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In this paper, we define the entropy number in probabilistic setting and determine the exact order of entropy number of finite-dimensional space in probabilistic setting. Moreover, we also estimate the sharp order of entropy number of univariate Sobolev space in probabilistic setting by discretization method.

In this paper, we prove a new generalized Mikhlin multiplier theorem whose conditions are given with respect to fractional derivatives in integral forms with two different integration intervals. We also discuss the connection between fractional derivatives and stable processes and prove a version of Mikhlin theorem under a condition given in terms of the infinitesimal generator of symmetric stable process. The classical Mikhlin theorem is shown to be a corollary of this new generalized version in this paper.

We present a proof of the weighted estimate of the Bergman projection that does not use extrapolation results. This estimate is extended to product domains using an adapted definition of Békollé-Bonami weights in this setting. An application to bounded Toeplitz products is also given.

The stochastic Fourier transform, or SFT for short, is an application that transforms a square integrable random function f(t, ω) to a random function defined by the following series; T ∈ , φ f ( t , w ) : = ∑ n ∈ n f n ( ω ) φ n ( t ) Where { ∈ n } is an l 2 -sequence such that ∈ n ≠ 0 , ∀ n and f ^ n is the SFC (short for “stochastic Fourier coefficient”) defined by f ^ n ( ω ) = ∫ 0 1 f ( t , ω ) φ n ( t ) ¯ d W t , a stochastic 0 integral with respect to Brownian motion Wt. We have been concerned with the question of invertibility of the SFT and shown affirmative answers with concrete schemes for the inversion. In the present note we aim to study the case of a special SFT called “natural SFT” and show some of its basic properties. This is a follow-up of the preceding article (Ogawa, S.,“A direct inversion formula for SFT”, Sankhya-A 77-1 (2015)).

Let $\\scr{A}$ denote the reduced amalgamated free product of a family A₁, A₂,..., ${\\rm A}_{n}$ of von Neumann algebras over a von Neumann subalgebra $\\scr{B}$ with respect to normal faithful conditional expectations ${\\rm E}_{k}\\colon {\\rm A}_{k}\\rightarrow \\scr{B}$. We investigate the norm in $L_{p}(\\scr{A})$ of homogeneous polynomials of a given degree d. We first generalize Voiculescu's inequality to arbitrary degree d ≥ 1 and indices 1 ≤ p ≤ ∞. This can be regarded as a free analogue of the classical Rosenthal inequality. Our second result is a length-reduction formula from which we generalize recent results of Pisier, Ricard and the authors. All constants in our estimates are independent of n so that we may consider infinitely many free factors. As applications, we study square functions of free martingales. More precisely, we show that, in contrast with the Khintchine and Rosenthal inequalities, the free analogue of the Burkholder-Gundy inequalities does not hold in $L_{\\infty}(\\scr{A})$. At the end of the paper we also consider Khintchine type inequalities for Shlyakhtenko's generalized circular systems.

Suppose that both you and your friend toss an unfair coin n times, for which the probability of heads is equal to α. What is the probability that you obtain at least d more heads than your friend if you make r additional tosses? We obtain asymptotic and monotonicity/convexity properties for this competing probability as a function of n, and demonstrate surprising phase transition phenomenon as the parameters d, r, and α vary. Our main tools are integral representations based on Fourier analysis.

Recent results concerning the convergence almost everywhere or divergence everywhere of Dirichlet series Σ a^sub n^n^sup it^ appeared in the literature, revealing significant differences with the case of trigonometric series Σ^a^sub n^e^sup int^. In this work, we prove in several cases the optimality of these results. We also discuss the statistical effect of a change of signs, by considering Σ±a^sub n^n^sup it^. According to the way (probabilistic or topological) this change of signs is made, the properties of the resulting series are quite different, and can also be applied to the theory of power series. [PUBLICATION ABSTRACT]

The paper is related to the following question of P. L. Ul'yanov. Is it true that for any 2[pi]-periodic continuous function f there is a uniformly convergent rearrangement of its trigonometric Fourier series? In particular, we give an affirmative answer if the absolute values of Fourier coefficients of f decrease. Also, we study how to choose m terms of a trigonometric polynomial of degree n to make the uniform norm of their sum as small as possible. [PUBLICATION ABSTRACT]

Consider a mean zero random variable X, and an independent sequence (Xn) distributed like X. We show that the random Fourier series ∑n≥ 1n-1Xnexp(2iπ nt) converges uniformly almost surely if and only if$E(|X|\\log\\log(\\max(e^e, |X|))) < \\infty$.

In this article it is shown that many classical results concerning absolute summability of Fourier series can be obtained by random methods. An estimate on the modulus of continuity of random Fourier series is obtained. This estimate is then applied to obtain some new sufficient conditions for the absolute summability of the Fourier series.