# Tagged Questions

A distribution is a continuous linear functional on the space $\mathcal{C}^{\infty}_c$ of smooth (indefinitely differentiable) functions with compact support. Though they appeared in formal computations in the physics and engineering literature in the late $19^{th}$ century, their formal setting ...

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Consider an oscillatory integral of the first kind $$I_\lambda(x)=\intop_{\mathbb{R}^{n}}e^{i\lambda\Phi(x,y)}a(x,y)\,d y,\quad \lambda\geq 0,\; a\in C_c^\infty(\mathbb{R}^{k+n}),\; \Phi\in C^\infty(\... 0answers 701 views ### Nonlinear PDE and Green functions This is somewhat of a curiosity that can hide somewhat deeper. For a Green function of a nonlinear PDE I mean something like$$ \partial^2\phi+V(\phi)=\delta^D(x). $$I do not know if a real ... 0answers 182 views ### Spectral gap of tempered distributions Hi, Let \Lambda\subset\mathbb{R} be an infinite discrete set of finite density (for simplicity one may take the density equals 1) and \delta_{\lambda} is a unit mass located at the point \lambda\... 0answers 42 views ### A question about Fourier transform of function of the type (1+P(x))^{z} Let$$f= (1+P(x))^{z},$$where P(x)\ge 0 is a real polynomial in \mathbb{R}^n of degree 2m>0, and z=a+ib with a<0. I want to study the behavior near the origin of the Fourier ... 0answers 149 views ### The dual of the space of smooth functions that vanish at infinity Let U \subset \Bbb R ^n be an open subset and let \mathcal C be the space of the smooth functions on U that vanish at infinity, endowed with the seminorms p_\alpha (f) = \sup \limits _{x \in U} ... 0answers 85 views ### Construct a PDE solution from a net of approximations Consider P a linear partial differential operator in \Bbb R ^n. Consider some boundary condition given in the generic form C(u) = 0, that guarantees a unique solution (if any) of Pu = 0. Let ... 0answers 63 views ### Bounds on functions pullbacked via exponential map Let us assume that M is a compact Riemannian manifold (without boundary). For any point x\in M, we can pullback C^\infty(M) functions to T_x M via the exponential map, by setting$$ (\exp_x^* ...
The classical Stone-Weierstrass theorem gives a necessary and sufficient condition for a class of continuous functions on a compact to approximate a larger class of continuous functions in $C^0$ ...
let be the differential equation in dimension $n > 3$ or $n =3$ $$-\Delta u =|grau|^{2}u$$ (1) with the constraint that the vector 'u' $|u|=1$ then how could i prove that the unitary ...