0
votes
1answer
20 views

characterization of unital Fourier multipliers on $L^\infty(\mathbb{R})$?

Does there exist a characterization of Fourier multipliers $T \colon L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ which are unital, i.e. $T(1)=1$? In the case of the torus $\mathbb{T}$, it is easy ...
1
vote
0answers
30 views

Fourier series and Riemann integral

On the heuristic level, one often says that given a periodic function with period L, its Fourier series converges when $L \rightarrow \infty$ towards a Riemann integral. In other words, the ...
2
votes
1answer
38 views

Sequence of trigonometric polynomials which converges to an integrable function

A function $f:\mathbb{R}\to \mathbb{C}$ is said to be a trigonometric polynomial if it has the form $$f(x)=\sum_{k=-N}^Na_ke^{ib_kx},$$ where $a_k\in \mathbb{C}$ and $b_k\in \mathbb{R}$. Can we find ...
1
vote
1answer
42 views

Inequality between Fourier coefficients implies inequality for $L^p$ norms on the circle

Given two functions from $L_p [-\pi,\pi]$, where $p\geq 2$, $p$ is an even integer, and $f_n>|g_n|$ for every $n$ (where $f_n$ is the $n$th Fourier coefficient), I need to prove that ...
1
vote
0answers
39 views

Want to prove certain sum representation of $\cot(x)$

So here is my problem, I would like to prove an identity I found in a book which was given without a proof. Namely $$-i\sum_{n\in\mathbb Z} \operatorname{sign}(n)\cdot e^{i2\pi nx}=\cot(\pi x)$$ I ...
1
vote
1answer
32 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 ...
1
vote
0answers
20 views

$f\mapsto \sum_{n\in \mathbb Z} |\widehat{F(f)}(n)|$ lower semi continuous?

Let $T$ 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 ...
1
vote
0answers
82 views

What is the Fourier series of $e^{\mu\cos\theta}$?

Motivation: I want to solve this convolution problem on the circle: find $f$, given $g$ and $$ g(\theta) = \int_{S^1} e^{\mu\cos(\theta-\phi)}g(\phi)\ d\phi. $$ To do this, I want to find the Fourier ...
-1
votes
1answer
70 views

Is it true that the Fourier coefficient of convolution is the product of the coefficients?

what I mean by the title is the following: if we define the convolution between two $2\pi$-periodic, $C^1$ functions as $f*g(x) = (2\pi)^{-1}\int_{-\pi}^\pi f(x-y)g(y)dy$, is it true that the Fourier ...
1
vote
0answers
19 views

Real part of a holomorphic function is bounded by polynomial then the holomorphic function is a polynomial [duplicate]

Let $u$ be a harmonic function on $\mathbb{R}^2\cong \mathbb{C}$ such that $Ref= u$ where $f$ is an entire function. If $|u(z)|\leq |z|^n$ for any $z\in\mathbb{C}$, then $f$ is a polynomial of degree ...
1
vote
0answers
34 views

The property of positive fourier series. [duplicate]

This is the problem in the book 'Classical and multilinear harmonic analysis, volume 1' Let $f(x)=\sum_{n=0}^{N}[a_{n}\cos{2\pi nx}+b_{n}\sin{2\pi nx}]$ be a nonnegative function defiend on $[0,1]$. ...
0
votes
1answer
118 views

How to use Parseval' s( Plancherel' s) identity?

Let $f\in L^{2} (\mathbb R)$ with $\lim_{t\to \pm \infty} f(t)=0.$ Put, $F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt, \ (n=1,2,...).$ Fix $\alpha \in (0, \infty)$ and we define $H_{n}(x)$ ...
5
votes
1answer
64 views

An $L^1$ function whose Fourier series converges but not to itself

Do we have an $L^1$ function whose Fourier series converges almost everywhere but not to itself?
3
votes
0answers
53 views

Is harmonicity preserved when taking limits (normal convergence) on the unit disk.

I'm reading Koosis's book on $H^p$ spaces and have a question. He is proving a $L^p$ version of the Dirichlet problem which states that if $F(t)$ is in $L^p$ on the unit circle then $$ ...
3
votes
0answers
588 views

Fourier Transform of spherical harmonics

I am trying seeking for definition (or some source) of the Fourier Transform of Spherical Harmonics (see https://en.wikipedia.org/wiki/Spherical_harmonics). Any help will be really appreciated. ...
2
votes
1answer
203 views

Fourier analysis questions

Can anyone give me a hand with the proof of this properties? Prove that: a) The linear span of the set $\left\{T_bh/b\in\mathbb{R}\right\}$ is dense in $L_2(\mathbb{R})$, where $h(x)=e^{-\pi x^2}$. ...
3
votes
2answers
124 views

Identity involving partial sums of Fourier series

Suppose $f$ is a continuous periodic function and $S_Nf(x) = \sum^N_{n=−N} \hat f(n) e^{inx}$, where $$\hat f(n)= \frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-inx} dx.$$ How can I show that ...
1
vote
0answers
45 views

Variation on a completeness relation

The completeness relation for the spherical harmonics is: $$\sum_{l=0}^{\infty} \sum_{m=-l}^{l} Y_{lm}^*\left(\theta_1,\phi_1\right)Y_{lm}\left(\theta_2,\phi_2\right) = ...
3
votes
0answers
179 views

How is study of fractals related to fourier/spectral/harmonic analysis?

In chap. 3 of "Fractal Geometry of Nature" Mandelbrot mentions that "part of the study of fractals is the geometric face of harmonic analysis" (spectral or Fourier, he specifies). But to my dismay ...
3
votes
1answer
160 views

Bounds on integral

I am calculating Fourier coefficients for certain functions and have come across an integral of the form $$I=\int_0^{2\pi} \int_0^1 r^2e^{2\pi i r(m\cos\theta+n\sin\theta)}drd\theta,$$ where ...
3
votes
1answer
173 views

Convergence in mean of differentiated Fourier series

Let a function $f: \mathbb{R} \rightarrow \mathbb{R}$ be integrable on $[-\pi,\pi]$ and $2\pi$-periodic. Let $$ \frac{a_0}{2}+\sum\limits_{n=1}^\infty (a_n \cos nx+b_n \sin nx) $$ be the Fourier ...
1
vote
3answers
698 views

Solution of Laplace's equation in an annulus with constant Dirichlet conditions?

What's the solution to Laplace's equation $\nabla^2V=0 $ in the annulus with centre 0, inner radius 1, and outer radius 2, with boundary conditions $V=0$ on the inner boundary and $V=1$ on the outer ...
2
votes
1answer
290 views

Asymptotic error of Fourier series partial sum of sawtooth function

In Iwaniec's book, Topics in Classical Automorphic Forms, pg. 4, he gives the statement: $$\{x\}=\frac{1}{2}-\sum_{n=1}^N\frac{\sin 2\pi nx}{\pi n}+O((1+||x||N)^{-1})$$ where $\{x\}$ denotes the ...