scholarly journals Sharp Inequalities for the Haar System and Fourier Multipliers

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Adam Osȩkowski
Keyword(s):  
Continuous Time ◽  
Classical Result ◽  
Weak Type ◽  
Haar System ◽  
Borel Subset ◽  
General Setting ◽  
Universal Constant ◽  
Fourier Multipliers ◽  
Arbitrary Subset ◽  
Best Constant

A classical result of Paley and Marcinkiewicz asserts that the Haar systemh=hkk≥0on0,1forms an unconditional basis ofLp0,1provided1<p<∞. That is, if𝒫Jdenotes the projection onto the subspace generated byhjj∈J(Jis an arbitrary subset ofℕ), then𝒫JLp0,1→Lp0,1≤βpfor some universal constantβpdepending only onp. The purpose of this paper is to study related restricted weak-type bounds for the projections𝒫J. Specifically, for any1≤p<∞we identify the best constantCpsuch that𝒫JχALp,∞0,1≤CpχALp0,1for everyJ⊆ℕand any Borel subsetAof0,1. In fact, we prove this result in the more general setting of continuous-time martingales. As an application, a related estimate for a large class of Fourier multipliers is established.

PORTFOLIO OPTIMIZATION WITH PERFORMANCE RATIOS

2019 ◽  
Vol 22 (05) ◽  
pp. 1950022
Author(s):  
HONGCAN LIN ◽  
DAVID SAUNDERS ◽  
CHENGGUO WENG
Keyword(s):  
Continuous Time ◽  
Original Problem ◽  
General Setting ◽  
Performance Measure ◽  
Existence Of Solution ◽  
Explicit Solutions ◽  
Martingale Method ◽  
Penalty Parameter ◽  
Objective Functions ◽  
Power Functions

We consider the portfolio selection problem of maximizing a performance measure in a continuous-time diffusion model. The performance measure is the ratio of the overperformance to the underperformance of a portfolio relative to a benchmark. Following a strategy from fractional programming, we analyze the problem by solving a family of related problems, where the objective functions are the numerator of the original problem minus the denominator multiplied by a penalty parameter. These auxiliary problems can be solved using the martingale method for stochastic control. The existence of solution is discussed in a general setting and explicit solutions are derived when both the reward and the penalty functions are power functions.

A note on the one-dimensional maximal function

1989 ◽  
Vol 111 (3-4) ◽  
pp. 325-328 ◽  
Author(s):  
Antonio Bernal
Keyword(s):  
Maximal Function ◽  
Weak Type ◽  
Type Inequality ◽  
Best Constant ◽  
Weak Type Inequality ◽  
One Dimensional ◽  
Littlewood Maximal Function ◽  
The One

SynopsisIn this note, we consider the Hardy-Littlewood maximal function on R for arbitrary measures, as was done by Peter Sjögren in a previous paper. We determine the best constant for the weak type inequality.

Remarks on the Hardy–Littlewood maximal function

1998 ◽  
Vol 128 (1) ◽  
pp. 1-9 ◽  
Author(s):  
J. M. Aldaz
Keyword(s):  
Maximal Function ◽  
Real Line ◽  
Weak Type ◽  
Best Constant ◽  
The Real ◽  
Littlewood Maximal Function

We answer questions of A. Carbery, M. Trinidad Menárguez and F. Soria by proving, firstly, that for the centred Hardy–Littlewood maximal function on the real line, the best constant C for the weak type (1, 1) inequality is strictly larger than 3/2, and secondly, that C is strictly less than 2 (known to be the best constant in the noncentred case).

Markovian couplings staying in arbitrary subsets of the state space

2002 ◽  
Vol 39 (01) ◽  
pp. 197-212 ◽  
Author(s):  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Transition Rates ◽  
Arbitrary Subset ◽  
State Spaces ◽  
Continuous Time Markov Chains ◽  
Countable State ◽  
Necessary And Sufficient

Let (X t ) and (Y t ) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (X t ) and (Y t ) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.

Some Sharp Estimates for the Haar System and Other Bases In $L^1(0,1)$

2014 ◽  
Vol 115 (1) ◽  
pp. 123
Author(s):  
Adam Osȩkowski
Keyword(s):  
Nonnegative Integer ◽  
Unconditional Basis ◽  
Haar System ◽  
Bellman Function ◽  
Real Numbers ◽  
Best Constant ◽  
Sharp Estimates

Let $h=(h_k)_{k\geq 0}$ denote the Haar system of functions on $[0,1]$. It is well known that $h$ forms an unconditional basis of $L^p(0,1)$ if and only if $1<p<\infty$, and the purpose of this paper is to study a substitute for this property in the case $p=1$. Precisely, for any $\lambda>0$ we identify the best constant $\beta=\beta_h(\lambda)\in [0,1]$ such that the following holds. If $n$ is an arbitrary nonnegative integer and $a_0$, $a_1$, $a_2$, $\ldots$, $a_n$ are real numbers such that $\bigl\|\sum_{k=0}^n a_kh_k\bigr\|_1\leq 1$, then \[ \Bigl|\Bigl\{x\in [0,1]:\Bigl|\sum_{k=0}^n \varepsilon_ka_kh_k(x)\Bigr|\geq \lambda\Bigr\}\Bigr|\leq \beta, \] for any sequence $\varepsilon_0, \varepsilon_1, \varepsilon_2,\ldots, \varepsilon_n$ of signs. A related bound for an arbitrary basis of $L^1(0,1)$ is also established. The proof rests on the construction of the Bellman function corresponding to the problem.

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