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A simple remark. Let $\mathbb H$ be an Euclidean space with dimension $N$ and let $u_1,\dots, u_{N-1}$ be an orthonormal set of vectors. Let $x$ be a vector in $\mathbb H$: it belongs to the hyperplan …

You should keep in mind analytic continuation. Let $f:\mathbb R\rightarrow\mathbb R$ be a $C^\infty$ function ; if there exists an entire function extending $f$, then (1) It is unique by analytic co …

The optimal space for the Fourier transform is the space of tempered distributions $\mathscr S'(\mathbb R^n)$, i.e. the dual space of the Schwartz functions $\mathscr S(\mathbb R^n)$. The latter is th …

Let $n\ge 2$ be an integer and let $f$ be an entire function on $\mathbb C^n$. Let $A$ be a subset of $\mathbb R^n$ with positive $n$-dimensional Lebesgue measure. Then if $f$ vanishes at $A$, this im …

Let $f$ be the entire function on $\mathbb C$ defined by $$ f(z)=\frac{z-\sin z}{z}. \tag{1}\label{1}$$ It is easy to see that $f$ is positive on $\mathbb R^*$ and has a zero of order 2 at 0. Does the …

Let me quote the Paley-Wiener-Schwartz theorem. Let $F$ be a tempered distribution on $\mathbb R^n$. Then the two following properties are equivalent. (i) $F$ is compactly supported with $\text{supp} …

More a comment than an answer, but too long anyway for a comment. There is nothing weird or mysterious about your first equalities: with $R>a>0$, we have from the residue formula, $$ \int_{[-R,R]}\fra …

Let us work first formally. You want to "calculate'' $ f(s)=\int_0^{+\infty}t^{1/2} e^{-st} dt. $ It is indeed possible to give a meaning to $ F(\tau)=\int_0^{+\infty}t^{1/2} e^{-i\tau t} dt $ as the …

Let $A$ be a $k\times k$ invertible matrix, i.e. in $Gl(k)$. Assume that the segment $[I,A]$ lies in $Gl(k)$. Let us define $$ \text{Log}A=\int_{[1,A]} \frac{d\xi}{\xi}=\int_0^1(I-tI+tA)^{-1}(A-I)dt. …

Take $k=1$ in your statement. There are two easy cases: the first one is when $f$ is real-valued, then you have only to use the implicit function theorem to get a normal form $t+a(x)$, up to a unit (a …

Let $\Omega$ be an open subset of $\mathbb R^d$, let $U$ be an open subset of $\mathbb C$ and let $f:\Omega\times U\rightarrow\mathbb C$ be a $C^\infty$ function which is holomorphic with respect to $ …

The function $\mathbb R\ni t\mapsto\zeta(\frac12+it)$ is analytic and smaller in absolute value than $C(1+\vert t\vert)^{1/6}$ (the $1/6$ may be replaced by $9/56$ and even by a slightly smaller numb …

You need to check Condition (ii) in Definition 8.2.2 in the first volume of Hörmander's ALPDO. Let us note $f(x+i0)$ the limit-distribution of your question and let $\Gamma$ be its wave-front-set. Let …