fa.functional analysis - Convolution in $\ell_p$ when $0<p<1$

BackgroundIt is known that given real sequences $a = (a_n)_{n \in \mathbb Z} \in \ell_p$ and $b = (b_n)_{n \in \mathbb Z} \in \ell_q$, their convolution defined as$$ a * b (n) = \sum_{k \in \mathbb Z} a_{n-k} b_k $$is in $\ell_r$ if $1 \le p, q < \infty$ and $\frac 1 r = \frac 1 p + \frac 1 q -1 $.QuestionWhat happens when $0 < p, q < 1$? Obviously, since $a$ and $b$ are in $\ell_1$, their convolution $a * b$ is in $\ell_1$. Can we say better, i.e. $a*b$ is in $\ell_r$ for some $r < 1$?...Read more

fa.functional analysis - About the critical points of quasi-convex functions

What do we know about the structure of critical points of quasi-convex functions? I am looking for statements like "the critical points of a quasi-convex function are always either a global minima or a saddle point" or "for a quasi-convex function the number of negative eigenvalues of its Hessian at a critical point is constant across intervals of values of the function evaluated at the critical points" Are statements like this known?...Read more

fa.functional analysis - solution uniqueness of non-linear Fredholm equations

the equation is $F(x)=G\left(\int k(x,y)f(y)dy\right)$ $(*)$where $f(x)=\frac{dF(x)}{dx}$ is unknown and it's required to be non-negative. With integration by parts we'll have the form of a non-linear Fredholm equation. In addition, k(x,y)>0 is square integrable, $k_x=\frac{dk(x,y)}{dx}<0$, $k_y=\frac{dk(x,y)}{dy}<0$; And $G$ is a differentiable, weakly increasing function ranging in $[0,1]$. I can prove uniqueness for the case when $k_{xy}>0$ and $G$ is convex (or conversely $k_{xy}<0$ while $G$ is concave). Yet I believe the resu...Read more

fa.functional analysis - Relationship between largest eigenvalue of a positive matrix $A$ and $A∘A^T$

I'm wondering whether there is certain relationship between the largest eigenvalue of a positive matrix(every element is positive, not neccesarily positive definite) $A$, $\rho(A)$ and that of $A∘A^T$, $\rho( A∘A^T)$, where $∘$ denotes hadamard product.Here's a result I find for many numerical cases. I create a matrix of size $n$ whose elements are uniformly drawn from $[0,M]$, as $n$ gets large (>20), $\rho(A)\rightarrow 2M\rho( A∘A^T)$.I've read some papers on the bound of eigenvalue of $A∘B$, yet none of them mention the special case of $A∘A...Read more

fa.functional analysis - Embeddings of Sobolev-Lorentz space

The classical Sobolev embedding theorem asserts that, under suitable conditions on the exponents $s,p$ and $n$, the Sobolev space $W^{s,p}(\mathbb{R}^n)$ embeds into an Holder space $C^{r,\alpha}(\mathbb{R}^n)$. Suppose now to work with the more general Sobolev-Lorentz space $$W^{s,(p,q)}(\mathbb{R}^n):=\{f\in L^{p,q}(\mathbb{R}^n)\,s.t.\nabla^sf\in L^{p,q}(\mathbb{R}^n)\}$$I wonder if is it possible to have (when $q<p$) some local logarithmic refinement of the Holder estimate, namely$$f\in W^{s,(p,q)}(\mathbb{R}^n)\Rightarrow \big(D^{\mu}f(...Read more

fa.functional analysis - Sets in constructive mathematics

It is not completely clear how Bridges, Richman and Youchuan treated sets in their paper. Example is in the following lemma (Lemma 7 on p. 7): Let $U$ and $V$ be (inhabited to mean $\exists u \in U, \exists v \in V$) sets of a Banach space such that $U \cup V$ is dense. Then, the following holds: If $u_0 \in U$ and $v_0 \in V$, then $\rho([u_0, v_0], \bar U \cap \bar V) = 0$ $\rho(x, \bar U \cap \bar V) = \rho(x, U) \wedge \rho(x, V)$ ... To do this, choose $w$ in $U \cup V$ within ...If a set $X$ is inhabited, then there is (at...Read more

fa.functional analysis - Estimating norms of derivatives

Let $(M,g)$ be a closed, compact Riemannian manifold. Let $u$ be a smooth function. Let $H^{-k}(M)$,, $k$ is a positive integer, be the dual Hilbert space of $H^{k}(M)$. Does it follow that $|| |\nabla^l u| ||^{-k}$, where $||\circ||^{-k}$ is the norm on $H^{-k}(M)$ and $l \in \Bbb{Z}_+$, can be bounded by $|| u ||^{l-k}$, where $||\circ||^{l-k}$ is the norm on $H^{l-k}(M)$?...Read more

fa.functional analysis - Characterization of the Laplace Transform

One of the main properties of the Laplace transform is given by the convolution theorem.$$\mathcal{L}(f*g)=\mathcal{L}(f)\cdot\mathcal{L}(g)$$Question: Is there a full characterization of the Laplace transform based on this property? I have in mind a theorem that reads: "Let $\mathcal{N}$ be an operator with the convolution property (given above). If $\mathcal{N}$ also satisfies properties A,B,C then $\mathcal{N}=\mathcal{L}$."...Read more

fa.functional analysis - Adjoint/transpose of wavelet transform

I'm using a wavelet transform in Matlab, so I think of it as a black-box. I'll represent it here as $W(x)$. There's a reconstruction function as well, which I'll write as $W^\dagger(y)$. I can decompose my image $x$ into coefficients and then get perfect reconstruction: $x = W^\dagger(W(x)) $. I'm using biorthogonal wavelets, but the only key thing (I think) is that they are not orthogonal, so $W$ is not a square matrix.My question: how can I get $W^T$, i.e. the adjoint of $W$? And I need it with a fast transform, similar to the speed of ...Read more

fa.functional analysis - Do we have that $\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}$ and $\| u \|_{W^{2,p}(\Bbb R^d)}$ are equivalent norms?

Is it possible that $$\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}~~~~and~~\| u \|_{W^{2,p}(\Bbb R^d)}$$ are equivalent norms?This results is pretty easy and straightforward for $p=2$ using techniques via Fourier transform and Plancherel.But what could we use in place of Fourier transform when $p\neq 2?$ Please prove or disprove or provide me with some good reference where I can fine.I fact I need to show that Domain of the generator of the Gauss-Weierstrass semi-group in $L^p(\Bbb R^d)$ is $W^{2,p}(\Bbb R^d)$. this result is count...Read more

fa.functional analysis - On the Nyman-Beurling equivalent form for RH

Now, I am new to functional analysis. So please dont be harsh.I was going through some papers by Balazard & Saias, Baez-Duarte, etc. that discussed and delved deep into details of approximating the characteristic function $\chi(0,1]$.Now, what appears to me is, if we can construct a function, using functions of the form $f(x) = \sum c_k \rho(\frac{\theta_k}{x})$ where $\rho(x)$ is the fractional part of $x$ that stay constant in the interval (0,1] and satisfying $\sum c_k \theta_k = 0$, then we have a proof of RH. But then I saw that they h...Read more

fa.functional analysis - Normalized tight frame that is not orthonormal

Does anybody know an example of a normalized tight frame (wavelet frame) that is not an orthonormal frame of $L^2( \mathbb{R})$?So in other words $\{\psi_{j,k}(x):=2^{j/2}\,\psi(2^j\,x-k)\}_{j,k \in \mathbb{Z}}$ is supposed to be a frame satisfying $||f||^2_2 = \sum_{j,k} |\langle f , \psi_{j,k} \rangle|^2 , $ but the $\psi_{j,k}$ shall not form an ONB. I am looking for such an example in the literature, but so far I did not find one....Read more

fa.functional analysis - Necessity of Lebegue's convergence criterion?

the well-known Lebesgue’s dominated convergence theorem states that pointwise convergence of a sequence of functions implies convergence of the sequence of integrals if an integrable function dominating the sequence of functions almost everywhere can be found. My question: is the existence of such a dominating function also a necessary condition for the convergence of integrals? Or can one think of an example where the sequence of integrals does converge to the expected limit, but no dominating function can be found for the sequence of function...Read more

fa.functional analysis - Alternative representations of Sobolev space

Is there a way to represent a Sobolev space as the image of a fractional integral operator over an $L^p$ Lebesgue space? Yes, as it was comment, there is an answer for that in the book "Singular Integrals and Differentiability Properties of Functions" by Elias M. Stein (Princeton, 1970), Section V.3, pp. 130ff.But, what i really want to know is if there exist an operator $D_k$ such that$W^{k,p}=\{f\in L^p: D_k f \in L^p\}$. And what will really help me if this operator can be defined for any real (positive) number $k$, in order to extend the de...Read more