The main result of this paper is a proof that, for any f ∈ L 1 [a, b], a sequence of its orthogonal projections (P ∆n (f )) onto splines of order k with arbitrary knots ∆ n , converges almost everywhere provided that the mesh diameter |∆ n | tends to zero, namelyThis extends the earlier result that, for f ∈ L p , we have convergence P ∆n (f ) → f in the L p -norm for 1 ≤ p ≤ ∞, where we interpret L ∞ as the space of continuous functions.
Abstract. Given any natural number k and any dense point sequence (t n ), we prove that the corresponding orthonormal spline system of order k is an unconditional basis in reflexive L p .
Given a normalized Orlicz function M we provide an easy formula for a distribution such that, if X is a random variable distributed accordingly and X 1 , . . . , Xn are independent copies of X, then 1 Cpwhere Cp is a positive constant depending only on p. In case p = 2 we need the function t → tM ′ (t)− M (t) to be 2-concave and as an application immediately obtain an embedding of the corresponding Orlicz spaces into L 1 [0, 1]. We also provide a general result replacing the ℓp-norm by an arbitrary N -norm. This complements some deep results obtained by Gordon, Litvak, Schütt, and Werner in [8]. We also prove a result in the spirit of [8] which is of a simpler form and easier to apply. All results are true in the more general setting of Musielak-Orlicz spaces.
We prove uniform estimates for the expected value of averages of order statistics of matrices in terms of their largest entries. As an application, we obtain similar probabilistic estimates for p norms via real interpolation.
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