Schauder System Research Articles

Orthorecursive Expansion with Respect to Modified Faber–Schauder System

Orthorecursive Expansion with Respect to Modified Faber–Schauder System

A Fourier Analysis Based New Look at Integration

Abstract We approach the problem of integration for rough integrands and integrators, typically representing trajectories of stochastic processes possessing only some Hölder regularity of possibly low order, in the framework of para-control calculus. For this purpose, we first decompose integrand and integrator into Paley–Littlewood packages along the Haar–Schauder system. By careful estimation of the components of products of packages of the integrand and derivatives of the integrator we obtain a characterization of Young’s integral. For the most interesting case of functions with Hölder regularities that sum up to an order below 1 we have to employ the concept of para-control of integrand and integrator with respect to a reference function for which a version of antisymmetric Lévy area is known to exist. This way we obtain an interpretation of the rough path integral. Lévy areas being known for most frequently used stochastic processes such as (fractional) Brownian motion, this integral serves as a basis for pathwise stochastic calculus, as the integral in classical rough path analysis.

A REVISIT TO STABILITY OF SCHAUDER BASES: FRACTALIZING MULTIVARIATE FABER–SCHAUDER SYSTEM

Let [Formula: see text] be a Banach space with a Schauder basis [Formula: see text], and [Formula: see text] be the identity operator on [Formula: see text]. It is known, at least in essence, that if [Formula: see text] is a sequence of bounded linear operators on [Formula: see text] such that [Formula: see text] then [Formula: see text] is also a basis. The first part of this work acts as an expository note to formally record the aforementioned stability result. In the second part, we apply this stability result to construct a Schauder basis consisting of bivariate fractal functions for the space of continuous functions defined on a rectangle. To this end, fractal perturbations of the elements in the classical bivariate Faber–Schauder system are formulated using a sequence of bounded linear fractal operators close to the identity operator in accordance with the stability result mentioned above. This illustration, although emphasized only for the bivariate case, can easily be extended to higher dimensions. Further, the perturbation technique used here acts as a companion for a few researches on fractal bases in the univariate setting.

Pointwise Rectangular Lipschitz Regularities for Fractional Brownian Sheets and Some Sierpinski Selfsimilar Functions

We consider pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities for continuous functions f on the unit cube I 2 in R 2 . Firstly, we provide characterizations by simple estimates on the decay rate of the coefficients (resp. leaders) of the expansion of f in the rectangular Schauder system, near the point considered. We deduce that pointwise rectangular Lipschitz regularity yields pointwise level coordinate axes Lipschitz regularities. As an application, we refine earlier results in Ayache et al. (Drap brownien fractionnaire. Potential Anal. 2002, 17, 31–43) and Kamont (On the fractional anisotropic Wiener field. Probab. Math. Statist. 1996, 16, 85–98), where uniform rectangular Lipschitz regularity of the trajectories of the fractional Brownian sheet over the total I 2 (or any cube) was considered. Actually, we prove that fractional Brownian sheets are pointwise rectangular and level coordinate axes monofractal. On the opposite, we construct a class of Sierpinski selfsimilar functions that are pointwise rectangular and level coordinate axes multifractal.

A higher order Faber spline basis for sampling discretization of functions

This paper is devoted to the question of constructing a higher order Faber spline basis for the sampling discretization of functions with higher regularity than Lipschitz. The basis constructed in this paper has similar properties as the piecewise linear classical Faber–Schauder basis (Faber, 1908) except for the compactness of the support. Although the new basis functions are supported on the real line they are very well localized (exponentially decaying) and the main parts are concentrated on a segment. This construction gives a complete answer to Problem 3.13 in Triebel’s monograph (Triebel, 2012) by extending the classical Faber basis to higher orders. Roughly, the crucial idea to obtain a higher order Faber spline basis is to apply Taylor’s remainder formula to the dual Chui–Wang wavelets. As a first step we explicitly determine these dual wavelets which may be of independent interest. Using this new basis we provide sampling characterizations for Besov and Triebel–Lizorkin spaces and overcome the smoothness restriction coming from the classical piecewise linear Faber–Schauder system. This basis is unconditional and coefficient functionals are computed from discrete function values similar as for the Faber–Schauder situation.

Quasiunconditional Basis Property of the Faber–Schauder System

We prove that, for any 0 1 − 𝛿, such that for any function f ∈ C[0, 1], one can find a function $$ \tilde{f} $$ ∈ C[0, 1] that coincides with f on E𝛿 such that the Fourier–Faber–Schauder series for this function $$ \tilde{f} $$ unconditionally converges in C[0, 1]. Moreover, the moduli of the nonzero Fourier–Faber–Schauder coefficients of the function $$ \tilde{f} $$ coincide with the elements of a given sequence {bn} satisfying the condition$$ {b}_n\downarrow 0,\kern1em \sum \limits_{n=1}^{\infty}\frac{b_n}{n}=+\infty . $$

A comparison of deep networks with ReLU activation function and linear spline-type methods

Deep neural networks (DNNs) generate much richer function spaces than shallow networks. Since the function spaces induced by shallow networks have several approximation theoretic drawbacks, this explains, however, not necessarily the success of deep networks. In this article we take another route by comparing the expressive power of DNNs with ReLU activation function to linear spline methods. We show that MARS (multivariate adaptive regression splines) is improper learnable by DNNs in the sense that for any given function that can be expressed as a function in MARS with M parameters there exists a multilayer neural network with O(Mlog(M∕ε)) parameters that approximates this function up to sup-norm error ε. We show a similar result for expansions with respect to the Faber–Schauder system. Based on this, we derive risk comparison inequalities that bound the statistical risk of fitting a neural network by the statistical risk of spline-based methods. This shows that deep networks perform better or only slightly worse than the considered spline methods. We provide a constructive proof for the function approximations.

The Fourier–Faber–Schauder Series Unconditionally Divergent in Measure

We prove that, for every e ∈ (0, 1), there is a measurable set E ⊂ [0, 1] whose measure |E| satisfies the estimate |E| > 1−e and, for every function f ∈ C[0,1], there is ˜ f ∈ C[0,1] coinciding with f on E whose expansion in the Faber–Schauder system diverges in measure after a rearrangement.

Expansion of Self-Similar Functions in the Faber–Schauder System

Let Ω = AN be a space of right-sided infinite sequences drawn from a finite alphabet A = {0,1}, N = {1,2,…}. Let ρ(x, y)Σk=1∞|x k − y k |2−k be a metric on Ω = AN, and μ the Bernoulli measure on Ω with probabilities p0, p1 > 0, p0 + p1 = 1. Denote by B(x,ω) an open ball of radius r centered at ω. The main result of this paper \(\mu (B(\omega ,r))r + \sum\nolimits_{n = 0}^\infty {\sum\nolimits_{j = 0}^{{2^n} - 1} {{\mu _{n,j}}} } (\omega )\tau ({2^n}r - j)\), where τ(x) = 2min {x,1 − x}, 0 ≤ x ≤ 1, (τ(x) = 0, if x 1 ), \({\mu _{n,j}}(\omega ) = (1 - {p_{{\omega _{n + 1}}}})\prod _{k = 1}^n{p_{{\omega _k}}} \oplus {j_k}\), \(j = {j_1}{2^{n - 1}} + {j_2}{2^{n - 2}} + ... + {j_n}\). The family of functions 1, x, τ(2 n r − j), j = 0,1,…, 2 n − 1, n = 0,1,…, is the Faber–Schauder system for the space C([0,1]) of continuous functions on [0, 1]. We also obtain the Faber–Schauder expansion for Lebesgue’s singular function, Cezaro curves, and Koch–Peano curves. Article is published in the author’s wording.

The Expansion of Self-similar Functions in the Faber–Schauder System

Let \(\Omega = A^{N}\) be a space of right-sided innite sequences drawn from a nite alphabet \(A = \{0,1\}\), \(N = \{1,2,\dots\}\). Let $$\label{rho} \rho(\boldsymbol{x},\boldsymbol{y}) =\sum_{k=1}^{\infty}|x_{k} - y_{k}|2^{-k} $$ - be a metric on \(\Omega = A^{N}\), and \(\mu\) - the Bernoulli measure on \(\Omega\) with probabilities \(p_0,p_1>0\), \(p_0+p_1=1\). Denote by \(B(\boldsymbol{x},\omega)\) an open ball of radius \(r\) centered at \(\boldsymbol{\omega}\). The main result of this paper is $$ \mu\left(B(\boldsymbol{\omega},r)\right) =r+\sum_{n=0}^{\infty}\sum_{j=0}^{2^n-1}\mu_{n,j}(\boldsymbol{\omega})\tau(2^nr-j), $$ where \(tau(x) =2\min\{x,1-x\}\), \(0\leq x \leq 1\), \(tau(x) = 0, if x<0 or x>1\), $$mu_{n,j}(\boldsymbol{\omega}) = \left(1-p_{\omega_{n+1}}\right) \prod_{k=1}^n p_{\omega_k\oplus j_k},\ \ j = j_12^{n-1}+j_22^{n-2}+\dots+j_n$$. The family of functions \(1,x,\tau(2^nx-j)\), \(j =0,1,\dots,2^n-1\), \(n=0,1,\dots\) is the Faber{Schauder system for the space \(C([0, 1])\) of continuous functions on \([0, 1]\). We also obtain the Faber{Schauder expansion for the Lebesgue's singular function, Cezaro curves, and Koch{Peano curves.

Unconditional C-strong property of Faber–Schauder system

Given a continuous function on [ 0 , 1 ] , it is always possible to alter its values on a small set so that the modified function is continuous and has an expansion which converges unconditionally in C [ 0 , 1 ] .

A Lévy-Ciesielski Expansion for Quantum Brownian Motion and the Construction of Quantum Brownian Bridges

We introduce “probabilistic” and “stochastic Hilbertian structures”. These seem to be a suitable context for developing a theory of “quantum Gaussian processes”. The Schauder system is utilised to give a Lévy-Ciesielski representation of quantum (bosonic) Brownian motion as operators in Fock space over a space of square summable sequences. Similar results hold for non-Fock, fermion, free and monotone Brownian motions. Quantum Brownian bridges are defined and a number of representations of these are given.

Subsystems of the Schauder system that are quasibases for 𝐿^{𝑝}[0,1],1≤𝑝<+∞

We show that if Φ = { φ i : i = 1 , 2 , … } \Phi =\{\varphi _i\colon i=1,2,\ldots \} is a subsystem of the Faber-Schauder system, and if Φ \Phi is complete in L 2 [ 0 , 1 ] L^2[0,1] , then Φ \Phi is a quasibasis for each space L p [ 0 , 1 ] L^p[0,1] , 1 ≤ p > + ∞ 1\le p>+\infty . Although it follows from the work of Ul’yanov that each element of L p [ 0 , 1 ] L^p[0,1] can be represented by a Schauder series that converges unconditionally to the function, in the metric of the space, it proves to be the case that none of the aforementioned systems is an unconditional quasibasis for any of the L p L^p -spaces herein considered.

Subsystems of the Schauder system whose orthonormalizations are Schauder bases for $L^p[0,1

Subsystems of the Schauder system whose orthonormalizations are Schauder bases for $L^p[0,1

On series with respect to a Schauder system

We establish some statements regarding series with respect to a Schauder system. In particular, we prove a representation theorem and we investigate the behavior of the Fourier coefficients of some classes of functions.

On orthogonalization of bases

An example of a basis for space C, close to the Schauder system, is constructed which, after orthogonalization by the Schmidt method, is not a basis for space LP for any p e [1, 2) +(2,∞].

Series with respect to the Schauder system

Unconditional convergence in the C(0, 1) metric and convergence almost everywhere of series with respect to the Schauder system are investigated.

On Schauder system series

On Schauder system series