Let $\mu _{M,D}$ be the self-similar measure generated by $M=RN^q$ and the product-form digit set $D=\{0,1,\ldots ,N-1\}\oplus N^{p_1}\{0,1,\ldots ,N-1\}\oplus \cdots \oplus N^{p_s}\{0,1,\ldots ,N-1\}$, where $R\geq 2$, $N\geq 2$, q, $p_i(1\leq i\leq s)$ are integers with $\gcd (R,N)=1$ and $1\leq p_1<p_2<\cdots <p_s<q$. In this paper, we first show that $\mu _{M,D}$ is a spectral measure with a model spectrum $\Lambda $. Then, we completely settle two types of spectral eigenvalue problems for $\mu _{M,D}$. In the first case, for a real t, we give a necessary and sufficient condition under which $t\Lambda $ is also a spectrum of $\mu _{M,D}$. In the second case, we characterize all possible real numbers t such that $\Lambda '\subset \mathbb {R}$ and $t\Lambda '$ are both spectra of $\mu _{M,D}$.

We construct two families of orthogonal polynomials associated with the universal central extensions of the superelliptic Lie algebras. These polynomials satisfy certain fourth-order linear differential equations, and one of the families is a particular collection of associated ultraspherical polynomials. We show that the generating functions of the polynomials satisfy fourth-order linear PDEs. Since these generating functions can be represented by superelliptic integrals, we have examples of linear PDEs of fourth order with explicit solutions without complete integrability.

The Moment-SOS hierarchy, first introduced in optimization in 2000, is based on the theory of the S-moment problem and its dual counterpart: polynomials that are positive on S. It turns out that this methodology can also be used to solve problems with positivity constraints ‘f(x) ≥ 0 for all $\mathbf{x}\in S$’ or linear constraints on Borel measures. Such problems can be viewed as specific instances of the generalized moment problem (GMP), whose list of important applications in various domains of science and engineering is almost endless. We describe this methodology in optimization and also in two other applications for illustration. Finally we also introduce the Christoffel function and reveal its links with the Moment-SOS hierarchy and positive polynomials.

Let $\{b_n\}_{n=1}^{\infty }$ be a sequence of integers larger than 1. We will study the harmonic analysis of the equal-weighted Moran measures $\mu _{\{b_n\},\{{\mathcal D}_n\}}$ with ${\mathcal D}_n=\{0,1,2,\ldots ,q_n-1\}$, where $q_n$ divides $b_n$ for all $n\geq 1.$ In this paper, we first characterize all the maximal orthogonal sets of $L^2(\mu _{\{b_n\},\{{\mathcal D}_n\}})$ via a tree mapping. By this characterization, we give some sufficient conditions for the maximal orthogonal set to be an orthonormal basis.

Given a Borel probability measure µ on $\mathbb{R}^n$ and a real matrix $R\in M_n(\mathbb{R})$. We call R a spectral eigenmatrix of the measure µ if there exists a countable set $\Lambda\subset \mathbb{R}^n$ such that the sets $E_\Lambda=\big\{{\rm e}^{2\pi i \langle\lambda,x\rangle}:\lambda\in \Lambda\big\}$ and $E_{R\Lambda}=\big\{{\rm e}^{2\pi i \langle R\lambda,x\rangle}:\lambda\in \Lambda\big\}$ are both orthonormal bases for the Hilbert space $L^2(\mu)$. In this paper, we study the structure of spectral eigenmatrix of the planar self-affine measure $\mu_{M,D}$ generated by an expanding integer matrix $M\in M_2(2\mathbb{Z})$ and the four-elements digit set $D = \{(0,0)^t,(1,0)^t,(0,1)^t,(-1,-1)^t\}$. Some sufficient and/or necessary conditions for R to be a spectral eigenmatrix of $\mu_{M,D}$ are given.

We consider the spectral analysis of several examples of bilateral birth–death processes and compute explicitly the spectral matrix and the corresponding orthogonal polynomials. We also use the spectral representation to study some probabilistic properties of the processes, such as recurrence, the invariant distribution (if it exists), and the probability current.

Let $\{R_{k}\}_{k=1}^{\infty }$ be a sequence of expanding integer matrices in $M_{n}(\mathbb {Z})$, and let $\{D_{k}\}_{k=1}^{\infty }$ be a sequence of finite digit sets with integer vectors in ${\mathbb Z}^{n}$. In this paper, we prove that under certain conditions in terms of $(R_{k},D_{k})$ for $k\ge 1$, the Moran measure

$$ \begin{align*} \mu_{\{R_{k}\},\{D_{k}\}}:=\delta_{R_{1}^{-1}D_{1}}\ast\delta_{R_{1}^{-1}R_{2}^{-1}D_{2}}\ast\cdots \end{align*} $$

$(R,D)$

This work is devoted to the study of uncertainty principles for finite combinations of Hermite functions. We establish some spectral inequalities for control subsets that are thick with respect to some unbounded densities growing almost linearly at infinity, and provide quantitative estimates, with respect to the energy level of the Hermite functions seen as eigenfunctions of the harmonic oscillator, for the constants appearing in these spectral estimates. These spectral inequalities allow us to derive the null-controllability in any positive time for evolution equations enjoying specific regularizing effects. More precisely, for a given index $\frac {1}{2} \leq \mu <1$, we deduce sufficient geometric conditions on control subsets to ensure the null-controllability of evolution equations enjoying regularizing effects in the symmetric Gelfand–Shilov space $S^{\mu }_{\mu }(\mathbb {R}^{n})$. These results apply in particular to derive the null-controllability in any positive time for evolution equations associated to certain classes of hypoelliptic non-self-adjoint quadratic operators, or to fractional harmonic oscillators.

$\bar {\partial } $-extension of the matrix Riemann–Hilbert method is used to study asymptotics of the polynomials $ P_n(z) $ satisfying orthogonality relations

$$ \begin{align*} \int_{-1}^1 x^lP_n(x)\frac{\rho(x)dx}{\sqrt{1-x^2}}=0, \quad l\in\{0,\ldots,n-1\}, \end{align*} $$

where $ \rho (x) $ is a positive $ m $ times continuously differentiable function on $ [-1,1] $, $ m\geq 3 $.

Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal. 158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set $\Lambda =\{0,{r}/{N}\}+2\mathbb Z$ is not necessarily a discrete subgroup of $\mathbb{R}$, and the translation factor is $2\textrm{N}$. Here r is an odd integer with $1\leq r\leq 2N-1$ such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set $\widetilde {\Lambda }=\{0,{r_1}/{N},{r_2}/{N},\ldots ,{r_q}/{N}\}+s\mathbb Z$, where s is an even integer, $q\in \mathbb {N}$, $r_i$ is an integer such that $1\leq r_i\leq sN-1,\,(r_i,N)=1$ for all i and $N\geq 2$. In this paper, we prove that the concept of NUMRA with the translation set $\widetilde {\Lambda }$ is possible only if $\widetilde {\Lambda }$ is of the form $\{0,{r}/{N}\}+s\mathbb Z$. Next we introduce $\Lambda _s$-nonuniform multiresolution analysis ($\Lambda _s$-NUMRA) for which the translation set is $\Lambda _s=\{0,{r}/{N}\}+s\mathbb Z$ and the dilation factor is $sN$, where s is an even integer. Also, we characterize the scaling functions associated with $\Lambda _s$-NUMRA and we give necessary and sufficient conditions for wavelet filters associated with $\Lambda _s$-NUMRA.

Let $M=$ diag $(\rho _1,\rho _2)\in M_{2}({\mathbb R})$ be an expanding matrix and Let $\{D_n\}_{n=1}^{\infty }$ be a sequence of digit sets with $D_n=\left \{(0, 0)^T,\,\,\,(a_n, 0 )^T, \,\,\, (0, b_n )^T \right \}$, where $a_n, b_n\in \{-1,1\}$. The associated Borel probability measure

$$ \begin{align*} \mu_{M,\{D_n\}}:=\delta_{M^{-1}D_1}\ast \delta_{M^{-2}D_2}\ast \delta_{M^{-3}D_3}\ast \cdots \end{align*} $$

$\mu _{M, \{D_n\}}$

$3\mid \rho _i$

$i=1, 2$

$a_n=b_n=1$

$n\in {\mathbb N}$

For integers $p,b\geq 2$, let $D=\{0,1,\ldots ,b-1\}$ be a set of consecutive digits. It is known that the Cantor measure $\unicode[STIX]{x1D707}_{pb,D}$ generated by the iterated function system $\{(pb)^{-1}(x+d)\}_{x\in \mathbb{R},d\in D}$ is a spectral measure with spectrum

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}(pb,S)=\bigg\{\mathop{\sum }_{j=0}^{\text{finite}}(pb)^{j}s_{j}:s_{j}\in S\bigg\},\end{eqnarray}$$

$S=pD$

$\unicode[STIX]{x1D70F}\in \mathbb{Z}$

$\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6EC}(pb,S)$

$\unicode[STIX]{x1D707}_{pb,D}$

Let $\{M_{n}\}_{n=1}^{\infty }$ be a sequence of expanding matrices with $M_{n}=\operatorname{diag}(p_{n},q_{n})$, and let $\{{\mathcal{D}}_{n}\}_{n=1}^{\infty }$ be a sequence of digit sets with ${\mathcal{D}}_{n}=\{(0,0)^{t},(a_{n},0)^{t},(0,b_{n})^{t},\pm (a_{n},b_{n})^{t}\}$, where $p_{n}$, $q_{n}$, $a_{n}$ and $b_{n}$ are positive integers for all $n\geqslant 1$. If $\sup _{n\geqslant 1}\{\frac{a_{n}}{p_{n}},\frac{b_{n}}{q_{n}}\}<\infty$, then the infinite convolution $\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}}=\unicode[STIX]{x1D6FF}_{M_{1}^{-1}{\mathcal{D}}_{1}}\ast \unicode[STIX]{x1D6FF}_{(M_{1}M_{2})^{-1}{\mathcal{D}}_{2}}\ast \cdots \,$ is a Borel probability measure (Cantor–Dust–Moran measure). In this paper, we investigate whenever there exists a discrete set $\unicode[STIX]{x1D6EC}$ such that $\{e^{2\unicode[STIX]{x1D70B}i\langle \unicode[STIX]{x1D706},x\rangle }:\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}\}$ is an orthonormal basis for $L^{2}(\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}})$.

The main result of this note implies that any function from the product of several vector spaces to a vector space can be uniquely decomposed into the sum of mutually orthogonal functions that are odd in some of the arguments and even in the other arguments. Probabilistic notions and facts are employed to simplify statements and proofs.

Suppose that $0<|\unicode[STIX]{x1D70C}|<1$ and $m\geqslant 2$ is an integer. Let $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}$ be the self-similar measure defined by $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\cdot )=\frac{1}{m}\sum _{j=0}^{m-1}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\unicode[STIX]{x1D70C}^{-1}(\cdot )-j)$. Assume that $\unicode[STIX]{x1D70C}=\pm (q/p)^{1/r}$ for some $p,q,r\in \mathbb{N}^{+}$ with $(p,q)=1$ and $(p,m)=1$. We prove that if $(q,m)=1$, then there are at most $m$ mutually orthogonal exponential functions in $L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$ and $m$ is the best possible. If $(q,m)>1$, then there are any number of orthogonal exponential functions in $L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$.

We study the heat semigroup maximal operator associated with a well-known orthonormal system in the $d$-dimensional ball. The corresponding heat kernel is shown to satisfy Gaussian bounds. As a consequence, we can prove weighted $L^{p}$ estimates, as well as some weighted inequalities in mixed norm spaces, for this maximal operator.

We consider upper‒lower (UL) (and lower‒upper (LU)) factorizations of the one-step transition probability matrix of a random walk with the state space of nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We provide conditions on the free parameter of the UL factorization in terms of certain continued fractions such that this stochastic factorization is possible. By inverting the order of the factors (also known as a Darboux transformation) we obtain a new family of random walks where it is possible to state the spectral measures in terms of a Geronimus transformation. We repeat this for the LU factorization but without a free parameter. Finally, we apply our results in two examples; the random walk with constant transition probabilities, and the random walk generated by the Jacobi orthogonal polynomials. In both situations we obtain urn models associated with all the random walks in question.

In this paper we consider the algorithm for recovering sparse orthogonal polynomials using stochastic collocation via ℓq minimization. The main results include: 1) By using the norm inequality between ℓq and ℓ 2 and the square root lifting inequality, we present several theoretical estimates regarding the recoverability for both sparse and non-sparse signals via ℓq minimization; 2) We then combine this method with the stochastic collocation to identify the coefficients of sparse orthogonal polynomial expansions, stemming from the field of uncertainty quantification. We obtain recoverability results for both sparse polynomial functions and general non-sparse functions. We also present various numerical experiments to show the performance of the ℓq algorithm. We first present some benchmark tests to demonstrate the ability of ℓq minimization to recover exactly sparse signals, and then consider three classical analytical functions to show the advantage of this method over the standard ℓ 1 and reweighted ℓ 1 minimization. All the numerical results indicate that the ℓq method performs better than standard ℓ 1 and reweighted ℓ 1 minimization.

In this work type II Hermite-Padé approximants for a vector of Cauchy transforms of smooth Jacobi-type densities are considered. It is assumed that densities are supported on mutually disjoint intervals (an Angelesco system with complex weights). The formulae of strong asymptotics are derived for any ray sequence of multi-indices.

Let μ λ be the Bernoulli convolution associated with λ ∈ (0, 1). The well-known result of Jorgensen and Pedersen shows that if λ = 1/(2k) for some k ∈ ℕ, then μ 1/(2k) is a spectral measure with spectrum Γ(1/(2k)). The recent research on the spectrality of μ λ shows that μ λ is a spectral measure only if λ = 1/(2k) for some k ∈ ℕ. Moreover, for certain odd integer p, the multiple set pΓ(1/(2k)) is also a spectrum for μ 1/(2k). This is surprising because some spectra for the measure μ 1/(2k) are thinning. In this paper we mainly characterize the number p that has the above property. By applying the properties of congruences and the order of elements in the finite group, we obtain several conditions on p such that pΓ(1/(2k)) is a spectrum for μ 1/(2k).