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The standard mollifier function is defined as follows $$f(x)=\begin{cases} 0 & \text{if } |x| \ge 1\\ \exp \left(-\cfrac{1}{1-x^2}\right) & \text{if } |x|<1.\end{cases}$$ It is well known that $f$ is …

Let $\phi$ be a smooth real function in one variable and say $w$ is a smooth function with compact support say $[- 1, 1]$. Let me define $$ I_{\lambda} = \int_{\mathbb{R}} w(t) e^{i \lambda \phi(t)} d …

Let $P$ be a large number. Let $[a,b]$ be a fixed interval. Is it possible to construct a function $w$ with the following properties?: $w$ is smooth, the support of $w$ is contained in $I = [a- 1/P, b …

Let $N$ be a large integer and $I = [aN, bN]$ for some $0 < a < b < 1$. Denote by $\chi_I(x) = 1$ if $x \in I$, $0$ otherwise. I was wondering if there exists a smooth function $w$ with the property …

Let $f(t) = \min(\frac{1}{\lvert t\rvert}, 1)$. I would like to find a smooth approximating function $g$ such that $f(t) \leq g(t)$ for all real $t$. Is there a nice function $g$ out there? Any sugges …

Let $a_i$ be a nonzero real number for each $1 \leq i \leq n$. $w$ a smooth nonnegative with compact support. I would like to understand the following integral. $$ I = \int_{\mathbb{R}} \int_{\mathbb{ …

Let $\mathbf{v}_1, \mathbf{v}_2$ be two vectors in $\mathbb{R}^n$. I would like to compute the following singular integral: $$\int_{-\infty}^{ \infty} \int_{-\infty}^{\infty} \int_{[-1,1]^n} e(\thet …

Suppose $\Lambda$ is a $3$ dimensional lattice inside $\mathbb{R}^4$ and let $E$ be the subspace $\mathbb{R}$-spanned by $\Lambda$. What exactly is meant by the unit ball in $E$? This is something tha …

Let $F$ be a homogeneous form with coefficients in $\mathbb{R}$. Suppose it defines a smooth projective variety, in other words at every point other than the origin at least one of the first partial d …

Let $T$ be a triangle in $\mathbb{R}^2$ defined by $y = \alpha x$, $y = \beta$ and $x = \gamma$ where $\alpha, \beta, \gamma \in \mathbb{R}_{>0}$. I am interested in obtaining an estimate for the numb …

Let $I_1$ and $I_2$ be two closed bounded intervals. Suppose $W(x,y)$ is a smooth function whose support is contained inside $I_1 \times I_2$. Suppose I have $\Phi= (\Phi_1(x,y), \Phi_2(x,y)) : \ma …

I have the following question, which intuitively seems it should be true but I wasn't sure how to prove it rigorously. Let $\delta, \varepsilon > 0$. Let $\Psi_i(w_1, w_2, \mathbf{v})$ be a polynom …

In the proof of the stationary phase method (at least the one I have seen) Morse lemma shows up, which states: Let $g:\mathbb R^n\to \mathbb R$ be a function of class $C^\infty$ for which $0$ is a no …

A statement of the stationary phase method I know is the following. Suppose $\phi(x_0) = \phi'(x_0) = 0$ and $\phi''(x_0) \not = 0$. If $\psi$ is a smooth function supported in a sufficiently small n …

Let $B \subseteq \mathbb{R}^n$ be a product of closed bounded intervals in $\mathbb{R}$. Fix $N>0$. Suppose I want to cover $B$ with $N$ open sets, $U_1, \ldots, U_N$, and get a smooth partition of un …