Let Um be an m×m Haar unitary matrix and U[m,n] be its n×n truncation. In this paper the large deviation is proven for the empirical eigenvalue density of U[m,n] as m/n → λ and n → ∞. The rate function and the limit distribution are given explicitely. U[m,n] is the random matrix model of quq, where u is a Haar unitary in a finite von Neumann algebra, q is a certain projection and they are free....

When studying “random” operators it is essential to be able to integrate over the Haar measure, both analytically and algorithmically. Unitary t-designs provide a method to simplify integrating polynomials of degree less than t over U(d). In particular, by replacing averages over the Haar measure by averages over a finite set, they allow applications in algorithms. We provide three equivalent d...

Frames which can be generated by the action of some operators (e.g. translation, dilation, modulation, ...) on a single element $f$ in a Hilbert space, called coherent frames. In this paper, we introduce a class of continuous frames in a Hilbert space $mathcal{H}$ which is indexed by some locally compact group $G$, equipped with its left Haar measure. These frames are obtained as the orbits of ...

Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U∗ nBnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the St...

The following question is proposed in [4, Question 1.20]: Let $G$ be a compact group, and suppose that $$\mathcal{N}_k(G) = \{(x1,\dots,x_{k+1}) \in G^{k+1} \;\|; [x_1,\dots, x_{k+1}] 1\}$$ has positive Haar measure $G^{k+1}$. Does have an open $k$-step nilpotent subgroup? case $k 1$ already known. We positively answer it for 2$.

Let X be a hypergroup. In this paper, we define a locally convex topology β on L(X) such that (L(X), β) with the strong topology can be identified with a Banach subspace of L(X). We prove that if X has a Haar measure, then the dual to this subspace is LC(X) ∗∗ = cl{F ∈ L(X);F has compact carrier}. Moreover, we study the operators on L(X) and L 0 (X) which commute with translations and convoluti...

this paper presents a rational haar wavelet operational method for solving the inverse laplace transform problemand improves inherent errors from irrational haar wavelet. the approach is thus straightforward, rather simple andsuitable for computer programming. we define that $p$ is the operational matrix for integration of the orthogonalhaar wavelet. simultaneously, simplify the formulaes of li...

The Minkowski Question Mark function relates the continued-fraction representation of the real numbers, to their binary expansion. This function is peculiar in many ways; one is that its derivative is ’singular’. One can show by classical techniques that its derivative must vanish on all rationals. Since the Question Mark itself is continuous, one concludes that the derivative must be non-zero ...