1
vote
0answers
34 views

Proof that these are Fourier coefficients

I proved that for $f \in \ell^1 (\mathbb Z)$ its Gelfand transform $\widehat{f}$ is a map $\widehat{f}: S^1 \to S^1$ defined by $$ \widehat{f}(z) = \sum_{n \in \mathbb Z}f(n) z^n$$ In Murphy's book ...
1
vote
0answers
21 views

Examples of semigroups of contractive Fourier multipliers but not positive?

Can you show me a concrete an example of semigroup $(T_t)_{t\geq 0}$ of Fourier multipliers such that each operator $T_t$ induces a contractive Fourier multiplier $T_t\colon L^p(\mathbb{T}) \to ...
3
votes
1answer
74 views

Fourier series to calculate infinite series

I try to show that $\sum_{i=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ using Fourier series and $f(x) = x$ on $L^2_{\mathbb{C}}[-\pi, \pi]$, with basis $e_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx}$. I ...
-1
votes
3answers
434 views

Does $f(x)\in L^1$ imply that $\lim_{\omega \to \infty } \, \int_{-\infty }^{+\infty } f(x) e^{-i x \omega } \, dx=0$?

Suppose that $f(x)$ is $L^1$ and R- integrable function, problem is to resolve if it is possible existence of such a $f(x)$ that: $$\lim_{\omega \to \infty } \, \int_{-\infty }^{+\infty } f(x) e^{-i x ...
2
votes
1answer
143 views

Structure of the functional space $\int_ {- \infty} ^ \infty f (x) dx = 1 $

Please, help me with studying of useful practical features of the following functional space: $$\int_{-\infty}^\infty f(x) \, dx = 1$$ For example: 1) What basis types are most convenient for ...
5
votes
1answer
58 views

How to make sense of Fourier series for a distribution?

In particular if I have an array of numbers say, $\{c_m\}_{m\in\mathbb{Z}^n}$. Under what conditions can we say that these are the Fourier coefficients of a distribution? [For examples Bessel's ...
1
vote
1answer
41 views

Smooth lower envelope of a function

I want to determine if this problem has a unique solution. Given a continuous function $f(x)$ in a bounded interval or the real line, say [-1/2,1/2],  the problem is to find a function $L(x)$ that ...
2
votes
1answer
98 views

Prove that the Fourier series of $\dfrac{1}{f}$ is absolutely convergent.

I have a problem: Let $f$ be a continuous function on the unit circle $(\Gamma)$: $$\Gamma=\{e^{i\theta}: \theta\in [0, 2 \pi]\}$$ Assume that $f \ne 0$ on $\Gamma$, and the Fourier ...
1
vote
0answers
33 views

Find the completeness radius of the prime numbers

As the title says, I'm trying to find the completeness radius of $\{2,3,5,7,11,\ldots\}$. The completeness radius of a sequence $\Lambda=\{\lambda_n\}$ is $R(\Lambda)=\sup\{A~|~\{e^{i\lambda_n ...
1
vote
2answers
47 views

Is $f$ is non-prime, Can we say $|f|$ is also non-prime ; in convolution algebra?

By Schwartz-inequality and Riesz–Fischer theorem, one can deduced that, $$L^{2}(\mathbb T) \ast L^{2}(\mathbb T) = A(\mathbb T)(:= \{f\in L^{1}(\mathbb T): \sum_{n\in \mathbb Z} |\hat{f}(n)| < ...
0
votes
2answers
130 views

The Fourier series converges absolutely $\implies$ it converges uniformly.

Let $S_N(f)$ be the $N$th partial sum of the Fourier series for $f$. I.e. $$S_N(f) = \sum_{n = -N}^{N} \hat{f}(n) e^{2\pi i n x / L}$$ Suppose that the Fourier series converges absolutely, i.e. ...
0
votes
2answers
56 views

How do they do this w.l.o.g. so freely (Fourier series).

Theorem 2.1. Suppose that $f$ is an integrable function on the circle with $\hat{f}(n) = 0$ for all $n \in \mathbb{Z}$. Then $f(\theta_0) = 0$ whenever $f$ is continuous at the point ...
1
vote
1answer
132 views

Green' s function for harmonic oscillator

Does someone know how to get a solution of differential equation for Green's function $(-d^2/dt^2 + \omega^2) G(t, s) = \delta(t-s) $? There is a periodicity of G, actually $\Delta (t-s) = G(t,s)$ ...
3
votes
0answers
57 views

An estimate For the Laplacian semi-group

Let $S(t)$ be the semi-group generated by the Dirichlet Laplacian in $L^2(0,1)$, which is given, for $y\in L^2(0,1)$, by $$S(t)y=\displaystyle\sum_{n=1}^\infty e^{-n^2\pi^2 t} \langle y,\sin(n\pi x) ...
1
vote
1answer
59 views

Fourier series convergence in $L^2$

Consider a function $g \in L^2(-\pi,\pi)$ such that it is continuous at $x \in (-\pi,\pi)$. Prove that if the Fourier series of g converges at x then that implies g(x) is its limit. I was thinking ...
2
votes
1answer
115 views

How can I proved, that $\left\{\sqrt{\tfrac{2}{\pi}}\sin(kx):k\in\mathbb{N}\right\}$ is an orthonormal basis of $L^2[0,\pi]$?

I want to prove that $S = \left\{\sqrt{\tfrac{2}{\pi}}\sin(kx):k\in\mathbb{N}\right\}$ forms an orthonormal basis of $L^2[0,\pi]$. I may use the fact, that $B = ...
5
votes
0answers
292 views

Show that the function is constant

Let $S^n$ be an $n$-dimentional unit sphere. Consider $f: S^n \longrightarrow R_+$ even continuous function. Denote $$ ...
5
votes
1answer
191 views

checking if a function is positive using Fourier coefficients

Given a function $$f(x) = \sum_{k=0}^N a_k\ \sin(k\pi x)$$ defined over the region $S = [0, 1]$, is there some way to check if $f(x) \geq 0$ for all $x \in S$ using the coefficients $\{ a_k;\ k \leq N ...
2
votes
2answers
212 views

Property of Fejer kernel

Let $$ F_n(x) = \frac{1}{n} \left( \frac{ \sin(\frac{1}{2} n x ) } { \sin(\frac{1}{2} x ) } \right)^2 $$ be the n-th Fejer-Kernel. Then $$ \forall \epsilon > 0, r < \pi : \exists N \in ...
1
vote
1answer
98 views

Non-negative function with a non-positive operator.

Question: I would like to know if there is any simple function that is $\geq 0$ but with its partial sums $S_{m} \leq 0$? Note: After much discussion, it would seem this question is not possible to ...
2
votes
1answer
124 views

Relation on fourier coefficients implies smoothness for a periodic continuous function

I just came across with the following question.. suppose we are given a periodic function of period $2\pi$. We define $a_n$ and $b_n$ to be the Fourier coefficients of $f$. To be precise, we have ...
1
vote
0answers
75 views

Bounds on Fourier coefficients of Euclidean distance functions

I am interested in the bounding the Fourier coefficients $a_{m,n}$ of the function $f(x,y)=\sqrt{x^2+y^2}$ defined on the interval $[-1,1]^2$. I am specifically interested in understanding the ...
3
votes
3answers
296 views

Easy Fourier series example: where is my mistake?

I'm doing exercise 15 on page 255 in Kreyszig: To illustrate that a Fourier series of a function $f$ may converge even at a point where $f$ is discontinuous, find the Fourier series of $$ f(x) = ...
3
votes
2answers
129 views

Function $f$ such that Fourier-series converges uniformly, but the series of the derivatives are divergent

I am studying Fourier-transformation right now, and I am asking if there exists a function $f$ such that is Fourier-series converges uniformly, the Fourier-series of $f'$ only in $L_2$ and that $f''$ ...
1
vote
1answer
58 views

norm of a variant of Fejer 's kernel

Let $K_N$ the Fejer's kernel on $\mathbb{T}$. Let $l$ be a positive integer. Let $Q$ the function defined by $$ Q(t)=K_N(lt). $$ In Hewitt/Ross "Abstract Harmonic Analysis 2" page 438, I can read that ...
4
votes
2answers
251 views

The relationship between Fourier coefficients of function $f$ and its continuity

How to prove that if Fourier series of function $f$ converge uniformly, then function is continuous?
5
votes
1answer
212 views

Completeness and Fourier series convergence

Consider the question: In an inner product space $V$, when does the Fourier series of $x$, $\sum\limits_{n=1}^k\langle e_n,x\rangle e_n$ converges to $x$ as $k\to\infty$? Well, certainly is converges ...
0
votes
1answer
63 views

Recovering an Operator on $L^2$ Given its Action as a Composition on the Spectrum of $f$

Let $\ell^2 = \ell^2(\mathbb{Z})$ denote the Hilbert space of square summable complex sequences on $\mathbb{Z}$ and suppose that $\sigma:\mathbb{Z} \to \mathbb{Z}$ is a function such that the linear ...
7
votes
3answers
309 views

For what sequences of real numbers $\left\{ k_{n}\right\}$ is the set of functions $\left\{ e^{ik_{n}x}\right\}$ a basis?

It is well known that the set of functions $\left\{ e^{^{inx}}\right\}$, for integer $n$, is an othonormal basis for the space of square integrable real functions in the interval $[-\pi,\pi]$. Now ...
0
votes
1answer
388 views

The Wiener algebra question

I want to show that the maximal ideal space of the Wiener algebra $W$ is $ \{ M_z : z \in \mathbb{T} \}$ where $M_z = \{ g \in W : g(z)=0 \}$ Could you please help me?
7
votes
1answer
227 views

When is the weighted space $\ell^p(\mathbb{Z},\omega)$ a Banach algebra ($p>1$)?

Let $\omega:\mathbb{Z}\to (0,\infty)$ and let $1\leq p<\infty$. Consider the space $\ell^p(\mathbb{Z},\omega)$ of complex valued sequences $f=(a_n)_{n \in \mathbb{Z}}$ such that ...