Well-definedness on C∞0(Rn)C_{0}^{\infty}(\mathbb{R}^{n})

Let T be a Calderon-Zygmund operator associated to a Calderon-Zygmund kernel K∈CZKα of order α>0 and b∈BMO(Rn). Then for f∈C∞0(Rn) define[b,T](f)=b⋅Tf−T(bf). Question. Is [b,T] is well-defined on C∞0(Rn) for all b∈BMO(Rn)? Answer AttributionSource : Link , Question Author : Timothy , Answer Author : Community

Fourier transform of C∞0C^\infty_0, smooth functions vanishing at infinity

Is there a proper description of the space {ˆf | f∈C∞ s.t.∀α∈Nn,∀ϵ>0∃K⊆Rn K compact; supEspecially I’m interested in the order of these distributions and their decay properties. To be precise are there constants C_{K,m}, for every K\subseteq\mathbb{R}^n compact and m\in\mathbb{R}, s.t. \left|\int \hat f(\xi)\phi(\xi-b)\ d \xi\right|\leq C_{K,m}(1+b^2)^{-m}\|\phi\|_{\infty}\quad \forall\phi\in C_c(K) or anything slightly worse? Edit: The assertion automatically holds for all m\in\mathbb{R} … Read more

Variability of finite Fourier coefficient with length

This is a restricted question related to the one here. Consider the unnormalized Fourier coefficients of subsets Dg of Z/nZ, denoted by ˆ1Dg(m,n)=∑d∈Dge(mdn), where e(x)=e2iπx and the Dg which are arithmetic series dilated by a geometric series, Dg=2g{1,2,…,s},g=1,…,v. Let n be odd, fix an m and a g and say that a certain n is … Read more

Fefferman’s article: Pointwise convergence of Fourier series, II

I have some problems reading Pointwise convergence of Fourier series by Fefferman https://www.jstor.org/stable/1970917 I got stuck in Chapter 6, Lemma 5. In the proof he split the P′ into three subcollections P′k, Pk″, \mathcal P”’_k. The first and third subcollections were estimated. However, for the second, he imposed a new assumption: \varphi_k(\xi_0′)=0, where \xi’_0 is … Read more

Convergence of integral formula for Fourier inversion (and Hilbert transform) for integrable piecewise-smooth functions

I asked the question below on Math Stack Exchange, https://math.stackexchange.com/questions/2592555/convergence-of-integral-formula-for-fourier-inversion-and-hilbert-transform-fo, but [despite it having 6 upvotes and two favourites] received no answer even after a 300-point bounty. Therefore, I am re-posting it here. Let f:R→R be a bounded L1 function that is piecewise-smooth (with the boundaries of the pieces having no accumulation points), but not … Read more

Is there a categorical foundation for manifolds of bounded geometry and bandlimited functions?

As an outsider to both, manifolds of bounded geometry and bandlimited functions appear rather connected: for example, bounded geometry is defined in terms of bounds on curvature and its derivatives, while if the derivative of a bandlimited function has L1 Fourier transform (and hence is bounded) then so do all higher derivatives. Isaac Pesenson et … Read more

A uniform Riemann sum approximation of the integral of the Fejer kernels

Let FN(t) denote the Fejer kernel FN(t):=1N+1sin2((N+1)t2)sin2(t2) . Consider Riemann sums approximation for ∫π−πFN(t)dt relative to uniform subdivisions with mesh 1n: 2π=∫π−πFN(t)dt =1n∑−πn<j<πnFN(jn)+ r(N,n) . I would like to obtain an estimate of the remainder term r(N,n) for n and N large, of the following form: assuming n=KN2+o(N2) as N→+∞, is it true that r(N,n)=cN+o(1N), and, most important, I … Read more

When does a continuous function’s “Fourier series” converge pointwise almost everywhere to the function?

Let $G$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $L^2(G)$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible representations of $G$. When $G$ is commutative, these irreducible representations are one-dimensional, and the Peter-Weyl theorem just says that the unitary characters of … Read more

Fourier coeffients of Cantor measure

For 0<θ<12, denote by μθ the uniform Cantor measure with dissection ratio θ. It is not hard to show that the Fourier–Stieltjes transform of μθ is ˆμθ(ξ)=∞∏k=1cos(πθkξ) (up to scaling and constant multiple). It is well known that ˆμθ(ξ)→0 as ξ→∞ if and only if θ is a Pisot number. Now we are concentrated on … Read more

L1L^1 norm of oscillatory integral operator

My question is about the L1x norm of an oscillatory integral like ∫Rnei(y⋅x+λϕ(y))f(y)dy, where λ∈R, f∈C∞c(Rn) is compactly supported and ϕ is real-valued and smooth (I have in mind ϕ(y)=√1+|y|2 but feel free to assume what you need). Of course, I am interested in the decay rate for λ in this norm. Is there any … Read more