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For the functions from sets $C_\beta^\psi C$ and $C_\beta^\psi L_s, \ 1\leq s\leq\infty$, generated by sequences $\psi(k)>0$ satisfying the condition d'Alembert $\mathop {\rm \lim}\limits_{k\rightarrow\infty}\frac{\psi(k+1)}{\psi(k)}=q, \ q\in[0,1)$, asymptotically unimprovable estimates for deviations of de la Vall\'{e}e Poussin sums in the uniform metric, which are represented in terms of values of the best approximations of $(\psi,\beta)$-differentiable functions of this sort by trigonometric polynomials in the metrics $L_s$ are obtained. Proved that received estimates are unimprovable on some important functional subsets.

In this paper, we establish separate necessary and sufficient John-Nirenberg (JN) type inequalities for functions in $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$ which imply Gagliardo-Nirenberg (GN) type inequalities in $Q_{\alpha}(\mathbb{R}^{n}).$ Consequently, we obtain Trudinger-Moser type inequalities and Brezis-Gallouet-Wainger type inequalities in $Q_{\alpha}(\mathbb{R}^{n}).$

Borwein and Erd\'elyi proved a Bernstein type inequality for rational functions on the unit circle and on the real line. Here we establish asymptotically sharp extensions of their inequalities for rational functions on analytic Jordan arcs and curves. In the proofs key roles are played by Borwein-Erd\'elyi inequality on the unit circle, Gonchar-Grigorjan type estimate of the norm of holomorphic part of meromorphic functions and Totik's construction of fast decreasing polynomials.

We prove a version of the doubling Bernstein inequalities for the trace of an analytic function of two variables on an analytic subset of $\mathbb{C}^2$. The estimate applies to the whole analytic set in question including its singular points. The proof relies on a version of the Cartan estimate for maps in $\mathbb{C}^2$ which we establish in this work.

In a previous article the author extended the Witten deformation to singular spaces with cone-like singularities and to a class of Morse functions called admissible Morse functions. The method applies in particular to complex cones and stratified Morse functions in the sense of the theory developed by Goresky and MacPherson. It is well-known from stratified Morse theory that the singular points of the complex cone contribute to the stratified Morse inequalities in middle degree only. In this article an analytic proof of this fact is given.

In an earlier work made by the first author with J. Turi (Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators, AMO, 2008), the solution of a stochastic variational inequality modeling an elasto-perfectly-plastic oscillator has been studied. The existence and uniqueness of an invariant measure have been proven. Nonlocal problems have been introduced in this context. In this work, we present a new characterization of the invariant measure. The key finding is the connection between nonlocal PDEs and local PDEs which can be interpreted with short cycles of the Markov process solution of the stochastic variational inequality.

It is well known that the cohomology groups of a closed manifold $M$ can be reconstructed using the gradient dynamical of a Morse-Smale function $f\colon M\to \R$. A direct result of this construction are Morse inequalities that provide lower bounds for the number of critical points of $f$ in term of Betti numbers of $M$. These inequalities can be deduced through a purely analytic method by studying the asymptotic behaviour of the deformed Laplacian operator. This method was introduced by E. Witten and has inspired a numbers of great achievements in Geometry and Topology in few past decades. In this paper, adopting the Witten approach, we provide an analytic proof for; the so called; equivariant Morse inequalities when the underlying manifold is acted on by the Lie group $G=S^1$ and the Morse function $f$ is invariant with respect to this action.

It is proved an inequality - integrated analogue of the Hardy inequality and as application simplified proof of the theorem of S. A. Vinogradov for the bounded Toeplitz operators on the space of functions analytic and bounded in the unit disc is given.

Let T : Lp --> Lp be a positive contraction, with p strictly between 1 and infinity. Assume that T is analytic, that is, there exists a constant K such that \norm{T^n-T^{n-1}} < K/n for any positive integer n. Let q strictly betweeen 2 and infinity and let v^q be the space of all complex sequences with a finite strong q-variation. We show that for any x in Lp, the sequence ([T^n(x)](\lambda))_{n\geq 0} belongs to v^q for almost every \lambda, with an estimate \norm{(T^n(x))_{n\geq 0}}_{Lp(v^q)}\leq C\norm{x}_p. If we remove the analyticity assumption, we obtain a similar estimate for the ergodic averages of T instead of the powers of T. We also obtain similar results for strongly continuous semigroups of positive contractions on Lp-spaces.

Self-testing refers to the phenomenon that certain extremal quantum correlations (almost) uniquely identify the quantum system under consideration. For instance observing the maximal violation of the CHSH inequality certifies that the two parties share a singlet. While self-testing results are known for several classes of states, in many cases they are only applicable if the observed statistics are almost perfect, which makes them unsuitable for practical applications. Practically relevant self-testing bounds are much less common and moreover they all result from a single numerical method (with one exception which we discuss in detail). In this work we present a new technique for proving analytic self-testing bounds of practically relevant robustness. We obtain improved bounds for the case of self-testing the singlet using the CHSH inequality (in particular we show that non-trivial fidelity with the singlet can be achieved as long as the violation exceeds $\beta^{*} = (16 + 14 \sqrt{2})/17 \approx 2.11$). In case of self-testing the tripartite GHZ state using the Mermin inequality we derive a bound which not only improves on previously known results but turns out to be tight. We discuss other scenarios to which our technique can be immediately applied.

Self-testing refers to the phenomenon that certain extremal quantum correlations (almost) uniquely identify the quantum system under consideration. For instance observing the maximal violation of the CHSH inequality certifies that the two parties share a singlet. While self-testing results are known for several classes of states, in many cases they are only applicable if the observed statistics are almost perfect, which makes them unsuitable for practical applications. Practically relevant self-testing bounds are much less common and moreover they all result from a single numerical method (with one exception which we discuss in detail). In this work we present a new technique for proving analytic self-testing bounds of practically relevant robustness. We obtain improved bounds for the case of self-testing the singlet using the CHSH inequality (in particular we show that non-trivial fidelity with the singlet can be achieved as long as the violation exceeds $\beta^{*} = (16 + 14 \sqrt{2})/17 \approx 2.11$). In case of self-testing the tripartite GHZ state using the Mermin inequality we derive a bound which not only improves on previously known results but turns out to be tight. We discuss other scenarios to which our technique can be immediately applied.

"Investigation of a Class of Fundamental Inequalities in the Theory of Analytic Functions" is an article from The Annals of Mathematics, Volume 21 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2007132

We give various equivalent formulations to the (partially) open problem about $L^p$-boundedness of Bergman projections in tubes over cones. Namely, we show that such boundedness is equivalent to the duality identity between Bergman spaces, $A^{p'}=(A^p)^*$, and also to a Hardy type inequality related to the wave operator. We introduce analytic Besov spaces in tubes over cones, for which such Hardy inequalities play an important role. For $p\geq 2$ we identify as a Besov space the range of the Bergman projection acting on $L^p$, and also the dual of $A^{p'}$. For the Bloch space $\SB^\infty$ we give in addition new necessary conditions on the number of derivatives required in its definition.

We give various equivalent formulations to the (partially) open problem about $L^p$-boundedness of Bergman projections in tubes over cones. Namely, we show that such boundedness is equivalent to the duality identity between Bergman spaces, $A^{p'}=(A^p)^*$, and also to a Hardy type inequality related to the wave operator. We introduce analytic Besov spaces in tubes over cones, for which such Hardy inequalities play an important role. For $p\geq 2$ we identify as a Besov space the range of the Bergman projection acting on $L^p$, and also the dual of $A^{p'}$. For the Bloch space $\SB^\infty$ we give in addition new necessary conditions on the number of derivatives required in its definition.

Let $ \{X_j, j\in \Z\}$ be a Gaussian stationary sequence having a spectral function $F$ of infinite type. Then for all $n$ and $z\ge 0$,$$ \P\Big\{\sup_{j=1}^n |X_j|\le z \Big\}\le \Big(\int_{-z/\sqrt{G(f)}}^{z/\sqrt{G(f)}} e^{-x^2/2}\frac{\dd x}{\sqrt{2\pi}} \Big)^n,$$ where $ G(f)$ is the geometric mean of the Radon Nycodim derivative of the absolutely continuous part $f$ of $F$. The proof uses properties of finite Toeplitz forms. Let $ \{X(t), t\in \R\}$ be a sample continuous stationary Gaussian process with covariance function $\g(u) $. We also show that there exists an absolute constant $K$ such that for all $T>0$, $a>0$ with $T\ge \e(a)$, $$\P\Big\{\sup_{0\le s,t\le T} |X(s)-X(t)|\le a\Big\} \le \exp \Big \{-{KT \over \e(a) p(\e(a))}\Big\} ,$$ where $\e (a)= \min\big\{b>0: \d (b)\ge a\big\}$, $\d (b)=\min_{u\ge 1}\{\sqrt{2(1-\g((ub))}, u\ge 1\}$, and $ p(b) = 1+\sum_{j=2}^\infty {|2\g (jb)-\g ((j-1)b)-\g ((j+1)b)| \over 2(1-\g(b))}$. The proof is based on some decoupling inequalities arising from Brascamp-Lieb inequality. Both approaches are developed and compared on examples. Several other related results are established.

Let $ \{X_j, j\in \Z\}$ be a Gaussian stationary sequence having a spectral function $F$ of infinite type. Then for all $n$ and $z\ge 0$,$$ \P\Big\{\sup_{j=1}^n |X_j|\le z \Big\}\le \Big(\int_{-z/\sqrt{G(f)}}^{z/\sqrt{G(f)}} e^{-x^2/2}\frac{\dd x}{\sqrt{2\pi}} \Big)^n,$$ where $ G(f)$ is the geometric mean of the Radon Nycodim derivative of the absolutely continuous part $f$ of $F$. The proof uses properties of finite Toeplitz forms. Let $ \{X(t), t\in \R\}$ be a sample continuous stationary Gaussian process with covariance function $\g(u) $. We also show that there exists an absolute constant $K$ such that for all $T>0$, $a>0$ with $T\ge \e(a)$, $$\P\Big\{\sup_{0\le s,t\le T} |X(s)-X(t)|\le a\Big\} \le \exp \Big \{-{KT \over \e(a) p(\e(a))}\Big\} ,$$ where $\e (a)= \min\big\{b>0: \d (b)\ge a\big\}$, $\d (b)=\min_{u\ge 1}\{\sqrt{2(1-\g((ub))}, u\ge 1\}$, and $ p(b) = 1+\sum_{j=2}^\infty {|2\g (jb)-\g ((j-1)b)-\g ((j+1)b)| \over 2(1-\g(b))}$. The proof is based on some decoupling inequalities arising from Brascamp-Lieb inequality. Both approaches are developed and compared on examples. Several other related results are established.

For $0\leq \alpha 0$) for $n\geq 3$ are obtained. Also a class of functions related to Carath\'eodory functions is considered.

89 p. 25 cm

In this unit, we consider a relatively simple example of a large sieve inequality, of the sort introduced by Linnik. This is a setup for the multiplicative large sieve inequality we will need for Bombieri-Vinogradov.