Showing papers on "Singular measure published in 2017"

Fourier Series for Singular Measures

Abstract: Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure μ on [ 0 , 1 ) , every f ∈ L 2 ( μ ) possesses a Fourier series of the form f ( x ) = ∑ n = 0 ∞ c n e 2 π i n x . We show that the coefficients c n can be computed in terms of the quantities f ^ ( n ) = ∫ 0 1 f ( x ) e − 2 π i n x d μ ( x ) . We also demonstrate a Shannon-type sampling theorem for functions that are in a sense μ -bandlimited.

Weak embedding property, inner functions and entropy

Abstract: Following Gorkin, Mortini, and Nikolski, we say that an inner function I in \(H^\infty (\mathbb {D})\) has the WEP property if its modulus at a point z is bounded from below by a function of the distance from z to the zero set of I. This is equivalent to a number of properties, and we establish some consequences of this for \(H^\infty /IH^\infty \). The bulk of the paper is devoted to wepable functions, i.e. those inner functions which can be made WEP after multiplication by a suitable Blaschke product. We prove that a closed subset E of the unit circle is of finite entropy (i.e. is a Beurling–Carleson set) if and only if any singular measure supported on E gives rise to a wepable singular inner function. As a corollary, we see that singular measures which spread their mass too evenly cannot give rise to wepable singular inner functions. Furthermore, we prove that the stronger property of porosity of E is equivalent to a stronger form of wepability (easy wepability) for the singular inner functions with support in E. Finally, we find out the critical decay rate of masses of atomic measures (with no restrictions on support) guaranteeing that the corresponding singular inner functions are easily wepable.

Singular measure traveling waves in an epidemiological model with continuous phenotypes

Abstract: We consider the reaction-diffusion equation \begin{equation*} u_t=u_{xx}+\mu\left(\int_\Omega M(y,z)u(t,x,z)dz-u\right) + u\left(a(y)-\int_\Omega K(y,z) u(t,x,z)dz\right) , \end{equation*} where $ u=u(t,x,y) $ stands for the density of a theoretical population with a spatial ($x\in\mathbb R$) and phenotypic ($y\in\Omega\subset \mathbb R^n$) structure, $ M(y,z) $ is a mutation kernel acting on the phenotypic space, $ a(y) $ is a fitness function and $ K(y,z) $ is a competition kernel. Using a vanishing viscosity method, we construct measure-valued traveling waves for this equation, and present particular cases where singular traveling waves do exist. We determine that the speed of the constructed traveling waves is the expected spreading speed $ c^*:=2\sqrt{-\lambda_1} $, where $ \lambda_1 $ is the principal eigenvalue of the linearized equation. As far as we know, this is the first construction of a measure-valued traveling wave for a reaction-diffusion equation.

Weak type (1,1) estimates for inverses of discrete rough singular integral operators

Abstract: We obtain weak type (1,1) estimates for the inverses of truncated discrete rough Hilbert transform. We include an ex- ample showing that our result is sharp. One of the ingredients of the proof are regularity estimates for convolution of singular measure associated with the sequence $[m^\alpha]$.

Lusin type theorems for Radon measures

Abstract: We add to the literature the following observation. If µ is a singular measure on the real line which assigns measure zero to every porous set and f : R ! R is a Lipschitz function which is non-differentiable µ-a.e. then for every C 1 function g : R ! R there holds µ{x 2 R : f(x) = g(x)} = 0. In other words the Lusin type approximation property of Lipschitz functions with C 1 functions does not hold with respect to a general Radon measure. Moreover we discuss, only in the one-dimensional setting, how the result contained in (Alberti, A Lusin type theorem for gradients, J. Funct. Anal., 100 (1991)) could be extended and improved when the Lebesgue measure is replaced by an arbitrary Radon measure.

A covering theorem for singular measures in the Euclidean space

Abstract: We prove that for any singular measure $\mu$ on $\mathbb{R}^n$ it is possible to cover $\mu$-almost every point with $n$ families of Lipschitz slabs of arbitrarily small total width More precisely, up to a rotation, for every $\delta>0$ there are $n$ countable families of $1$-Lipschitz functions $\{f_i^1\}_{i\in\mathbb{N}},\ldots, \{f_i^n\}_{i\in\mathbb{N}},$ $f_i^j:\{x_j=0\}\subset\mathbb{R}^n\to\mathbb{R}$, and $n$ sequences of positive real numbers $\{\varepsilon_i^1\}_{i\in\mathbb{N}},\ldots, \{\varepsilon_i^n\}_{i\in\mathbb{N}}$ such that, denoting $\hat x_j$ the orthogonal projection of the point $x$ onto $\{x_j=0\}$ and $$I_i^j:=\{x=(x_1,\ldots,x_n)\in \mathbb{R}^n:f_i^j(\hat x_j)-\varepsilon_i^j< x_j< f_i^j(\hat x_j)+\varepsilon_i^j\},$$ it holds $\sum_{i,j}\varepsilon_i^j\leq \delta$ and $\mu(\mathbb{R}^n\setminus\bigcup_{i,j}I_i^j)=0$ We apply this result to show that, if $\mu$ is not absolutely continuous, it is possible to approximate the identity with a sequence $g_h$ of smooth equi-Lipschitz maps satisfying $$\limsup_{h\to\infty}\int_{\mathbb{R}^n}{\rm{det}}( abla g_h) d\mu<\mu(\mathbb{R}^n)$$ From this, we deduce a simple proof of the fact that every top-dimensional Ambrosio-Kirchheim metric current in $\mathbb{R}^n$ is a Federer-Fleming flat chain