Universal function for a weighted space $$L^1_{\mu }[0,1]$$

It is shown that there exist such a function \(g\in L^1[0,1]\) and a weight function \(0<\mu (x)\le 1\) that g is universal for the weighted space \(L^1_\mu [0,1]\) with respect to signs of its Fourier–Walsh coefficients.     

Classifying First Extremal Points for a Fractional Boundary Value Problem with a Fractional Boundary Condition

Let \(n\in \mathbb {N}\), \(n\ge 2\), \(\beta >0\) fixed, and \(0<b\le \beta \). For \(n-1<\alpha \le n\), we look to classify extremal points for the fractional differential equation \(D_{0^+}^{\alpha }u+p(t) u=0\), satisfying the boundary conditions \(u^{(i)}(0)=0\), \(i=0,\ldots ,n-2\), \(D_{0^+}^\gamma u(b)=0\), where p(t) is a continuous nonnegative function on \([0,\beta ]\) which does not vanish identically on any nondegenerate compact subinterval of \([0,\beta ]\). Using the theory of Krein and Rutman, first extremal points of this boundary value problem are classified. As an application, the results are applied, along with a fixed-point theorem, to show the existence of a solution of a nonlinear fractional boundary value problem.     

Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices

For \(k,m,n\in {\mathbb {N}}\), we consider \(n^k\times n^k\) random matrices of the form

$$\begin{aligned} {\mathcal {M}}_{n,m,k}({\mathbf {y}})=\sum _{\alpha =1}^m\tau _\alpha {Y_\alpha }Y_\alpha ^T,\quad {Y}_\alpha ={\mathbf {y}}_\alpha ^{(1)}\otimes \cdots \otimes {\mathbf {y}}_\alpha ^{(k)}, \end{aligned}$$

where \(\tau _{\alpha }\), \(\alpha \in [m]\), are real numbers and \({\mathbf {y}}_\alpha ^{(j)}\), \(\alpha \in [m]\), \(j\in [k]\), are i.i.d. copies of a normalized isotropic random vector \({\mathbf {y}}\in {\mathbb {R}}^n\). For every fixed \(k\ge 1\), if the Normalized Counting Measures of \(\{\tau _{\alpha }\}_{\alpha }\) converge weakly as \(m,n\rightarrow \infty \), \(m/n^k\rightarrow c\in [0,\infty )\) and \({\mathbf {y}}\) is a good vector in the sense of Definition 1.1, then the Normalized Counting Measures of eigenvalues of \({\mathcal {M}}_{n,m,k}({\mathbf {y}})\) converge weakly in probability to a nonrandom limit found in Marchenko and Pastur (Math USSR Sb 1:457–483, 1967). For \(k=2\), we define a subclass of good vectors \({\mathbf {y}}\) for which the centered linear eigenvalue statistics \(n^{-1/2}{{\mathrm{Tr}}}\varphi ({\mathcal {M}}_{n,m,2}({\mathbf {y}}))^\circ \) converge in distribution to a Gaussian random variable, i.e., the Central Limit Theorem is valid.

     

Continued fractions and irrational rotations

Let \(\alpha \in (0, 1)\) be an irrational number with continued fraction expansion \(\alpha =[0; a_1, a_2, \ldots ]\) and let \(p_n/q_n= [0; a_1, \ldots , a_n]\) be the nth convergent to \(\alpha \). We prove a formula for \(p_nq_k-q_np_k\) \((k<n)\) in terms of a Fibonacci type sequence \(Q_n\) defined in terms of the \(a_n\) and use it to provide an exact formula for \(\{n\alpha \}\) for all n.     

One-sided invertibility of discrete operators and their applications

For \(p\in [1,\infty ]\), we establish criteria for the one-sided invertibility of binomial discrete difference operators \({{\mathcal {A}}}=aI-bV\) on the space \(l^p=l^p(\mathbb {Z})\), where \(a,b\in l^\infty \), I is the identity operator and the isometric shift operator V is given on functions \(f\in l^p\) by \((Vf)(n)=f(n+1)\) for all \(n\in \mathbb {Z}\). Applying these criteria, we obtain criteria for the one-sided invertibility of binomial functional operators \(A=aI-bU_\alpha \) on the Lebesgue space \(L^p(\mathbb {R}_+)\) for every \(p\in [1,\infty ]\), where \(a,b\in L^\infty (\mathbb {R}_+)\), \(\alpha \) is an orientation-preserving bi-Lipschitz homeomorphism of \([0,+\infty ]\) onto itself with only two fixed points 0 and \(\infty \), and \(U_\alpha \) is the isometric weighted shift operator on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f= (\alpha ^\prime )^{1/p}(f\circ \alpha )\). Applications of binomial discrete operators to interpolation theory are given.     

Transience and Recurrence of Random Walks on Percolation Clusters in an Ultrametric Space

We study transience and recurrence of simple random walks on percolation clusters in the hierarchical group of order N, which is an ultrametric space. The connection probability on the hierarchical group for two points separated by distance k is of the form \(c_k/N^{k(1+\delta )}, \delta >0\), with \(c_k=C_0+C_1\log k+C_2k^\alpha \), non-negative constants \(C_0, C_1, C_2\), and \(\alpha >0\). Percolation occurs for \(\delta <1\), and for the critical case, \(\delta =1\), \(\alpha >0\) and sufficiently large \(C_2\). We show that in the case \(\delta <1\) the walk is transient, and in the case \(\delta =1,C_2>0,\alpha >0\) there exists a critical \(\alpha _\mathrm{c}\in (0,\infty )\) such that the walk is recurrent for \(\alpha <\alpha _\mathrm{c}\) and transient for \(\alpha >\alpha _\mathrm{c}\). The proofs involve ultrametric random graphs, graph diameters, path lengths, and electric circuit theory. Some comparisons are made with behaviours of simple random walks on long-range percolation clusters in the one-dimensional Euclidean lattice.     

Complete mappings and Carlitz rank

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any \(d\ge 2\) and any prime \(p>(d^2-3d+4)^2\) there is no complete mapping polynomial in \(\mathbb {F}_p[x]\) of degree d. For arbitrary finite fields \(\mathbb {F}_q\), we give a similar result in terms of the Carlitz rank of a permutation polynomial rather than its degree. We prove that if \(n<\lfloor q/2\rfloor \), then there is no complete mapping in \(\mathbb {F}_q[x]\) of Carlitz rank n of small linearity. We also determine how far permutation polynomials f of Carlitz rank \(n<\lfloor q/2\rfloor \) are from being complete, by studying value sets of \(f+x.\) We provide examples of complete mappings if \(n=\lfloor q/2\rfloor \), which shows that the above bound cannot be improved in general.     

Convergence of powers of composition operators on certain spaces of holomorphic functions defined on the right half plane

This paper studies the asymptotic behaviour of the powers \(C_\varphi ^n\) of a composition operator \(C_\varphi \) on certain spaces of holomorphic functions defined on the right half plane \(\mathbb {C}_+\). It is shown that for composition operators on the Hardy spaces and the standard weighted Bergman spaces, if the inducing map \(\varphi \) is not of parabolic type, then either the powers \(C_\varphi ^n\) converge uniformly only to 0 or they do not converge even strongly.     

Combinatorial invariance of Kazhdan–Lusztig–Vogan polynomials for fixed point free involutions

When \(\mathrm {Sp}(2n,\mathbb {C})\) acts on the flag variety of \(\mathrm {SL}(2n,\mathbb {C})\), the orbits are in bijection with fixed point free involutions in the symmetric group \(S_{2n}\). In this case, the associated Kazhdan–Lusztig–Vogan polynomials \(P_{v,u}\) can be indexed by pairs of fixed point free involutions \(v\ge u\), where \(\ge \) denotes the Bruhat order on \(S_{2n}\). We prove that these polynomials are combinatorial invariants in the sense that if \(f:[u,w_0]\rightarrow [u',w_0]\) is a poset isomorphism of upper intervals in the Bruhat order on fixed point free involutions, then \(P_{v,u} = P_{f(v),u'}\) for all \(v\ge u\).     

Conditionally Negative Definite Functions

Let \(f:[0,\infty )\rightarrow [0,\infty )\) be an operator monotone function and \(g: \mathbb {R}\rightarrow [0,\infty )\) be a conditionally negative definite(in short cnd) function. We obtain that \(f\circ g:\mathbb {R}\rightarrow [0,\infty )\) is also conditionally negative definite. This generalizes and subsumes several existing results. A versatile direct connection between cnd functions and functions having Weierstrass factorization is established and consequently a reasonable account for cnd functions is presented.     

Two Novel Characterizations of Self-Decomposability on the Half-Line

Two novel characterizations of self-decomposable Bernstein functions are provided. The first one is purely analytic, stating that a function \(\varPsi \) is the Bernstein function of a self-decomposable probability law \(\pi \) on the positive half-axis if and only if alternating sums of \(\varPsi \) satisfy certain monotonicity conditions. The second characterization is of probabilistic nature, showing that \(\varPsi \) is a self-decomposable Bernstein function if and only if a related d-variate function \(C_{\psi ,d}\), \(\psi :=\exp (-\varPsi )\), is a d-variate copula for each \(d \ge 2\). A canonical stochastic construction is presented, in which \(\pi \) (respectively \(\varPsi \)) determines the probability law of an exchangeable sequence of random variables \(\{U_k\}_{k\in {\mathbb {N}}}\) such that \((U_1,\ldots ,U_d) \sim C_{\psi ,d}\) for each \(d \ge 2\). The random variables \(\{U_k\}_{k\in {\mathbb {N}}},\) are i.i.d. conditioned on an increasing Sato process whose law is characterized by \(\varPsi \). The probability law of \(\{U_k\}_{k \in {\mathbb {N}}}\) is studied in quite some detail.     

Regularity for Inhomogeneous Dirichlet Problems of Some Schrödinger Equations on Domains

Let \(n\ge 3, \Omega \) be a bounded, simply connected and semiconvex domain in \({\mathbb {R}}^n\) and \(L_{\Omega }:=-\Delta +V\) a Schrödinger operator on \(L^2 (\Omega )\) with the Dirichlet boundary condition, where \(\Delta \) denotes the Laplace operator and the potential \(0\le V\) belongs to the reverse Hölder class \(RH_{q_0}({\mathbb {R}}^n)\) for some \(q_0\in (\max \{n/2,2\},\infty ]\). Assume that the growth function \(\varphi :\,{\mathbb {R}}^n\times [0,\infty ) \rightarrow [0,\infty )\) satisfies that \(\varphi (x,\cdot )\) is an Orlicz function and \(\varphi (\cdot ,t)\in {\mathbb {A}}_{\infty }({\mathbb {R}}^n)\) (the class of uniformly Muckenhoupt weights). Let \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) be the Musielak–Orlicz–Hardy space whose elements are restrictions of elements of the Musielak–Orlicz–Hardy space, associated with \(L_{{\mathbb {R}}^n}:=-\Delta +V\) on \({\mathbb {R}}^n\), to \(\Omega \). In this article, the authors show that the operators \(VL^{-1}_\Omega \) and \(\nabla ^2L^{-1}_\Omega \) are bounded from \(L^1(\Omega )\) to weak-\(L^1(\Omega )\), from \(L^p(\Omega )\) to itself, with \(p\in (1,2]\), and also from \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) to the Musielak–Orlicz space \(L^\varphi (\Omega )\) or to \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) itself. As applications, the boundedness of \(\nabla ^2{\mathbb {G}}_D\) on \(L^p(\Omega )\), with \(p\in (1,2]\), and from \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) to \(L^\varphi (\Omega )\) or to \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) itself is obtained, where \({\mathbb {G}}_D\) denotes the Dirichlet Green operator associated with \(L_\Omega \). All these results are new even for the Hardy space \(H^1_{L_{{\mathbb {R}}^n},\,r}(\Omega )\), which is just \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) with \(\varphi (x,t):=t\) for all \(x\in {\mathbb {R}}^n\) and \(t\in [0,\infty )\).     

On Filtered Polynomial Approximation on the Sphere

This paper considers filtered polynomial approximations on the unit sphere \(\mathbb {S}^d\subset \mathbb {R}^{d+1}\), obtained by truncating smoothly the Fourier series of an integrable function f with the help of a “filter” h, which is a real-valued continuous function on \([0,\infty )\) such that \(h(t)=1\) for \(t\in [0,1]\) and \(h(t)=0\) for \(t\ge 2\). The resulting “filtered polynomial approximation” (a spherical polynomial of degree \(2L-1\)) is then made fully discrete by approximating the inner product integrals by an N-point cubature rule of suitably high polynomial degree of precision, giving an approximation called “filtered hyperinterpolation”. In this paper we require that the filter h and all its derivatives up to \(\lfloor \tfrac{d-1}{2}\rfloor \) are absolutely continuous, while its right and left derivatives of order \(\lfloor \tfrac{d+1}{2}\rfloor \) exist everywhere and are of bounded variation. Under this assumption we show that for a function f in the Sobolev space \(W^s_p(\mathbb {S}^d),\ 1\le p\le \infty \), both approximations are of the optimal order \( L^{-s}\), in the first case for \(s>0\) and in the second fully discrete case for \(s>d/p\), conditions which in both cases cannot be weakened.     

Monochromatic Solutions for Multi-Term Unknowns

Given integers \(k\ge 2\), \(n \ge 2\), \(m \ge 2\) and \( a_1,a_2,\ldots ,a_m \in {\mathbb {Z}}{\backslash }{\{0\}}\), and let \(f(z)= \sum _{j=0}^{n}c_jz^j\) be a polynomial of integer coefficients with \(c_n>0\) and \((\sum _{i=1}^ma_i)|f(z)\) for some integer z. For a k-coloring of \([N]=\{1,2,\ldots ,N\}\), we say that there is a monochromatic solution of the equation \(a_1x_1+a_2x_2+\cdots +a_mx_m=f(z)\) if there exist pairwise distinct \(x_1,x_2,\ldots ,x_m\in [N]\) all of the same color such that the equation holds for some \(z\in \mathbb {Z}\). Problems of this type are often referred to as Ramsey-type problems. In this paper, it is shown that if \(a_i>0\) for \(1\le i\le m\), then there exists an integer \(N_0=N(k,m,n)\) such that for \(N\ge N_0\), each k-coloring of [N] contains a monochromatic solution \(x_1,x_2,\ldots ,x_m\) of the equation \(a_1x_1+a_2x_2+ \cdots +a_mx_m= f(z)\). Moreover, if n is odd and there are \(a_i\) and \(a_j\) such that \(a_ia_j<0\) for some \(1 \le i\ne j\le m\), then the assertion holds similarly.     

Cliques in $$C_4$$-free graphs of large minimum degree

A graph G is called \(C_4\)-free if it does not contain the cycle \(C_4\) as an induced subgraph. Hubenko, Solymosi and the first author proved (answering a question of Erd?s) a peculiar property of \(C_4\)-free graphs: \(C_4\)-free graphs with n vertices and average degree at least cn contain a complete subgraph (clique) of size at least \(c'n\) (with \(c'= 0.1c^2\)). We prove here better bounds \(\big ({c^2n\over 2+c}\) in general and \((c-1/3)n\) when \( c \le 0.733\big )\) from the stronger assumption that the \(C_4\)-free graphs have minimum degree at least cn. Our main result is a theorem for regular graphs, conjectured in the paper mentioned above: 2k-regular \(C_4\)-free graphs on \(4k+1\) vertices contain a clique of size \(k+1\). This is the best possible as shown by the kth power of the cycle \(C_{4k+1}\).     

Funktionen mit Stammfunktion

The aim of this note is to analyze the class \(St[a,b]\) of all real functions on \([a,b]\) with primitive with respect to its size and its relations to other function classes. Apart from recalling previously known results and discussing new results we put a particular emphasis on examples and counterexamples.     

On Weak Invariance Principles for Partial Sums

Given a sequence of random functionals \(\bigl \{X_k(u)\bigr \}_{k \in \mathbb {Z}}\), \(u \in \mathbf{I}^d\), \(d \ge 1\), the normalized partial sums \(\check{S}_{nt}(u) = n^{-1/2}\bigl (X_1(u) + \cdots + X_{\lfloor n t \rfloor }(u)\bigr )\), \(t \in [0,1]\) and its polygonal version \({S}_{nt}(u)\) are considered under a weak dependence assumption and \(p > 2\) moments. Weak invariance principles in the space of continuous functions and càdlàg functions are established. A particular emphasis is put on the process \(\check{S}_{nt}(\widehat{\theta })\), where \(\widehat{\theta } \xrightarrow {\mathbb {P}} \theta \), and weaker moment conditions (\(p = 2\) if \(d = 1\)) are assumed.     

On a problem of Janusz Matkowski and Jacek Wesołowski

We study the problem of the existence of increasing and continuous solutions \(\varphi :[0,1]\rightarrow [0,1]\) such that \(\varphi (0)=0\) and \(\varphi (1)=1\) of the functional equation

$$\begin{aligned} \varphi (x)=\sum _{n=0}^{N}\varphi (f_n(x))-\sum _{n=1}^{N}\varphi (f_n(0)), \end{aligned}$$

where \(N\in {\mathbb {N}}\) and \(f_0,\ldots ,f_N:[0,1]\rightarrow [0,1]\) are strictly increasing contractions satisfying the following condition \(0=f_0(0)<f_0(1)=f_1(0)<\cdots<f_{N-1}(1)=f_N(0)<f_N(1)=1\). In particular, we give an answer to the problem posed in Matkowski (Aequationes Math. 29:210–213, 1985) by Janusz Matkowski concerning a very special case of that equation.

     

Fourier Spectrum Characterizations of $$H^{p}$$ Spaces on Tubes Over Cones for $$1\le p \le \infty $$1≤p≤∞

We give Fourier spectrum characterizations of functions in the Hardy \(H^p\) spaces on tubes for \(1\le p \le \infty .\) For \(F\in L^p(\mathbb {R}^n), \) we show that F is the non-tangential boundary limit of a function in a Hardy space, \(H^{p}(T_\Gamma ),\) where \(\Gamma \) is an open cone of \(\mathbb {R}^n\) and \(T_\Gamma \) is the related tube in \(\mathbb {C}^n,\) if and only if the classical or the distributional Fourier transform of F is supported in \(\Gamma ^*,\) where \(\Gamma ^*\) is the dual cone of \(\Gamma .\) This generalizes the results of Stein and Weiss for \(p=2\) in the same context, as well as those of Qian et al. in one complex variable for \(1\le p\le \infty .\) Furthermore, we extend the Poisson and Cauchy integral representation formulas to the \(H^p\) spaces on tubes for \(p\in [1, \infty ]\) and \(p\in [1,\infty ),\) with, respectively, the two types of representations.