0
votes
0answers
12 views

the multidimensional Hilbert transforms or partial Hilbert transforms

The one-dimensional Hilbert transform can be defined by the convolution $Hf:=f*\frac{1}{\pi x}$, or can be given by Fourier multiplier $(Hf)\hat{\,}(\xi)=-i\mathrm{sgn}(\xi)\hat{f}(\xi)$. ...
1
vote
0answers
34 views

The Fourier sine transform of $f(x)/\sin x$

Is the following result $$\lim_{\lambda \to \infty} \frac{2}{\pi} \int_0^\infty \frac{f(x)}{\sin x} \sin(\lambda x) \, dx = f(0) + 2\sum_{k = 1}^\infty f(k\pi),$$ where $\lambda$ is an odd integer, ...
1
vote
0answers
45 views

Square-summable sequence and Fourier series

Every square-summable sequence $(a_{n})_{n}$ is represented by $a_{n}=\widehat{f}(4^n)$, where $\widehat{f}(i)$ is Fourier coefficient of continuous function $f$. Where can I find proof of this ...
2
votes
1answer
40 views

Fourier series with a weighted mean square norm

I am interested in Fourier series with a non-uniformly weighted error norm. What I mean by this is that the usual Fourier series of a periodic function is a minimizer of the mean squared error: $$ J_N ...
0
votes
1answer
55 views

Recommend Fourier Analysis Workbook or online examples

I am studying a graduate level course in Fourier Analysis, however my Functional Analysis background is extremely weak, I have also never met Lebesgue Integration and it has been a while since I ...
0
votes
0answers
26 views

Fourier-Stieltjes Transform of the Cantor measure

I am looking for an elementary derivation of the formula for the Fourier-Stieltjes Transform of Cantor measure on the Cantor middle-third set as an infinite product of cosines (with all the details ...
6
votes
1answer
72 views

Gowers' proof of Szemerdi's theorem

Are there any good books or other resources (expository notes) which explains Gowers' proof of Szemerdi's theorem in detail?
0
votes
1answer
41 views

A basic book on (discrete) 2D - Fourier transforms?

In the context of image manipulation i need to learn about 2D fourier transforms, especially about the discrete version. Can somebody recommend a book that starts at the basics and treats some ...
2
votes
1answer
111 views

Need to learn wavelet, suggest steps and resources

I am looking for a good introduction to wavelets and wavelet transforms. that covers the following: Basics Vector Spaces – Properties– Dot Product – Basis – Dimension, Orthogonality and ...
4
votes
0answers
98 views

Involutive fourier transform

The writer here states I am introducing a viewpoint (the involutive convention) which makes the Fourier transform its own inverse (i.e., the Fourier transform so defined is an involution). ...
3
votes
1answer
191 views

Exercises about Distributions

I'm looking for references (books or pdf) about the following themes (especially the first two) : Fourier Series of Distributions. Distributional solutions of ordinary differential equations. ...
6
votes
1answer
79 views

percentage of numbers starting with $2$ in $\{2^n\}$

I have once heard a professor telling (during a course on Fourier theory) that there is a way to determine the numbers starting with a $2$ in the sequence $\{2^n\colon n\in\mathbb{N}\}$. I asked him ...
3
votes
2answers
170 views

Derivative of Integral (in Fourier transform)

I've taken some analysis, but somehow Fourier transforms were never brought up until they were assumed to be familiar. Fun. Anyway, in a class example (showing the integral of a Gaussian is again a ...
5
votes
3answers
261 views

Further studies on Fourier Series and Integrals.

If you had to choose two books from the following list, which pair would you chose, and why? If you haven't read any, would you pick any pair among the list based on the author of the book? I am ...
2
votes
1answer
419 views

How to solve differential equations using fft?

Can anyone point me to the principles and books/websites about it? Which properties must the differential equation have that a solution with fft is possible? Why can it be solved that way?
5
votes
2answers
116 views

State-of-art of the Discrete Fourier Transform

I would like to know what is the state-of-art in the research of the discrete Fourier transform. I have listed some questions to help answering, please add your own to make the list more ...
0
votes
1answer
51 views

Littlewood Paley characterization of BMO spaces

I know that there is a Littlewood-Paley characterization of Hardy spaces (for instance, this is found in Grafakos, Modern Fourier Analysis, section 6.4.6). I'd like to know if a similar ...
2
votes
5answers
294 views

Fourier Analysis

I am interested in Fourier Analysis. But I don't get why the coefficients are choosen that way and why the Fourier series will converge to a given function? Can someone provide me simple information ...
3
votes
0answers
210 views

What is a spherical Gaussian kernel?

In this paper (page 8, Example 3), a spherical Gaussian kernel is defined by the formula $$K(\mathrm x,\mathrm y)=e^{-2\epsilon(1- \mathrm x\cdot\mathrm y)}$$ where $\mathrm{x,y}\in ...
0
votes
1answer
61 views

Convolution Error Estimate Reference Request

I am looking for a reference for the $L_p$ error of the difference of a Sobolev function and its convolution with a band-limited mollifier. The type of estimate that is quoted in a paper without a ...
1
vote
0answers
19 views

Almost Everywhere Convergence of Walsh Series of $L^2$ functions

I am currently reading the Hunt's papar (http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0655.0662.ocr.pdf), and am wondering if there is some notes which presents his argument more ...
1
vote
1answer
456 views

Can the differentiating and squaring process in the cochlea explain a reported dichotic stimulation experiment?

On this math.stackexchange on url What is Octave Equivalence? in an answer on the related ( octave equivalence ) question is stated: Mathematically, this signifies that the mammalian cochlea ...
10
votes
5answers
653 views

Notes for Beginner Fourier Analysis?

Are there any good lecture notes or books on basic fourier analysis that authors publish freely online? It is very difficult to find rigorous mathematical theory of fourier analysis because google is ...
3
votes
1answer
152 views

Poisson integral on $\mathbb{H}$ for boundary data which is orientation-preserving homeomorphism of $\mathbb{R}$

Let $f$ be a real-valued function (in my case, an orientation-preserving homeomorphims of $\mathbb{R}$) on the real line $\mathbb{R}$ which is not in any $L^p$ -space. Let us take the simplest example ...
3
votes
3answers
407 views

Need a crash course in fourier analysis, recommend resources

I need to be able to understand everything about fourier analysis asap. Could you recommend one or two references or books that are considered 'the book' to learn this subject?
3
votes
1answer
645 views

Is there a time-domain proof of Nyquist sampling theorem?

For a continuous-time signal $x(t)$ that is bandlimited (in the baseband) to $[-W,W]$, the standard proof of Nyquist sampling theorem proceeds in the frequency domain by examining the Fourier ...
1
vote
1answer
203 views

Reference for studying the method of stationary phase

I would like to learn about the stationary phase method, as part of the theory of Fourier Integral Operators. From what I can see so far, Hormander's book "The Analysis of Linear Partial Differential ...
6
votes
1answer
232 views

An elegant non-technical account on the work of Joseph Fourier.

It would seem difficult for a naive person to understand the beauty of work done by Fourier. So as far as I know, one can use the Fourier transforms, analysis and series to apply them for heat ...
2
votes
0answers
81 views

Scale invariance and $1/f^2$ power spectrum

In the paper Occlusion Models for Natural Images : A Statistical Study of a Scale-Invariant Dead Leaves Model; Lee, A. B. Mumford, D. B. Huang, J.; International Journal of Computer Vision I read ...
1
vote
0answers
93 views

Introductory book on Distribution theory [duplicate]

Possible Duplicate: Distribution theory book Is there a good alternative to Friedlander Introduction to the Theory of Distributions ? Many thanks !
1
vote
0answers
291 views

How to solve a linear PDEs with trigonometric function as coefficients

Is a general method for solving a system of linear partial differential equation with trigonometric function as coefficients exist ? For example something like that: $q$ is the unknown function, $2 ...
1
vote
3answers
2k views

What are some good Fourier analysis books?

I have taken real analysis, but never learned Fourier analysis. What is a good book to get started? I'm not sure the Stein book would be good.
4
votes
2answers
110 views

inequality with roots of unity

Do you know proofs or references for the following inequality: There exists a positive constant $C>0$ such that for any complex numbers $a_1,\ldots,a_n$ $$ |a_1|+\cdots+|a_n| \leq ...
5
votes
3answers
298 views

Applications of Fourier analysis for math students

Does anyone have recommendations for a good book which explains how Fourier analysis is applied in engineering and physics that does not assume any knowledge of the above topics? I'd like to see the ...
2
votes
1answer
202 views

Fourier transform

Suppose $1< p<\infty$. Let $f$ be a continuous function with compact support defined on $\mathbb{R}$. Does it exist a function $g \in L^p(\mathbb{T})$ such that: $$ ...
3
votes
0answers
68 views

Reference of explanation of this result about Fourier transforms in euclidean space

The paper says that since $$u = - \nabla \times (\Delta^{-1} \omega)$$ "The Fourier transforms of $\nabla u$ and $\omega$ satisfy $(\nabla u)^{\text{^}}(\xi) = S(\xi)\hat\omega(\xi)$ where $S$ is a ...
2
votes
1answer
163 views

reference for Fourier series for periodic functions of the form $f : \mathbb{R} \to \mathbb{C}$

I am in need of a good reference which has a complete treatment (with all the convergence proofs) for Fourier series representation for periodic functions of the form $f : \mathbb{R} \to \mathbb{C}$. ...
0
votes
2answers
302 views

reference request for proof of Gibbs phenomenon at jump discontinuities

Please suggest a reference for a proof of Gibbs phenomenon at jump discontinuities of a function.
1
vote
2answers
456 views

Pointwise convergence of Fourier series of a piecewise continuous (and Lipschitz continuous everywhere) function - a reference request

Pointwise convergence of Fourier series of a piecewise continuous (and Lipschitz continuous everywhere) function.I basically want to understand a proof for convergence of a Fourier series of $f(x)$ to ...
6
votes
6answers
1k views

Fourier Analysis textbook recommendation

I am taking a fourier analysis course at the graduate level and I am unhappy with the textbook (Stein and Shakarchi). What I am looking for is a book that is less conversational and more to the ...