By Weber M.

This e-book provides in a concise and available manner, in addition to in a standard surroundings, numerous instruments and strategies bobbing up from spectral idea, ergodic concept and stochastic methods idea, which shape the foundation of and give a contribution interactively greatly to the present learn on almost-everywhere convergence difficulties. Researchers operating in dynamical platforms and on the crossroads of spectral idea, ergodic concept and stochastic techniques will locate the instruments, tools, and effects offered during this booklet of significant curiosity. it's written in a method obtainable to graduate scholars.

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**Sample text**

Vm (θ ) − Vn (θ )| 2 n|θ | 2π 4π Q(θ, y)dy = 1/m (2) |θ | ≤ 1 m. Then, for the same reasons 1/n |θ| dy = (m − n)|θ | 2 1/m y 4 2 ≥ |Vm (θ ) − Vn (θ )| ≥ |Vm (θ ) − Vn (θ )|2 . π π Q(θ, y)dy = 1/m (3) 1 n 1/n > |θ | ≥ 1 m. 1/n This case is obvious since we have |Vm (θ ) − Vn (θ )| ≤ 2. Let f ∈ H , with spectral measure μf . Introduce a new measure, the spectral regularization of the measure μf with respect to the kernel Q, defined by μˆ f (dy) = 4π π −π Q(θ, y)μf (dθ ) dy + 4μf (dy). 2) It is easy to verify that μˆ f ([0, 1]) ≤ 4(2π + 1)μf ([−π, π]) ≤ 4(2π + 1) f 2 .

Hence I (μf ) ≤ (q + 1)μf (−π, π] = (q + 1) f 2 . The second assertion of the proposition then follows from the previously established result. 3. 6) is optimal. One can indeed prove the following statement, providing a minoration of the same order for the Littlewood–Paley square function associated to the moving averages. 6 Theorem. Let U be a unitary operator. Let φ(·) be a regular varying function with Karamata index α ∈ (1, 2), and assume that the function φ(u)/u is increasing. Then there exists a constant c = c(φ), and a nondecreasing sequence of positive integers {np , p ≥ 1}, such that for any element f ∈ H , one has ∞ Bnφp+1 (f ) − Bnφp (f ) 2 ≥c (θ )μf (dθ ).

4 |θ|. (iii) For any integers m ≥ n ≥ 1, |Vn (θ ) − Vm (θ )| ≤ π 4 |θ | (m − n). (iv) |Vn (θ) − Vm (θ )| ≤ 2 (m − n) /m. Proof. First observe that |Vx (θ )| ≤ 2 . x|eiθ −1| As for any −π ≤ θ ≤ π , |eiθ − 1| = 2|θ | π 2| sin |θ| 2 | ≥ π , we deduce that |Vx (θ )| ≤ x|θ | , if −π ≤ θ ≤ π and x > 0. And |Vx (θ)| ≤ 1 if x is an integer. Hence (i). Now let −π ≤ θ ≤ π and put for any real x > 0, eixθ − 1 ϕθ (x) = . x Then ϕθ (x) = iθ xeixθ −eixθ +1 , x2 and noting δ(u) := |iueiu − eiu + 1|2 , we have δ(u) = (1 − u sin u − cos u)2 + (u cos u − sin u)2 = 2[1 − u sin u − cos u] + u2 .