# Tagged Questions

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For a bounded complex Borel measure $\mu$ on $\mathbb R$, we define, its Fourier-Stieltjes transform, $\hat{\mu}(y)= \int_{\mathbb R} e^{-2\pi ix\cdot y} d\mu(x); (y\in \mathbb R).$ Let $1\leq p \leq ... 0answers 70 views ### Tauberian theorem from generalized Gelfand transform Wiener's theorem gives the necessary and sufficient conditions for the set of translates of a set of functions to be dense in$L^1(\mathbb{R}^n)$, which translates algebraically into a statement about ... 0answers 88 views ### Special elements in$L^{\infty}(G)^*$Let$G$be a locally compact group. Let$M(G)$denote the measure algebra and$L^1(G)$denote the group algebra on$G$. Then$M(G)$acts on$L^1(G)$by convolution. So by duality$M(G)$acts on ... 1answer 146 views ### Closed sets in the space of Fourier transforms$\mathcal{F}L^{1}$Consider the space of all Fourier transforms of$L^{1}(\mathbb R),$that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ... 0answers 45 views ### Hermite coefficients of a positive density It is well known that a necessary condition for a function in$L_2$to be a.e. positive is that its fourier transform is positive-definite (in fact, due to Bochner's theorem, this is also a sufficient ... 0answers 91 views ###$f,g , |f|f, |g|g \in A(\mathbb R) \ \text{(Banach algebra)} \implies \left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$? Let$f\in L^{1}(\mathbb R)$and it Fourier transform,$\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$and consider Fourier algebra $$A(\mathbb R):= \{f\in ... 0answers 132 views ### Scattering for rapidly decaying solutions of NLS Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions of the Nonlinear SchrÃ¶dinger Equation" the following property. Given the problem \left\{ \begin{array}{rl} ... 1answer 147 views ### Can we expect \left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g|| in the Banach algebra A(\mathbb T) ? Let f\in L^{1}(\mathbb T) and define the Fourier coefficient of f : \hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z).Consider the space,$$A(\mathbb T):= \{f\in ... 1answer 51 views ### uniqueness for Poisson equation in R^d with mildly regular data I'm interested in Poisson's equation$-\Delta u=f$set in the whole space$R^d$(let's say$d\geq 3$for simplicity) when$f$has very little integrability, specifically$f\in L^{1+\varepsilon}$for ... 0answers 85 views ### How to use,$(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space$C([0,T]; M^{p,1})$? (May be this is very basic question for MO) (For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ... 1answer 136 views ### Does Hilbert Transform commute with Function Multiplication modulo Compact on$L^p(R)$? Define Hilbert Transform (HT) as the convolution with the function$1/x$. E. Stein proves in his book Singular Integrals and Differentiability Properties of Functions that HT, when understood as a ... 1answer 220 views ### When one can expect$\widehat{(fg)} = \hat{f} \ast \hat{g}$;$f, g\in L^{1} (G)$? Let$f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of$f$as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ... 0answers 158 views ### Laplace Transform in the context of Gelfand/Pontryagin Question: Do quasi-characters properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the Fourier transform and decompose$L^1$... 0answers 64 views ### Harmonic/functional analysis question: Uniform bounds for$(1 - \varepsilon\Delta)^{-1}$as$\varepsilon\to 0$? Is there a space$X$, compactly embedded in$H^{-1}(R^2)$, such that the operators$(1 - \varepsilon \Delta)^{-1}$are bounded from$L^1(R^2)$into$X$, with operator norms that are in turn bounded by ... 1answer 294 views ### Is every distribution a linear combination of Dirac deltas? My question is whether Dirac-type distributions over an Abelian group define a basis of the Schwartz-Bruhat space$\mathcal{S}(G)^\times$of tempered distributions on$G$, so that any distribution ... 0answers 228 views ### Carleson's Theorem on Manifolds Let$M$be an oriented, compact, differentiable manifold with some Riemmanian metric$g$, so that$(M,g)$has a nice volume form and one can define$L^2(M,g)$as the completion of$C^\infty(M)$under ... 1answer 147 views ### Understanding Bruhat's notion of Schwartz function I am trying to understand Bruhat's generalized Schwartz functions over (Hausforff) locally compact Abelian groups [1], following this paper [2] by Osborne. There, the Schwarz-Bruhat space ... 2answers 850 views ### Fourier transform of the unit sphere The Fourier transform of the volume form of the (n-1)-sphere in$\mathbf R^n$is given by the well-known formula $$\int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ... 0answers 107 views ### Weak amenability and quasi central bounded approximate identity Let \cal A, \cal B be a non commutative Banach algebras, and \cal A be weakly amenable and has a quasi central bounded approximate identity. Let T:\cal A\to \cal B be an algebra ... 1answer 158 views ### eigenfunctions of the Fourier transform in the Schwartz space Recall that L^2(\mathbb R) decomposes into the direct sum of the eigenspaces of the Fourier transform corresponding to its four eigenvalues, namely the four fourth roots of unity. If f\in ... 0answers 79 views ### Differentiability of f*g on the circle, for integrable f, bounded g, and some decay of the Fourier coefficients of f If f\in L^1(\mathbb{T}) and g\in L^\infty(\mathbb{T}) where \mathbb{T} is the circle, such that \hat{f}\in L^{p}(\mathbb{Z}) for some 1\leq p<\infty, do we have that f*g is ... 0answers 120 views ### A problem concerning measures on locally compact spaces I am stuck on a question for quite sometime now, although in the text it is said to be "apparent". The problem goes as the following : Let X and Y be locally compact Hausdorff spaces. Then M(X) ... 2answers 318 views ### L^p-norm of Fourier series in terms of coefficients, p \neq 2 It is known that the L^2-norm of a Fourier series equals the l^2-norm of the coefficients. Are there similar results in the case of L^p-norm for p\neq 2? Can it be expressed explicitly in ... 0answers 80 views ### Applying Anderson's theorem to Spherically symmetric distribution in Stein estimation The question appears Example 3.1 of the paper "Stein Estimation for Spherically Symmetric Distributions: Recent Developments" ... 1answer 171 views ### A proof of energy functional appearing in the regularity of elliptic and parabolic equations I have got trapped in this problem for nearly two years when I dealt with regularity of solutions of elliptic and parabolic equations. I have not found a nice proof to support this assertion. Now I am ... 0answers 112 views ### Tauberian measures on a locally compact abelian group Given a locally compact abelian group G with Haar measure m, it is well-known that there exist measures \mu\in M(G) which are singular (with respect to m), but the convolution product ... 1answer 168 views ### Does the topological Varopoulos algebra consist of functions that are continuous and Varopoulos norm bounded? Let X_1,\dots,X_n be compact Hausdorff spaces. Let's define the Varopoulos algebra as the projective tensor product:$$V(X_1,\dots,X_n) := C(X_1) \hat{\otimes} \dots \hat{\otimes} C(X_n),$$i.e. the ... 0answers 66 views ### Fourier multiplier with a singularity on a convex curve Let h be a strictly convex function such that h(0) = h'(0)=0. Let \Phi: \mathbb{R}^2 \to \mathbb{R} be a C^{\infty}-function with compact support (say, \Phi is supported on ... 1answer 500 views ### How differentiable is the convolution of two continuous functions? The question is really simple: Given$$ f, g\in C^\alpha_c(\mathcal{R}^d) $$is$$ f*g\in C^d_c? $$I came up with a formal argument using the decay of the Fourier transform of continuous functions, ... 1answer 266 views ### Poisson Summation Formulas for Cut and Project Quasicrystals In Lagarias' paper "Mathematical Quasicrystals and the Problem of Diffraction" http://www.math.lsa.umich.edu/~lagarias/doc/diffraction.pdf he discusses various ways one might get Poisson summation ... 1answer 111 views ### Matched pair of locally compact groups In measure theoretic language there is a notion of matched pair of locally compact (l.c.) groups due to Baaj-Skandalis-Vaes. A pair (G_{1}, G_{2}) is called a matched pair of l.c. groups if there ... 1answer 94 views ### local moments of measures whose Fourier transform vanish in an interval Assume h is a measure whose Fourier transform vanishes in an interval [-\Omega,\Omega]. I'm interested in obtaining inequalities of the form \begin{equation*} \int_{-\delta}^{+\delta}|h|(dt)\le ... 2answers 179 views ### On lower bounds of exponential frames in l1 norm Let \{t_k\}_{k=-\infty}^\infty be a sequence of real numbers. I'm interested in finding the largest number A such that \begin{equation*} \int_{-\Omega}^\Omega|\sum_{k=-\infty}^{+\infty}c_ke^{2\pi i ... 2answers 141 views ### Non-global oscillation of banded Fourier transform Can we say something like monotonicity, growth rate and oscillation of the Fourier transform of a banded function f with support [0, N]$$\mathcal{F}f(\xi) = \int_{0}^N f(x)e^{-ix\xi}dx.$$Of ... 0answers 98 views ### How to bound Haar coefficients in terms of total variation? I'm trying to get the basic idea behind the proof of Theorem 8.1 of this paper, but I'm having difficulty. Specifically, it says: We shall show that there is a set \Lambda_n\subset\mathcal{D} ... 1answer 275 views ### Could we interpolate the compactness of compact operators? Classical theorems of Marcinkiewicz and Riesz and their extensions to general Banach spaces by CalderÃ³n, Lions, Peetre, et al. allow us to interpolate the continuity of two operators, viz., the ... 2answers 387 views ### For what spaces is the Hardy-Littlewood maximal operator of strong type (p,p) if and only if p > p_0 > 1? (This is essentially a continuation of my previous question, here.) Let (X,d,\mu) be a metric measure space, i.e. \mu is a Borel measure on the metric space (X,d). Further assume (though you ... 1answer 154 views ### Description of Bessel potential spaces Hi, let 1 < p <\infty , 0 < \alpha < 1, and \mathscr{L}^p_\alpha(R^n) be the usual Bessel potential space defined by$$ \mathscr{L}^p_\alpha = (1-\triangle)^{-\alpha/2}L^p(R^n). ... 0answers 244 views ### What information about a locally compact group$G$is encoded in$C_r^\ast(G)$which is not in$L^1(G)$? Let$G$be a locally compact group and let$ C_r^\ast(G) $denote its reduced group$C^\ast$-algebra. Many features of a$G$can be realized from$L^1(G)$or$C_r^\ast(G)$. For example,$G$is ... 1answer 212 views ### About the boundedness of a multiplication operator. Let be$f$a$2\pi-$periodic function and$\hat{f}(k)=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-ikx}dx$. Consider the operator: Tf(x)=\sum_{k\in\mathbb{Z}}sign(k)\ \hat{f}(k)\ e^{ikx}. ... 1answer 234 views ### For which metric measure spaces is the Hardy-Littlewood maximal operator not of weak type (1,1)? Let$(X,d,\mu)$be a metric measure space, i.e.$\mu$is a Borel measure on the metric space$(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind ... 1answer 235 views ### An interpolation inequality. For all$s>0$define for$\epsilon\in(0,1)$the function: $$g(\epsilon)=\sum_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k.$$ Prove that$\exists C>0$and$\phi(s)$such ... 0answers 102 views ### Growth of inner functions on the disk Recall that an inner function on the disk$D$is a bounded analytic function on$D$having radial limits of modulus one almost everywhere. There has been many works on the growth of the inner ... 0answers 238 views ### Continuity of multiplicative character Let$G$be a discrete group and$\beta (G)$denote the Stone-Cech compactification of$G$, a right topological semigroup. By a multiplicative character, I mean a mapping that preserves multiplication ... 2answers 248 views ### Orthonormal basis for$L^2(G/H)$. Let$G$be a locally compact group and$H$be a closed subgroup of$G$. Is there any way to define a reasonable orthonormal basis for$L^2(G/H)$? By "reasonable" I mean elements of the orthonormal ... 1answer 244 views ### Characters separating points on Maximal Torus modulo Weyl group? Let G be a compact Lie group, for example, SU(n). Let T be its maximal torus. Let W be its Weyl group. Every finite-dimensional representation of G has a character, which is a function on G, T and ... 2answers 386 views ### Corona Theorem in several variables Hallo, I have read about the Corona Theorem (see link:http://en.wikipedia.org/wiki/Corona_theorem). From this one ca deduce that: Let$f_{1}, ..., f_{n}$be holomorphic bounded functions on the unit ... 2answers 194 views ### Alternate definitions of$C^{1,\alpha}$and$C^{1,\alpha}(\bar{D})$maps My question is about the precise definition regarding the following: Let$f$be an orientation-preserving$C^1$diffeomorphism of the unit circle$S^1$. So$f'(b)$exists and can be thought as a ... 2answers 218 views ### Structure of the unitary representation$L^2(N/M)$when$N\$ is a nilpotent Lie group

Hi All, I am new to this (though I seem to be a latecomer); so forgive me if this is not your most favorite question: I am trying to understand the structure (e.g., decomposition) of the unitary ...
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### When is Prim(A) of an infinite discrete group hausdorff ?

Does anyone know, if the following result has been proved ? Let G be an infinite discrete group. A = L1(G) it's algebra and Prim(A) the set of prime ideals with spectral topology. The result is : ...