# Tagged Questions

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### Convergence of Fourier series in $L^\infty$

So if $f\in L^1(\mathbb{T})$ and $S_Nf\rightarrow f$ in $L^\infty(\mathbb{T})$ ($S_Nf$ is the partial sum of the fourier series of $f$), then $f$ is continuous. How do we show that this is true? In ...
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### 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 ...
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### Complex exponential is not Fourier multiplier on $L^p$

I am having difficulty to show that the function $m(\xi):= e^{i|\xi|^2}$ is not a Fourier multiplier on $L^p$ when $p\neq 2$. Note that $m:\mathbb{R}^n\to \mathbb{C}$ is called an $L^p$ Fourier ...
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### 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 ...
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### $f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty$. We put, $$X_{p}= \{f\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) :\hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R)\};$$ and we consider the algebra of Fourier ...
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### 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 ...
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### 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 Banach sapace $C([0, T];M^{p,1})$?

(For details of this question you may see the paper,(proof of theorem 1, page.9), MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; Bull. Lond. Math. Soc. 41 ...
Let $G$ be a locally compact Hausdorff abelian group (LCA). Then the Fourier transform gives a map from functions on G to functions on $\hat G$ $$f\mapsto \hat f(\xi) = \int_G f(x)\Psi(x,\xi) ... 0answers 71 views ### Closure of Schwartz space in homogeneous Besov space Let \dot{B}^s_{\infty,\infty}(\mathbb{R}^d) denote the homogeneous Besov space of order s with second and third index \infty, i. e. the homogeneous Zygmund space. Let \mathcal{S}(\mathbb{R}^d) ... 1answer 183 views ### How to use Parseval' identity( Plancherel)? [duplicate] (May be this is very basic question for MO) Let f\in L^{2} (\mathbb R) with \lim_{t\to \pm \infty} f(t)=0. Put$$F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt \ (n=1,2,...)$$Fix ... 1answer 118 views ### How to use Parseval' s( Plancherel' s) identity? Let f\in L^{2} (\mathbb R) with \lim_{t\to \pm \infty} f(t)=0. Put, F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt, \ (n=1,2,...). Fix \alpha \in (0, \infty) and we define H_{n}(x) ... 0answers 38 views ### How to choose \lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha); so \lambda ^{-1}\in L^{1}(\mathbb R)? We define,$$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$where \mu (\alpha) is a non-decreasing function such that the integral converges for ... 1answer 50 views ### How to apply the method of stationary phase here? Consider the following oscillating integral$$ I(\xi;t) = \int\limits_{\mathbb R}\int\limits_{\mathbb R^n} e^{it(\xi y - \theta f(y))}a(y) \, dy \, d\theta, \quad \xi \in \mathbb R^n \setminus 0, ...
Let $\{c_{n}\}_{n\in \mathbb Z}\subset \mathbb C$ and $\sum_{n\in \mathbb Z} |c_{n}| < \infty$ (that is, the series $\sum c_{n}$ is absolutely converges); we define $F:\mathbb R \to \mathbb C$ ...