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Susanna Spektor

Publications and source records attributed to Susanna Spektor.

At least 19 recordsLinked to original sources
arXiv2026

The Hardy averaging operator between Orlicz spaces and a new gauge functional characterizing a given Orlicz space

We prove that the largest Orlicz space into which the Hardy averaging operator maps coincides with the largest rearrangement-invariant (r.i.) space mapping into that space; in other words, the optimal Orlicz domain is automatically the optimal r.i.\ domain. More generally, we show that every Luxemburg--Orlicz norm is equivalent to a sublinear functional which makes more computable a certain expression for a dual norm arising in the $\K$-theory of interpolation. As applications we obtain Orlicz-space mapping properties for the Hardy--Littlewood maximal function, approximate identities and Calderón--Zygmund singular integral operators. These mapping properties are shown to be optimal for the Hardy-Littlewood maximal function and the approximate identities.

math.AP↗
arXiv2023

Orlicz-Lorentz Gauge Functional Inequalities for Positive Integral Operators. Revised Version

Let $f \in M_+(\R_+)$, the class of nonnegative, Lebesgure-measurable functions on $\R_+=(0, \infty)$. We deal with integral operators of the form \[ (T_Kf)(x)=\int_{\R_+}K(x,y)f(y)\, dy, \quad x \in \R_+, \] with $K \in M_+(\R_+^2)$. We are interested in inequalities \[ ρ_{1}((T_Kf)^*)\leq Cρ_2(f^*), \] in which $ρ_1$ and $ρ_2$ are functionals on functions $h \in M_+(\R_+)$, and \[ h^*(t)=μ_h^{-1}(t), \quad t \in \R_+, \] where \[ μ_h(λ)=|\{x \in \R_+: \, h(x)> λ\}|, λ\in \R_+. \] Specifically, $ρ_1$ and $ρ_2$ are so-called Orlicz-Lorentz gauge functionals of the type \[ ρ(h)=ρ_{Φ, u}(h)=\inf\left\{λ>0:\, \int_{\R_+}Φ\left(\frac{h(x)}λ\right)u(x)\, dx \leq 1\right\}, \quad h \in M_+(\R_+); \] here $Φ(x)=\int_0^xϕ(y)\, dy$, $ϕ$ an increasing function mapping $\R_+$ onto itself and $u\in M_+(\R_+)$.

math.FA↗
arXiv2021

Functional Response Designs via the Analytic Permutation Test

Vast literature on experimental design extends from Fisher and Snedecor to the modern day. When data lies beyond the assumption of univariate normality, nonparametric methods including rank based statistics and permutation tests are enlisted. The permutation test is a versatile exact nonparametric significance test that requires drastically fewer assumptions than similar parametric tests. The main downfall of the permutation test is high computational cost making this approach laborious for complex data and sophisticated experimental designs and completely infeasible in any application requiring speedy results such as high throughput streaming data. We rectify this problem through application of concentration inequalities and thus propose a computation free permutation test -- i.e. a permutation-less permutation test. This general framework is applied to multivariate, matrix-valued, and functional data. We improve these concentration bounds via a novel incomplete beta transform. We extend our theory from 2-sample to $k$-sample testing through the use of weakly dependent Rademacher chaoses and modified decoupling inequalities. We test this methodology on classic functional data sets including the Berkeley growth curves and the phoneme dataset. We further consider analysis of spoken vowel sound under two experimental designs: the Latin square and the randomized block design.

math.ST↗
arXiv2017

Estimation of wavelet coefficients on some classes of functions

Let $Ψ_m^D$ be orthogonal Daubechies wavelets that have m zero moments and let $$ W_{2,p}^k=\{f \in L_2(R):\|(I ω)^k\hat f(ω)\|_p\leq 1\}, \, k \in N. $$ We prove that $$ \lim_{m \to \infty}\, \sup\left\{\frac{|(Ψ_m^D)|}{\|(\hat Ψ_m^D)\|_q}: f \in W_{2, p'}^k\right\}=\frac{\frac{(2π)^{1/p-1/2}}{π^k}\left(\frac{1-2^{1-pk}}{pk-1}\right)^{1/p}}{(2π)^{1/q-1/2}}. $$

math.FA↗
arXiv2017

An estimate of the root mean square error incurred when approximating an $f \in L^2({\mathbb{R}})$ by a partial sum of its Hermite series

Let $f$ be a band-limited function in $L^2({\mathbb{R}})$. Fix $T >0$ and suppose $f^{\prime}$ exists and is integrable on $[-T, T]$. This paper gives a concrete estimate of the error incurred when approximating $f$ in the root mean square by a partial sum of its Hermite series. Specifically, we show, for $K=2n, \quad n \in Z_+,$ $$ \left[\frac{1}{2T}\int_{-T}^T[f(t)-(S_Kf)(t)]^2dt\right]^{1/2}\leq \left(1+\frac 1K\right)\left(\left[ \frac{1}{2T}\int_{|t|> T}f(t)^2dt\right]^{1/2} +\left[\frac{1}{2T} \int_{|ω|>N}|\hat f(ω)|^2dω\right]^{1/2} \right) +\frac{1}{K}\left[\frac{1}{2T}\int_{|t|\leq T}f_N(t)^2dt\right]^{1/2} +\frac{1}π\left(1+\frac{1}{2K}\right)S_a(K,T), $$ in which $S_Kf$ is the $K$-th partial sum of the Hermite series of $f, \hat f $ is the Fourier transform of $f$, $\displaystyle{N=\frac{\sqrt{2K+1}+% \sqrt{2K+3}}{2}}$ and $f_N=(\hat f χ_{(-N,N)})^\vee(t)=\frac{1}π\int_{-\infty}^{\infty}\frac{\sin (N(t-s))}{t-s}f(s)ds$. An explicit upper bound is obtained for $S_{a}(K,T)$.

math.CA↗
arXiv2017

Explicit Bernstein type inequalities for wavelet coefficients in $L_p(R^n)$

In this paper, we investigate the wavelet coefficients for function spaces $\mathcal{A}_k^p=\{f: \|(i ω)^k\hat{f}(ω)\|_p\leq 1\}, k\in N, p\in(1,\infty)$ using an important quantity $C_{k,p}(ψ)$. In particular, Bernstein type inequalities associated with wavelets are established. We obtained a sharp inequality of Bernstein type for splines, which induces a lower bound for the quantity $C_{k,p}(ψ)$ with $ψ$ being the semiorthogonal spline wavelets. We also study the asymptotic behavior of wavelet coefficients for both the family of Daubechies orthonormal wavelets and the family of semiorthogonal spline wavelets. Comparison of these two families is done by using the quantity $C_{k,p}(ψ)$.

math.FA↗
arXiv2017

An asymptotically sharp form of Ball's integral inequality

We solve the open problem of determining the second order term in the asymptotic expansion of the integral in Ball's integral inequality. In fact, we provide a method by which one can compute any term in the expansion. We also indicate how to derive an asymptotically sharp form of a generalized Ball's integral inequality.

math.FA↗
arXiv2017

Quantitative version of a Silverstein's result

We prove a quantitative version of a Silverstein's Theorem on a condition for convergence in probability of the norm of random matrix. More precisely, we show that for a random matrix whose entries are i.i.d. random variables, $w_{i,j}$, satisfying certain natural conditions, is not small with large probability.

math.PR↗
arXiv2017

An asymptotically sharp form of Ball's inequality by probability methods

To prove by probabilistic methods that every $(n-1)$-dimensional section of the unit cube in $R^n$ has volume at most $\sqrt 2$, K. Ball made essential use of the inequality $$ \frac{1}π\int_{-\infty}^{\infty} \left(\frac{\sin^2 t}{t^2}\right)^pdt\leq \frac{\sqrt 2}{\sqrt p}, \quad p\geq 1, $$ in which equality holds if and only if $p=1$. The right side of above inequality has the correct rate of decay though the limit of the ratio of the right to left side is ${\sqrt{\frac{3}π}}$ rather then $\sqrt 2$. Applying Ball's methods we put all of this into the improved form of the Ball's inequality.

math.FA↗
arXiv2017

On Khintchine type inequalities for $k$-wise independent Rademacher random variables

We consider Khintchine type inequalities on the $p$-th moments of vectors of $N$ $k$-wise independent Rademacher random variables. We show that an analogue of Khintchine's inequality holds, with a constant $N^{1/2-k/2p}$, when $k$ is even. We then show that this result is sharp for $k=2$; in particular, a version of Khintchine's inequality for sequences of pairwise Rademacher variables \emph{cannot} hold with a constant independent of $N$. We also characterize the cases of equality and show that, although the vector achieving equality is not unique, it is unique (up to law) among the smaller class of exchangable vectors of pairwise independent Rademacher random variables. As a fortunate consequence of our work, we obtain similar results for $3$-wise independent vectors.

math.PR↗
arXiv2017

Restricted Khinchine inequality

We prove a Khintchine type inequality under the assumption that the sum of Rademacher random variables equals zero. As an application we show a new tail-bound for a hypergeometric random variable.

math.PR↗
arXiv2015

The approximation of almost time and band limited functions by their expansion in some orthogonal polynomials bases

The aim of this paper is to investigate the quality of approximation of almost time and almost band-limited functions by its expansion in three classical orthogonal polynomials bases: the Hermite, Legendre and Chebyshev bases. As a corollary, this allows us to obtain the quality of approximation in the L 2 --Sobolev space by these orthogonal polynomials bases. Also, we obtain the rate of the Legendre series expansion of the prolate spheroidal wave functions. Some numerical examples are given to illustrate the different results of this work.

math.CA↗