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Fourier transform of an $L^{1}$ function $f$ is bounded by $\|f\|_1$. $f_n \to f$ in $L^{1}(\mathbb R^{n})$ implies that $\|f_n\|_1$ is a bounded sequence. So $\int |\hat {f_n}-\hat f|^{p} \leq (\sup_x |\hat {f_n}(x)-\hat f(x)|)^{p-2} \|f_n-f\|_2^{2} \to 0$.

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You can use this version of the Riemann-Lebesgue lemma: If $g$ is Riemann integrable on $[a,b]$, then $$\int\limits_a^b g(x)\cos (nx)\, dx\rightarrow 0$$ as $n\rightarrow\infty$. It is clear that $g(x)=f(x)/x$ is continuous on $[0,\pi]$ if we take $g(0)=0$ (the only problem area is zero, and the condition that $|f(x)|\leq x^4$ takes care of that). Since ...

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I can provide an affirmative answer if $a>1/2$. Since $f\in C(\mathbb{T})\subset L^2({\mathbb{T}})$ and $(n^a a_n)$ is a bounded sequence by assumption, an application of Parseval's Theorem implies \begin{align*} ||\partial_t^a g||_2 = ||n^a g_n||_{\ell^2} = ||n^a a_n f_n||_{\ell^2} \leq C ||f_n||_{\ell^2} = C ||f||_2<\infty. \end{align*} It ...

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Since we do a 4 sample DFT, we will be looking at powers of the complex 4th root of unity. $w$ is the complex fourth root of unity $w^4 = 1$, we can pick $w = i$ or $w=-i$ It seems this aufgabe chooses the negative one. Now substitute $-i$ everywhere and you will get the matrix. $$\cases{(-i)^0 = 1\\(-i)^1 = -i\\(-i)^2 = -1\\ (-i)^3=i}$$ And then it will ...

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In general, Good kernels serve as a bridge between the continuous and the discontinuous, and even the discrete. The mathematical tools of analysis are generally defined for functions that are at least continuous, and often smooth. But when you model a physical object, that model is generally discontinuous. For example, density suddenly drops from a non-...

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This is an old question, but it's not an uncommon exercise, so I think that it's worth having an approach sketched. The orthonormal basis for $L^2([0,1])$ is given by elements of the form $e_n=e^{2\pi i nx},$ with $n\in\mathbb{Z}$ (not in $\mathbb{N}$). Clearly, this family is an orthonormal system with respect to $L^2$, so let's focus on the basis part. ...

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