# Tagged Questions

The representation of functions (or objects which are in some generalize the notion of function) as constant linear combinations of sines and cosines at integer multiples of a given frequency, as Fourier transforms or as Fourier integrals.

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### A problem on real valued functions in $\mathbb{R}^2$ with least variation

Let $\alpha(s) = (x(s),y(s))$ be the arc length parametrization of a plane, smooth, closed, convex curve, of length $L$. Let $J:(0,L)\to\mathbb{R}$ be a smooth and Bounded variation (BV introduced ...
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### If $\mathcal{F}$ is the Fourier transform, what can be said about $\mathcal{F}(L^1(\mathbb{R})) \cap L^1(\mathbb{R})$?

The Fourier transform gives a map of the Schwartz space to itself which turns out to be a linear homeomorphism of period 4. However, when the domain is extended to $L^1(\mathbb{R})$, the situation is ...
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### Real rank zero of group $C^*$-algebras

The concept of real rank zero of a $C^*$-algebra is introduced as non-commutative analogue of dimension ( topological dimension ). For example, it shown (by Brown-Pedersen) such that, if $X$ is a ...
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### Ideal structure of group $C^*$-agebras [closed]

Let $G$ be a locally compact groups and $C_r^*(G)$ be a reduce group $C^*$-algebra. $\ Question:$What is the ideal structure of reduce group $C_r^*(G)$?
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### The inverse of Laplacian operator for different orders

I post this question in MSE couple of days before and get no response. So I repost it here for better luck. Thank you! Let $u,v\in C_c^\infty(\Omega)$ and $\Omega\subset \mathbb R^N$ is open ...
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### Multiplier operators on anisotropic weighted $L^2$ spaces

Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$, $\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$ where the scalar complex function ...
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### What's the relationship between the roots of a function and that of a filtered Fourier series representation?

Suppose $M$ is a piecewise constant function on an interval $T$ taking values $+1$ and $-1$, and that $M$ exhibits all the properties sufficient to ensure the existence of some converging Fourier ...
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### Is the Fourier transform of $e^{-|x|^n}$ positive?

Let $$\Phi(x) = \int_{\mathbf{R}^n} e^{-|y|^n +i (x,y)} dy.$$ Is $\Phi$ positive everywhere in $\mathbf{R}^n$? Could someone helps me answer this question or gives a reference for it? Thanks.
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Consider the properties of band-limited functions $f_N:[-\pi,\pi]\to\mathbb{R}$ defined through their Fourier series $f_N(x)=\sum_{n=-N}^N c_n e^{inx}$ where $c_n=a_n+i b_n$ and both $a_n,b_n\sim\cal{... 0answers 119 views ### Nonclassical polynomials, circles, and groups Tao and Ziegler have introduced a generalization of polynomials over a prime field called nonclassical polynomials, useful for studying the Gowers norm. A nonclassical polynomial of degree$d$is a ... 0answers 193 views ### Precise relationship between “finite” Fourier analysis and Galois theory in solving the cubic? Suppose we want to solve$$x^3 - ax^2 + bx - c = 0.$$We know a priori that this can be factored as$(x - r_0)(x - r_1)(x - r_2)$; by Vieta's formulas, we know$$a = r_0 + r_1 + r_2,\quad b = r_0r_1 + ... 2answers 550 views ### Fourier transform of the critical line of zeta? This was asked on MSE and got a lot of upvotes but no answers, so I'm posting it here. Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along ... 1answer 142 views ### separable BV space for PDE's, Whats stopping us? [closed] Consider the metric space BV(0,1) with the following metric$$ d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)| $$. It is separable, compact, uniformly bounded and complete. So What is the really obvious thing ... 2answers 238 views ### Fourier transform localisation (still unanswered, but apparently off-topic?) [closed] In the context of Pólya's theorem I was reading these notes here on p. 19. In the last paragraph the authors claim (it is the sentence starting like "standard Fourier theory shows...") that the ... 0answers 99 views ### Integrating a series expansion of \mbox{frac}(x)\lfloor x\rfloor coming from Fourier series of sawtooth function Let me preface this question by saying that I am not exactly sure it counts as research level. It is crossposted on mathstackexchange: http://math.stackexchange.com/questions/1519724/integrating-a-... 1answer 67 views ### What are the known conditions for the log of the Fourier transform of a 2D real discrete signal to have no branch cuts? Suppose we sample a real 2D signal, f(x,y), at N by N evenly spaced points in x and y. Then we compute the Fourier transform of the sampled signal, F(u,v), and then take the log of F. There will be a ... 3answers 435 views ### Completeness of nonharmonic Fourier Series I have the following question: The Exponential System (\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}} constitutes an orthonormal basis of L^2([-1/2,1/2]). Thus, certainly the oversampled system \Phi:... 1answer 120 views ### A metric on the set of BV functions, is it mentioned/studied in literature? I'd like to propose the following metric which operates on the set M of all square integrable functions that are also of bounded variation, of the form f : (0,1) \to \mathbb{R}. Given any x,y \in ... 0answers 135 views ### Proof without distributions I was wondering whether there is a way to show this identity$$\pi \int_{\mathbb{R}^3} \frac{f(x)}{|x|} dx = \int_{\mathbb{R}^3} \frac{\widehat{f(x)}}{|x|^2} dx $$without using distributions for f ... 1answer 105 views ### Lower bounds from Fourier dimension? According to Mattila, Geometry of sets and measures in Euclidean spaces, p. 168, the Fourier dimension \text{dim}_F(A) of A\subseteq \mathbb R^n is the unique number in [0,n] such that for any ... 0answers 72 views ### Combining oscillatory integrals of the first and second kind Consider an oscillatory integral of the first kind$$ I_\lambda(x)=\intop_{\mathbb{R}^{n}}e^{i\lambda\Phi(x,y)}a(x,y)\,d y,\quad \lambda\geq 0,\; a\in C_c^\infty(\mathbb{R}^{k+n}),\; \Phi\in C^\infty(\... 1answer 291 views ### Maximal$L_1$norm of Fourier Transform of a Subset Let$A_n$be the following$n\times n$matrix:$(A_n)_{i,j}= \frac{1}{\sqrt{n}}\omega_n^{i\cdot j}$for all$0 \le i,j <n$, where$\omega_n=e^{\frac{2\pi i}{n}}$. I want to understand how$A_n$... 0answers 92 views ### Error term for a Fourier integral There is a well-known theorem that states that for$f$continuous and$f,\hat f$integrable, $$f(0)=\frac{1}{\pi}\lim_{T\to\infty}\int_{-\infty}^\infty f(x)\frac{\sin(Tx)}{x}dx.$$ So it should be that ... 2answers 295 views ### Bounding exponential sum with square roots It is well known that for each$m\in\mathbb{N}$$$\lim_{N\to\infty}\frac1N\sum_{n=1}^Ne^{2\pi i\sqrt{nm}}=0$$ My question is whether there is some uniformity in the variable$m$. More precisely, is it ... 1answer 117 views ### Fractional Sobolev spaces on the circle with a Littlewood-Paley characterisation Fractional Sobolev space$H^s_p(\mathbb R), s>0, 1<p<\infty$is a space of tempered distributions$f$that satisfy$F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L_p(\mathbb R)$. Here,$F$denotes the ... 2answers 530 views ### Can exponential sums be small on a whole interval? This is almost certainly routine to an analyst, so forgive me in advance. Let$\alpha_i\in \mathbb{R}$. Consider the functional $$\varphi: L^1[0.9A,A]\to \mathbb{C}$$ via $$f\mapsto \sum_i \hat{f}(\... 2answers 170 views ### Name for an orthogonal decomposition of L^2 (\mathbb{R}^2; \mathbb{C}) The space L^2 (\mathbb{R}^2; \mathbb{C}) can be decomposed as$$ L^2 (\mathbb{R}^2; \mathbb{C}) = \bigoplus_{k \in \mathbb{Z}} L^2_k (\mathbb{R}^2; \mathbb{C}), $$where$$ L^2_k (\mathbb{R}^2; \... 1answer 83 views ### Spaces$C^\infty(\mathbb T^n\times \mathbb R^n)$,$C^\infty_0(\mathbb T^n\times \mathbb R^n)$and$\mathscr{S}(\mathbb T^n\times \mathbb R^n)$? [closed] Is there any characterization of the space$C^\infty(\mathbb T^n\times \mathbb R^n)$that I can take as a definition of it? I assume it would be something like this: $$C^\infty(\mathbb T^n\times \... 0answers 118 views ### Looking for some “nontrivial” examples of pseudodifferential operators/symbols I'm reading up on \Psi DO's and trying to find some examples of symbols that are not quite so trivial. Obviously, the first example of a symbol that most people talk about is just a polynomial in \... 1answer 82 views ### Help with notations from 2D to 3D FFT representations as 1D FFT I have this question on mathematics forum too, Notations, I thought of posting here, which ever place I get an answer, I will try to close it in the other. I need some help and clarifications for my ... 1answer 68 views ### Locality of homogeneous pseudo-differential operator Let P be a polynomial in several variables, and let P(D) be the corresponding differential operator. Obviously, P(D) is a local operator, in the sense that I need only to know the function u ... 0answers 38 views ### Wiener amalgam space W(\mathcal{F}L^{2}, L^{1}) \subset L^{1}? (I have asked this question on SE but could not get any answer and hope this is o.k for MO) Let X=\mathcal{F}L^{p}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{p}(\mathbb R)\}, and \|f\|_{X}= \|\hat{... 0answers 91 views ### Swapping sums and integration for a kernel in Fourier space (the non absolutely convergent case) Under what conditions on c_{r}^{m} does$$\int_0^{2\pi} k(p,q)\exp(-inq)dq=\sum_{r=0}^{\infty}c_{r}^{m}\exp\left(-imp\right) \text{ in } L^2_{per}$$hold for$$k(p,q)=\sum_{r=0}^{\infty}\sum_{l=... 0answers 70 views ### Is there a space in which the$\vec a$in$\sin(a_1\cdot x)+\sin(a_2\cdot x)$is linear? Suppose one has equations of the form$\sin(a_1\cdot x_i)+\sin(a_2\cdot x_i)=y_i$for$i = 1, \dots, n$(there are also amplitudes and phase shifts, but let's ignore these for now). I want to solve ... 1answer 386 views ### Proof of a Fourier pair with Bessel functions? How can we prove that the Fourier transform of the function $$f(x) = \begin{cases} (a^2-x^2)^{c/2} BesselJ[c,b\sqrt{a^2-x^2}] & \text{for }x^2 < a^2\\ 0 & \text{otherwise} \end{cases}$$ ... 1answer 77 views ### Elaboration of a certain section of a paper by Thanigasalam In section 11 of this paper by Thanigasalam, it says "... we get$G(10)\le 105$, and this implies that$H(10) \le 107$". However, it is very unclear how this follows. Why is it the case that$G(10)\le ...
I'm trying to answer this problem: Consider a real function f, bandlimited by frequency $\omega$, which satisfy $$\int_{-\infty}^\infty f(x)^2dx=c.$$ (For pure mathematicians: "bandlimited" means ...