On alternative quantization for doubly weighted approximation and integration over unbounded domains

Abstract

It is known that for a $\varrho$-weighted $L_q$ approximation of single variable functions defined on a finite or infinite interval, whose $r$th derivatives are in a $\psi$-weighted $L_p$ space, the minimal error of approximations that use $n$ samples of $f$ is proportional to ${\Vert {\omega }^{1∕\alpha }\Vert }_{L_1}^{\alpha }{\Vert f^{\left(r\right)}\psi \Vert }_{L_p}{n}^{-r+{(1∕p-1∕q)}_{+}},$ where $\omega =\varrho ∕\psi$ and $\alpha =r-1∕p+1∕q,$ provided that ${\Vert {\omega }^{1∕\alpha }\Vert }_{L_1}<+\infty .$ Moreover, the optimal sample points are determined by quantiles of ${\omega }^{1∕\alpha }.$ In this paper, we show how the error of the best approximation changes when the sample points are determined by a quantizer $\kappa$ other than $\omega .$ Our results can be applied in situations when an alternative quantizer has to be used because $\omega$ is not known exactly or is too complicated to handle computationally. The results for $q=1$ are also applicable to $\varrho$-weighted integration over finite and infinite intervals.

Introduction

In various applications, continuous objects (signals, images, etc.) are represented (or approximated) by discrete counterparts. That is, we deal with quantization. From a pure mathematics point of view, quantization often leads to approximating functions from a given space by step functions or, more generally, by piecewise polynomials of certain degree. Then it is important to know which quantizer should be used, or how to select $n$ break points (knots) to make the error of approximation as small as possible.

It is well known that for $L_q$ approximation on a compact interval $D=[a,b]$ in the space $F_p^r\left(D\right)$ of real-valued functions $f$ such that $f^{(r-1)}$ are absolutely continuous and $f^{\left(r\right)}\in L_p\left(D\right)$, the choice of an optimal quantizer is not a big issue, since equidistant knots lead to approximations with optimal $L_q$ error

$$c{(b-a)}^{\alpha }{\Vert f^{\left(r\right)}\Vert }_{L_p}{n}^{-r+{(1∕p-1∕q)}_{+}}\text{with}\alpha ≔r-\frac{1}{p}+\frac{1}{q},$$

where $c$ depends only on $r$, $p$, and $q$, and where $x_{+}≔max(x,0)$, see, e.g., [5]. The problem becomes more complicated if we switch to weighted approximation on unbounded domains. A generalization of (1) to this case was given in [6], and it reads as follows. Assume for simplicity that the domain $D={\mathbb{R}}_{+}≔[0,+\infty )$. Let $\psi ,\varrho :D\to [0,+\infty )$ be two a.e. positive and integrable weight functions. For a positive integer $r$ and $1\le p,q\le +\infty$, consider the $\varrho$-weighted $L_q$ approximation in the linear space $F_{p,\psi }^r\left(D\right)$ of functions $f:D\to \mathbb{R}$ with locally absolutely continuous $(r-1)$st derivative and such that the $\psi$-weighted $L_p$ norm of $f^{\left(r\right)}$ is finite, i.e., ${\Vert f^{\left(r\right)}\psi \Vert }_{L_p}<+\infty$. Note that the spaces $F_{p,\psi }^r\left(D\right)$ have been introduced in [8], and the role of $\psi$ is to moderate their size.

Denote

$$\omega ≔\frac{\varrho }{\psi },$$

and suppose that $\omega$ and $\psi$ are nonincreasing⁴ on $D$, and that

$${\Vert {\omega }^{1∕\alpha }\Vert }_{L_1}≔{\int }_D{\omega }^{1∕\alpha }\left(x\right)\mathrm{d}x<+\infty .$$

It was shown in [6, Theorem 1] that then one can construct approximations using $n$ knots with $\varrho$-weighted $L_q$ error at most

$$c_1{\Vert {\omega }^{1∕\alpha }\Vert }_{L_1}^{\alpha }{\Vert f^{\left(r\right)}\psi \Vert }_{L_p}{n}^{-r+{(1∕p-1∕q)}_{+}},$$

where the factor $c_1$ is given in (10). This means that if (3) holds true, then the upper bound on the worst-case error is proportional to ${\Vert {\omega }^{1∕\alpha }\Vert }_{L_1}^{\alpha }{n}^{-r+{(1∕p-1∕q)}_{+}}$. The convergence rate $n^{-r+{(1∕p-1∕q)}_{+}}$ is optimal and a corresponding lower bound implies that if (3) is not satisfied then the rate $n^{-r+{(1∕p-1∕q)}_{+}}$ cannot be reached (see [6, Theorem 3]).

The optimal knots

$$0=x_0^{\ast }<x_1^{\ast }<\cdots <x_{n-1}^{\ast }<x_n^{\ast }=+\infty$$

are determined by quantiles of ${\omega }^{1∕\alpha }$, to be more precise,

$${\int }_0^{x_i^{\ast }}{\omega }^{1∕\alpha }\left(t\right)\mathrm{d}t=\frac{i}{n}{\Vert {\omega }^{1∕\alpha }\Vert }_{L_1}.$$

In order to use the optimal quantizer (5) one has to know $\omega$; otherwise one has to rely on some approximations of $\omega$. Moreover, even if $\omega$ is known, it may be a complicated function and therefore difficult to handle computationally. Driven by this motivation, the purpose of the present paper is to generalize the results of [6] even further to see how the quality of best approximations will change if the optimal quantizer $\omega$ is replaced in (5) by another quantizer $\kappa$.

A general answer to the aforementioned question is given in Theorem 1, Theorem 3 of Section 2. We show, respectively, tight (up to a constant) upper and lower bounds for the error when a quantizer $\kappa$ with ${\Vert {\kappa }^{1∕\alpha }\Vert }_{L_1}<+\infty$ instead of $\omega$ is used to determine the knots. To be more specific, define

$${\mathcal{E}}_p^q(\omega ,\kappa )={\Vert \frac{\omega }{\kappa }\Vert }_{L_{\infty }}\text{for }p\le q,$$

and

$${\mathcal{E}}_p^q(\omega ,\kappa )={\left({\int }_D\frac{{\kappa }^{1∕\alpha }\left(x\right)}{{\Vert {\kappa }^{1∕\alpha }\Vert }_{L_1}}{\left(\frac{\omega \left(x\right)}{\kappa \left(x\right)}\right)}^{\frac{1}{1∕q-1∕p}}\mathrm{d}x\right)}^{1∕q-1∕p}\text{for }p>q\text{.}$$

(Note that (6), (7) are consistent since ${lim}_{(1∕q-1∕p)\to 0^{+}}{\mathcal{E}}_p^q(\omega ,\kappa )={\Vert \omega ∕\kappa \Vert }_{L_{\infty }}$.) If ${\mathcal{E}}_p^q(\omega ,\kappa )<+\infty$ then the best achievable error is proportional to

$${\Vert {\kappa }^{1∕\alpha }\Vert }_{L_1}^{\alpha }{\mathcal{E}}_p^q(\omega ,\kappa ){\Vert f^{\left(r\right)}\psi \Vert }_{L_p}{n}^{-r+{(1∕p-1∕q)}_{+}}.$$

This means, in particular, that for the error to behave as $n^{-r+{(1∕p-1∕q)}_{+}}$ it is sufficient (but not necessary) that $\kappa \left(x\right)$ decreases no faster than $\omega \left(x\right)$ as $\left|x\right|\to +\infty$. For instance, if the optimal quantizer is Gaussian, $\omega \left(x\right)=exp(-x^2∕2)$, then the optimal rate is still preserved if its exponential substitute $\kappa \left(x\right)=exp(-a\left|x\right|)$ with arbitrary $a>0$ is used. It also shows that, in case $\omega$ is not exactly known, it is much safer to overestimate than to underestimate it, see also Example 5.

The use of a quantizer $\kappa$ as above results in approximations whose errors are worse than those for the optimal approximations by the factor of

$$\mathrm{FCTR}(p,q,\omega ,\kappa )=\frac{{\Vert {\kappa }^{1∕\alpha }\Vert }_{L_1}^{\alpha }}{{\Vert {\omega }^{1∕\alpha }\Vert }_{L_1}^{\alpha }}{\mathcal{E}}_p^q(\omega ,\kappa )\ge 1.$$

In Section 3, we calculate the exact values of this factor for various combinations of weights $\varrho$, $\psi$, and $\kappa$, including: Gaussian, exponential, log-normal, logistic, and $t$-Student. It turns out that in many cases $\mathrm{FCTR}(p,q,\omega ,\kappa )$ is quite small, so that the loss in accuracy of approximation is well compensated by simplification of the quantizer.

The results for $q=1$ are also applicable to the problem of approximating the $\varrho$-weighted integral ${\int }_{D}f\left(x\right)\varrho \left(x\right)\mathrm{d}x$ for functions $f\in F_{p,\psi }^r\left(D\right)$, see Remark 6, Remark 7. In this case, the best possible convergence rate is $n^{-r}$ and it can be achieved if and only if (3) holds with $\alpha =r+1-1∕p$. These results are especially important for unbounded domains, e.g., $D={\mathbb{R}}_{+}$ or $D=\mathbb{R}$. For such domains, the integrals are often approximated by Gauss–Laguerre rules and Gauss–Hermite rules, respectively, see, e.g., [1], [3], [7]; however, their efficiency requires infinitely smooth integrands and the results are asymptotic. Moreover, it is not clear which Gaussian rules should be used when $\psi$ is not a constant function. But, even for $\psi =1$, it is likely that the worst-case errors (with respect to $F_{p,\psi }^r$) of Gaussian rules are much larger than $O\left(n^{-r}\right)$, since the Weierstrass theorem holds only for compact $D$. An extension of Gaussian rules to functions with singularities has been proposed in [2]. However, the results of [2] are also asymptotic and it is not clear how the proposed rules behave for functions from spaces $F_{p,\psi }^r$. In the present paper, we deal with functions of bounded smoothness ($r<+\infty$) and provide worst-case error bounds that are minimal. We stress here that the regularity degree $r$ is a fixed but arbitrary positive integer. The paper [4] proposes a different approach to the weighted integration over unbounded domains; however, it is restricted to regularity $r=1$ only.

The paper is organized as follows. In the following section, we present ideas and results about alternative quantizers. The main results are Theorem 1, Theorem 3. In Section 3, we apply our results to some specific cases for which numerical values of $\mathrm{FCTR}(p,q,\omega ,\kappa )$ are calculated.