# Questions tagged [cv.complex-variables]

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

2,832
questions

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### Conditions for an improper integral of a real-analytic function to be real-analytic

Let $U$ be a complex domain and suppose $f:U \times \mathbb{R} \rightarrow \mathbb{R}$ is continuous and, for all $t$, is real-analytic in $s$.
That is, $f(s, t)$ is continuous and $g_t(s):=f(s, t)$ ...

2
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### Existence of analytic function in disk algebra

Does there exist an analytic function $f\in A(\mathbb{D})$, where $A(\mathbb{D})$ is the disk algebra, such that $f(0)=0$ and the real part of $f(z)$ is strictly positive?

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### Generalization of residue theorem to arbitrary complex manifolds

If $f : \mathbb{C}\rightarrow \mathbb{C}$ is an everywhere complex-analytic function, then along a closed contour $C$, $\int_{C} f(z)dz = 0$. Is there a generalization of this theorem to arbitrary ...

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### Computing $\int_0^{2\pi}\frac{e^{ikt}}{|e^{it}-e^{it_0}|^m}~\text{d}t$, where $k\in\mathbb{Z}$, $t_0\in\mathbb{C}$, and $m=1,3,5,\dots$

I am working on a project on accurate numerical quadrature where I need to compute the following integral in order to find my quadrature weights,
$$
\int_{0}^{2\pi}\frac{{\rm e}^{{\rm i}kt}}{\,\left\...

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1
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50
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### Vector recurrences (asymptotic property)

Fix $m\in \mathbb{N}.$
For each $n\in \mathbb{N},$ let $A_n\in \mathbb{M}_{m}(\mathbb{C}),$ $X_n\in \mathbb{C}^m,$ and $B_n\in \mathbb{C}^m.$ Suppose that
$$X_{n+1}=A_n X_n+B_n,$$
$$\lim_{n\rightarrow ...

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233
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### How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic?

Let $\langle;\rangle$ be the usual scalar product on $L^2(\Bbb R^2)$.
How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic on $\Bbb C^+_*=\{z\in\Bbb C:\text{...

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77
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### Difference of a curve with itself [closed]

I asked this question on stack exchange but did not get any response.To re iterate,Consider a closed ,smooth curve $\gamma(t),a\leq t \leq b.$ For any $t \in [a,b]$,let us define :$$ z(t)=x(t)+iy(t)$$ ...

18
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958
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### Laurent series in several complex variables

Is there a good generalisation of Laurent series for several complex variables?
I am interested in generalised power series that have some terms with negative powers, but not too many. In single ...

2
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1
answer

126
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### Reference for asymptotic estimates

In the way of studying an enumerative problem I have found that I have to estimate the Taylor coefficients of functions of the following form. For two polynomials $P(x)$ and $Q(x)$ with $P(0)=Q(0)=1$, ...

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105
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### How to prove an equality involving Laguerre polynomials

Assume $\mu<0$. Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$ and $f\in L^2(\Bbb C)$.
How to prove that $$\sum^\infty_{k=0}\int_{\Bbb C} f(z)\frac{1}{2(2k+1)-\mu}L^0_k(\lvert z\rvert^2)...

3
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97
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### Converse theorem for zeta universality

Voronin's Universality Theorem for $\zeta(s)$ is that the zeta function can uniformly approximate any non-vanishing holomorphic function to any degree of accuracy in the right-half of the critical ...

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101
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### Bott-Chern cohomology for singular complex spaces

I'm reading the book 'An Introduction to the Kahler-Ricci Flow' (Lecture Notes in Mathematics 2086). They discuss Bott-Chern cohomology on complex spaces:
Let $X$ be a complex space(i.e. analytic ...

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32
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### For fixed $B > 0$, asymptotically describe the upper half plane complex zeros of $B\cos(\omega z) = iz$ as $\omega \to \infty$

Long story short, I am seeking to evaluate an oscillatory integral via complex methods, whose partial fraction expansion has $B\cos(\omega z) - iz$ in the denominator, where $B, \omega$ are positive ...

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162
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### On $\sum_{\rho\in D} \text{dist}(\rho)=\frac{1}{2\pi i} \int_{\partial{D}}\log \zeta(s)\ ds$

Let $D$ denote a closed two dimensional figure as: $D=2+iT\to 2\to 2-\delta\to 2-\delta+i(T-\delta)\to \frac{1}{2}+\epsilon+i(T-\delta)\to\frac{1}{2}+\epsilon\to\frac{1}{2}-\epsilon\to \frac{1}{2}-\...

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### Criterion to decide whether a function is algebraic

For Christol's theorem see 1, if a power series is algebraic over every fields of characteristics $p$, is it algebraic over fields of $0$, by Robinson's principle?
Update: The power series is in $\...

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25
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### Application of the $\operatorname{BMO}$, $H^1$ duality

Let $f\in \operatorname{BMO}(\partial \Delta)$, then there exists a Carleson measure $\mu$ in $\Delta$ such that
$$f(\zeta)-\int_{\Delta}P_{z}(\zeta)d\mu(z)\in L^{\infty}(\partial \Delta),\ \zeta\in\...

3
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### Non-holomorphicity of Hecke regularization of weight-2 level-1 Eisenstein series

Originally asked on stackexchange since I figured it was a very elementary and silly error, but since I haven't had any response or interest in a couple days, I thought I'd post it here.
I have a &...

3
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127
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### How to correctly renormalize this function at the pole $x=1$? Evaluating: $\sum_{n=1}^{\infty} e^{e^n}$

So I was considering the divergent everywhere but 0 power series
$$ f(x) = \sum_{n=0}^{\infty} e^{e^n} x^n $$
Now one can do the following "questionable" manipulation
$$ f(x) = \sum_{n=0}^{\...

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57
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### To find a DFT for complex functions on a semigroup

For a certain commutative semigroup of integer size $n$, $G=(\{1,2,\dots,n\},\circ: x\circ y\mapsto \min(n,x+y))$, consider all complex functions on it, denoted by $\mathbb C[G]$ or $\mathbb CG$. ...

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616
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### On the equation $\zeta(s) = F(s)+F(s+1)$

Define the function $F(s)$ as the Dirichlet series
$$
F(s) = \sum_{n=1}^\infty \frac{1}{(n+1)n^{s-1}},
$$
which converges for $\operatorname{Re}(s)>1$.
Has anyone seen/studied this function before? ...

3
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2
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117
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### Is every planar bounded $C^2$ domain finitely connected?

Let $\Omega \subset \mathbb R^2$ be a bounded $C^2$ domain. Is $\Omega$ then finitely connected? As I learned recently a domain in $\mathbb R^2$ is finitely connected iff “[its] complement has ...

2
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50
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### Analytic continuation of Fredholm integral equations

I am considering the equation of Fredholm type
$$
f(x) = \lambda \int_\zeta \prod(y-l_i(x))^{\alpha_i} f(y) dy\label{1}\tag{1}
$$
where
$\zeta$ is an element of 1st homology of the complement to ...

1
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0
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39
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### Holomorphic "quasi-interpolation" of a function sequence

I am interested in some sort of analytic interpolation. A toy version of my problem is as follows. Let $V \subset \mathbb{C}$ be a complex neighborhood of $[0,1]$. Assume there is some bounded ...

3
votes

1
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233
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### Relationship between two kinds of classifications of Riemann surfaces

There are two kinds of classifications of Riemann surfaces.
Classification 1: Let $M$ be a Riemann surface. We will call $M$:
elliptic iff $M$ is compact (= closed);
parabolic iff $M$ is not compact ...

11
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3
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654
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### Can computers find zeros of order $2$?

We assume we are given an entire function $f: \mathbb C \to \mathbb C$ with $f(0)=1$ and $f'(0)=0$ and $f$ is real on the real axis.
We assume (as a fact about $f$, that we want to demonstrate ...

7
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1
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187
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### Real analytic subvariety in complex manifold which is complex outside of its singular set

Let $M$ be a complex manifold, and $Z \subset M$ a closed real analytic subvariety. Suppose that the set of smooth points in $Z$ is complex analytic in $M$. Will it follow that $Z$ is complex analytic?...

2
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45
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### Semilinear elliptic equations in complex plane

Let $D$ denote the closed unit disk centered at the origin in the complex plane. Let $F: D \times \mathbb C \to \mathbb C$ be a smooth function. Is there any theory for well-posedness (in the sense of ...

3
votes

3
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174
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### Evaluating the series $\sum_{n=0}^{\infty} n! x^n$ and inverse variable-fractional-derivatives

So I was interested in formally assigning values to the completely divergent series $G(x) = \sum_{n=0}^{\infty} n!x^n $. I guess the question COULD end here if you already have an idea of how to ...

13
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3
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### Is anything known about the series $\sum_{n=0}^{\infty} x^{\sqrt{n}} $?

It's well known that there are a shocking number of identities for the usual Jacobi theta function $$ \theta_3(x) = \sum_{n=-\infty}^{\infty} x^{n^2}. $$
So I wanted to turn my attention to slowly ...

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0
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122
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### What can be said about cluster sets for power series of two variables?

I'm still trying to prove the continuity of a function $u$ which can be interpreted as the restriction of a power series of two variables, which I haven't managed to approach the right way yet. To ...

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1
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117
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### Seeking an integral formulation for an algebraic function

While working with a generating function for the Catalan numbers, I came across the integral representation
$$\frac1{1+\sqrt{1-4x}}=\frac1{2\pi}\int_0^{\infty}\frac{\sqrt{t}}{(t+\frac14)(t-x+\frac14)}\...

2
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84
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### The dual of the Lefschetz operator under a perturbation

Let $(X, \omega)$ be a compact Kähler, or more generally, Hermitian manifold. Let $L_{\omega} : \Omega^k(X) \to \Omega^{k+2}(X)$ denote the Lefschetz operator given by $$L_{\omega}(\alpha) : = \omega \...

4
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2
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278
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### Borel summation and the Abel function of $e^z-1$

This is a question that has bothered myself and Gottfried Helms a fair amount of late. He has made his case for the following result, but a proof escapes both of us. The question is deceptively simple,...

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0
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74
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### Construction of an $L^p$ function

Is possible to construct a function $\chi$ defined in unit ball of $\mathbb{C}^n$ which is in $L^p$ for $p$ large such that
$(f\ast \chi)(z)\geq f(z)$, where $f\in L^1$ ?

2
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1
answer

131
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### Regularity of boundary of a level set of a $C^{1,\alpha}$ function

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^{1,\alpha}$ function. Denote $S_C=\{x\in\mathbb{R}^2\mid f(x)=C \}$ the level set of $f$ with value $C$.
What i want to ask is, if $S_C$ is nonempty for some $...

3
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1
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278
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### Can a power series of several variables be discontinuous on a compact set if it converges in every point of this set?

Say we have a power series of two variables, with an associated function $f$ defined as
$$
\begin{split}
f(x, y) =\, & \sum_{n,m} a_{n,m}x^ny^m,\\
& a_{n,m} \geq 0 \quad \forall n, m \in\...

2
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0
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122
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### "Circulant-Vandermonde" matrix: in search of a formula

An $n\times n$ circulant matrix $\mathbf{X}_n$ has the form
\begin{align}
\mathbf{X}_n= \begin{bmatrix}
x_1 & x_2 & \cdots & x_{n-1} & x_n \\
x_2 & x_3 & \cdots & x_n&...

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1
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217
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### Significance of $N_0(T+1)-N_0(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$

Let $N(T)$ be the number of zeros of Riemann zeta function upto height $T$ in the critical strip and $N_0(T)$ be the number of zeros on the critical line.
What will be the significance of proving ...

3
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56
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### Boundary behavior of conformal map on domain satisfying an exterior sphere condition

I'm in the middle of a project concerning a Bernoulli-type free boundary problem in $\mathbb{R}^2$ and, as part of this project, I would like to understand the boundary behavior of conformal maps on ...

3
votes

1
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117
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### On well separated circular regions in the Riemann sphere and complex polynomials

It started with a conjecture I had, see A statement on complex polynomials, which was false for $n \geq 3$, as shown by Noam D. Elkies in his answer there. The present post is an attempt to salvage ...

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### When a strictly positive log pluriharmonic function $g$ is equal to the norm of holomorphic function?

Suppose $V$ is a local analytic variety (singular). Suppose $g$ a strictly positive log pluriharmonic function on $V$, i.e. $\log g$ is pluriharmonic. I wonder when $g=|f|^2$, where $f$ is a ...

1
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2
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104
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### $\log(t)$ term in the small time expansion of $\mathrm{Tr}( A e^{-tB} )$

Assume $A$ is an operator on a Hilbert space with discrete spectrum. Assume $B$ is a positive operator on the same Hilbert space also with a discrete spectrum. Also assume $A$ and $B$ commute.
I'm ...

5
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1
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771
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### A statement on complex polynomials

I have a feeling the following is true.
Assume that there are $n$ mutually disjoint closed disks $D_i$ in the complex plane and $n$ complex polynomials $p_i(z)$ of degree $n - 1$, with both types of ...

5
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0
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199
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### Picard-Lefschetz formula for the quotient of a degenerating family of curves by a cyclic group

$\newcommand{\cD}{\mathcal{D}}\newcommand{\cX}{\mathcal{X}}$(This is a slight rephrasing and modification of the original question)
Let $D\subset\mathbb{C}$ be the complex unit disk. Let $X$ be a ...

2
votes

1
answer

100
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### Finding the repelling fixed point of an exponential, knowing only its attracting one

This question has been bugging me for a while, I have an answer that is working sufficiently for the program I'm using, but it is a tad slow, and let's say imprecise. It is not an overtly difficult ...

5
votes

1
answer

349
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### Example illustrating necessity of considering birational equivalence and not biholomorphic equivalence in MMP

The minimal model program attempts to classify algebraic varieties up to birational equivalence. For compact Riemann surfaces, Riemann's uniformization theorem tells us that the geometry of the curve ...

9
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2
answers

1k
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### Zeros of a complex function

I wonder whether $\sum_{k=0}^n \exp(r_k z)$ has a complex zero for any $n\in \mathbb{Z}_n^*,0=r_0<r_1<r_2<\dotsb<r_n$. I think the answer is affirmative.

1
vote

2
answers

230
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### Abscissa of convergence for a very specific Dirichlet series / Euler product

I am interested in the convergence of the following Euler product:
$$
\prod_p \frac{1}{1-\chi(p)\cdot p^{-s}}.
$$
The product is over all primes (in increasing order), with $\chi(p)=+1$ if $p \bmod 4 =...

4
votes

0
answers

190
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### Geometric interpretation of Theta functions and the Jacobi inversion problem

A great part of complex geometry and, algebraic geometry has been developed to address the theory of abelian integrals/functions. A very special problem that kept many great mathematicians busy was ...

6
votes

1
answer

274
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### On frequency decay of an integral transform of a function

Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that
$$
\bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$
for all $\tau \...