Convergence Almost Everywhere of Operator Averages

Almost Everywhere Convergence of Series in L

We answer positively a question of J. Rosenblatt (1988), proving the existence of a sequence (ci) with ∑∞ i=1 |ci| = ∞, such that for every dynamical system (X,Σ, m, T ) and f ∈ L1(X), ∑∞i=1 cif(T ix) converges almost everywhere. A similar result is obtained in the real variable context.

Almost Everywhere Convergence of Riesz Means Related to Schrödinger Operator with Constant Magnetic Fields

and Applied Analysis 3 Lemma 4. For λ > 0, one has 󵄩󵄩󵄩󵄩󵄩 K δ,l,j λ f (x) 󵄩󵄩󵄩󵄩󵄩 2 2 ≤ C2 −2M(j+l) δ 2M󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩 2 2 , (19) where the constant C is independent of λ and δ. Proof. With the method similar to the proof of Lemma 4 in [9], we write h(t) = φ(t) − φ(2t) and expandm into a Taylor series around λt. Then, ?̂? δ,l,j λ (t) = ∫m δ (λ(t − 2 −(j+l) δ 2 r λ )) ĥ (r) dr = ∫m δ (λt − 2 −(j+l) δ 2 ...

Almost Everywhere Convergence of Poisson Integrals on Generalized Half-planes

Almost Everywhere Convergence of Orthogonal Expansions of Several Variables

For weighted L space on the unit sphere of R, in which the weight functions are invariant under finite reflection groups, a maximal function is introduced and used to prove the almost everywhere convergence of orthogonal expansions in h-harmonics. The result applies to various methods of summability, including the de la Vallée Poussin means and the Cesàro means. Similar results are also establi...

On Almost Everywhere Strong Convergence of Multidimensional Continued Fraction Algorithms

We describe a strategy which allows one to produce computer assisted proofs of almost everywhere strong convergence of Jacobi-Perron type algorithms in arbitrary dimension. Numerical work is carried out in dimension three to illustrate our method. To the best of our knowledge this is the rst result on almost everywhere strong convergence in dimension greater than two.