Acta Mathematica Hungarica最新文献

An affine version of the linear subspace concentration inequality as proposed by Wu in [11] is established for centered convex bodies. This generalizes results from [11] and [8] on polytopes to convex bodies.

In this paper, we investigate the compactness of semicommutators of Toeplitz operators on Hardy spaces and Bergman spaces, focusing on the operators of the form (T^{H}_{|f|^{2}}-T^{H}_{f}T^{H}_{overline{f}}) and (T^{H}_{|tilde{f}|^{2}}-T^{H}_{tilde{f}}T^{H}_{overline{tilde{f}}} ), where (tilde{f}(z)=f(z^{-1})). We establish that the compactness of these operators can be characterized through the convergence of the sequence ({T^{H}_{n}(|f|^{2})-T^{H}_{n}(f)T^{H}_{n}(overline{f})}) in the sense of singular value clustering. This provides a method for determining the compactness of semicommutators by examining the corresponding Toeplitz matrices derived from the Fourier coefficients of the symbol functions.Furthermore, we identify the function space (VMO cap L^{infty}(mathbb{T})) as the largest (C^{*})-subalgebra of (L^{infty}(mathbb{T})) such that, for any (f, g in VMO cap L^{infty}(mathbb{T}) ), sequence ({T^{H}_{n}(fg)-T^{H}_{n}(f)T^{H}_{n}(g)}) converges in terms of singular value clustering. It is already known that ( VMO cap L^{infty}(mathbb{T})) is the largest (C^{*})-subalgebra of (L^{infty}(mathbb{T})) such that, for any (f, g in VMO cap L^{infty}(mathbb{T}) ), the operator (T^{H}_{fg}-T^{H}_{f}T^{H}_{g}) is compact. Similar considerations are made for Bergman spaces (A^{2}(mathbb{D})), where we obtain partial results. This work links operator theory, numerical linear algebra, and function spaces, providing new insights into the compactness properties of Toeplitz operators and their semicommutators.

We study the cardinality of orthogonal exponential functions in (L^{2}(mu_{{R,D}})), where (mu_{{R,D}} ) is the self-affine measure generated by an expanding real matrix ( R = {rm diag}[rho_{1},rho_{2},dots,rho_{n}] ) and a finite digit set ( Dsubsetmathbb{Z}^{n} ). Let ( m ) be a prime and ( mathcal{Z}(m_{D}) ) be the set of zeros of mask polynomial ( m_{D} ) of ( D ). Suppose (mathcal{Z}(m_{D})) can be decomposed into the union of finite (mathcal{Z} _{i}(m),) where (mathcal{Z} _{i}(m)) satisfies( (mathcal{Z} _{i}(m)-mathcal{Z} _{i}(m))backslashmathbb{Z}^{n}subsetmathcal{Z} _{i}(m)subset(m^{-1}mathbb{Z}backslash mathbb{Z})^{n} ) and ( mathcal{Z} _{i}(m)nsubseteq(m_{1}^{-1}mathbb{Z}backslash mathbb{Z})^{n} ) for all integer ( m_{1}in(0,m) ), then we show that ( L^{2}(mu_{{R,D}})) admits infinite orthogonal exponential functions if and only if ( rho_{i}=(frac{m p_{i}}{q_{i}})^{frac{1}{r_{i}}} ) for some ( r_{i},p_{i},q_{i}inmathbb{N} ) with ( gcd(p_{i},q_{i})=1 ), ( i=1,2,dots,n ). Furthermore, if ( L^{2}(mu_{{R,D}})) does not admit infinite orthogonal exponential functions, we estimate the number of orthogonal exponential functions in some cases.

The Kepler set of a sequence ((a_n)_{n=0}^infty) is the closure of the set of consecutive ratios ({a_{n+1}/a_{n} : ngeq 0}). Following several studies, dealing with Kepler sets of recurrence sequences of order 2, we study here the case of recurrences of any order.

Let ( p ) be a prime and let ( mathcal{D}={d_1, d_2, dots, d_s} ) be a subset of ( left { 1, 2, dots, p-1 right } .)If ( mathcal{F} ) is a Hamming symmetric family of subsets of ([n]) such that ( lvert F bigtriangleup F' rvert ( bmod p ) in mathcal{D} ) and ( n- lvert F bigtriangleup F' rvert ( bmod p ) in mathcal{D} ) for any pair of distinct ( F ), ( F' in mathcal{F} ), then

This result can be considered as a modular version of Hegedüs's Theorem [6] about Hamming symmetric families. We also improve the above upper bound on the size of Hamming symmetric families in the non-modular version when the size of any member of ( mathcal{F} ) is restricted.

We prove the consistency of the relation (left(begin{matrix}nu mu lambda end{matrix}right) rightarrow left(begin{matrix} nu mu lambda end{matrix}right)) when (lambda < mu = text{cf}(mu) < nu = text{cf} (nu) leq 2^{mu}).

Fix (k geq 2). For any (N geq 1), let (F_k(N)) denote the cardinality of the largest subset of ({1,dots,N}) that does not contain (k) distinct elements whose product is a square. Erdős, Sárközy, and Sós showed that (F_2(N) = (frac{6}{pi^2}+o(1)) N), (F_3(N) = (1-o(1))N), (F_k(N) asymp N/log N) for even (k geq 4), and (F_k(N) asymp N) for odd (k geq 5). Erdős then asked whether (F_k(N) = (1-o(1)) N) for odd (k geq 5). Using a probabilistic argument, we answer this question in the negative.

We introduce a new concept of Lebesgue points for higher dimensionalfunctions. Every continuity point is a Lebesgue point and almost everypoint is a Lebesgue point of an integrable function. Given a strictly increasingcontinuous function(delta), we prove that the Fejér or Cesàro means(sigma_n^{alpha}f) of the Fourierseries of a two-dimensional function (fin L_1(mathbb{T}^2)) converge to (f) at each Lebesguepoint as (nto infty) and n is in the cone around the graph of (delta). We also prove thisresult for higher dimensional functions and for other summability means. This isa generalization of the classical one-dimensional Lebesgue’s theorem for the Fejérmeans.

We prove that the diameter of a Sidon set (also known as a Babcock sequence, Golomb ruler, or (B_2) set) with (k) elements is at least (k^2-b k^{3/2}-O(k)) where (ble 1.96365), a comparatively large improvement on past results. Equivalently, a Sidon set with diameter (n) has at most (n^{1/2}+0.98183n^{1/4}+O(1)) elements. The proof is conceptually simple but very computationally intensive, and the proof uses substantial computer assistance. We also provide a proof of (ble 1.99058) that can be verified by hand, which still improves on past results. Finally, we prove that (g)-thin Sidon sets (aka (g)-Golomb rulers) with (k) elements have diameter at least (g^{-1} k^2 - (2-varepsilon)g^{-1}k^{3/2} - O(k)), with (varepsilonge 0.0062g^{-4}).