This is connected to my question here. Let $K$ be a normal compact subgroup in a locally compact group $G$, $\widehat{K}$ the dual object for $K$, and $\mu_K$ the normed Haar measure on $K$ ...
6
votes
1answer
217 views

Shortcut from discrete Fourier transform F{x} to zero-padded F{x:0…0}

Summary: Given $X$ (the discrete Fourier transform of some unknown vector $x$ of length $N$), is there any shortcut to computing $X'$ (the Fourier transform of $x$ after padding it with $N$ zeros)? ...
3
votes
0answers
109 views

Estimates of a bilinear oscillatory integral

Consider the operator $T_\lambda f(x)= \int_{\mathbb{R}^3}{\frac{\sin\lambda|x-y|}{|x-y|}\phi(x-y) f(y) dy}$, where $\phi\in C_0^{\infty}$, $\phi(x)=1$, when $|x|<1$ . My question is that do we ...
13
votes
2answers
261 views

Can a continuous function on a compact group $G$ be interpreted as the sum in $C(G)$ of its Fourier series?

I asked this in math.stackexchange, but it disappeared from the "main list" almost immediately, so I hope it will be appropriate as a separate question in MO. For a given function $f\in C(G)$ on a ...
2
votes
1answer
127 views

Solvability of a Fredholm system in $L^2$

Suppose $\lambda\not=0\in\mathbb{C}$. Does the following system have a non trivial solution in $L^2 [0,1]$? \begin{array} {lcl} \int_0 ^1 f(y)\log|x-y|dy=\lambda f(x) \\\int_0 ^1f(x)dx=0& ...
2
votes
1answer
134 views

To understand integral :$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$

I wants to understand the integrals of the form $$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$$ where $\mu(\alpha)$ is a non decreasing function ...
7
votes
1answer
414 views

Major arcs in the proof that every odd number is the sum of at most 5 primes

In his proof that all odd numbers greater than 1 are the sum of at most 5 primes, Terence Tao uses one large major arc around 0 rather than small ones around the rationals, which I am more accustomed ...
18
votes
2answers
2k views

Is this statement which relates the Fourier transform of a function to its singularities correct?

I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...
2
votes
0answers
126 views

pointwise limit of uniformly bounded sequence in $A(\mathbb T)$ is again in $A(\mathbb T)$?

Let $\mathbb T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$ Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in ...
3
votes
1answer
246 views

Can the Fourier series of a continuous function diverge on an uncountable set of measure zero?

I know that there exist continuous function $f: [0,2\pi]\rightarrow\mathbb{R}$ whose Fourier series diverges at all rational points of $[0,2\pi]$(c.f. Katznelson).We also know that the set of ...