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I guess it is possible to gain some understanding at least understand rough bounds. I will just consider the case $d = 1$. First observe that $$ \|p\| _{\ell^2(X)} \leq \|p\| _{\infty}. $$ Second we …

Just, a basic property one establish, when one assumes that $$ w^*(x) \leq C w(x) $$ is that $w(x)$ grows subexponentially, if furthermore $w(x)$ is continuous. Define $A = C \cdot \sup_{|x| \leq 1} …

There is the characterization from http://www.math.caltech.edu/SimonPapers/324.pdf or the work of Lubinsky. Of course the conditions are slightly different then yours. I am not sure of the meaning o …

The Schwartz space $\mathcal{F}$ is just one space, one could use to define distributions. Two other common examples are smooth functions $C^{\infty}$ and smooth functions with compact support $C^{\in …

Define $\psi_n(x) = c_n H_n(x) e^{-x^2/2}$ as in http://en.wikipedia.org/wiki/Hermite_polynomials . Also define the differential operator $H u = - u'' + x^2 u$. Then the $\psi_n$ form an othonormal ba …

You cannot split $$\left(1-\left(\frac{x}{n}\right)^2\right)\tag{1}$$ into $$\left(1 -\frac{x}{n}\right) \left(1 + \frac{x}{n}\right)\tag{2}$$ since the products no longer converge.

Here's how to carry out direct proof: By Weyl's criterion it suffices to show $$ S_N = \frac{1}{N} \sum_{n=1}^{N} e(k n^{\rho}) \to 0 $$ for $k \in \mathbb{Z} \setminus \{0\}$ and $\rho \in (1,2)$. …

Here is what is done with spectral theory: The boundary condition is just encoded in the domain. That is instead of considering the operator $-\Delta$ with domain $H_0^{2}(\Omega)$, one considers oth …

Of course, yes. Given a real number $\lambda$ define the distribution $$ E_{\lambda}(u) = \int e^{i \lambda x} u(x) dx $$ for all test functions $u$ (that is $u$ is smooth and compactly supported). T …

Continuing the idea from Gerry's answer. The quantity, you are looking for is just $$ D(N) = 2 \left( \# \{1 \leq n \leq N: \theta n \pmod{1} \in [0,\frac{1}{2}) \}- \frac{N}{2} \right) $$ If $\the …

AS for physical meaning of Hardy's inequality for p=2. Consider the Schroedinger operator $$ H = -\frac{d^2}{dx^2} - \frac{c}{x^2} $$ on $(0,\infty)$ with Dirichlet boundary condition at $0$. A natur …

Dear all. Let $$ f(x) = \sum_{k \in \mathbb{Z}} \hat{f}(k) \exp(2\pi \mathrm{i} kx) $$ be a function given by usual fourier series. Since my original question hasn't got any answer yet, and I cam …

I think the following is another proof. It suffices to approximate a polynomial of degree n by an exponential polynomial of degree n + 1. Now, define $$ g_{\lambda}(x) = \frac{e^{\lambda x}}{\lambda} …

Here is how to obtain a formula for $R_{k,m}$. Denote by $x_1, \dots, x_m$ the zeros of $H_m$. We have that $$ H_k(x) = P(x) H_{m}(x) + R_{k,m}(x) $$ for all $x$, so in particular for $x_j$ that $$ …

There is first Polya's estimate that if $f$ is a monic polynomial, then $$ |\{x\in \mathbb{R}:\quad |f(x)|\leq 2\}| \leq 4. $$ A proof can be found in the book "Proofs from the book". One can obtain …