Notes on Approximation Theory

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Peter Occil

The notes in this page catalog results that may be useful in finding bounds on how close a single-variable function is to a polynomial or rational function (or, more generally, a “simpler” version of the other function).

The aim is to find error bounds that are explicit, with no hidden constants and without introducing transcendental or trigonometric functions. If an error bound is explicit, it can be computed offline, without performing an approximation first, so that it can be known, for example, which degree polynomial to build in order to come close to a function with a given accuracy.

The mapping from a function to a function (for example, from a single-variable function to a polynomial “close” to it) is called an operator, and operators involved in these bounds are often linear operators, whose behavior is relatively simple to examine.

Contents

• Contents
• About This Document
• Notation and Definitions
• Bernstein Form and Bernstein Polynomials
• “Moments” of Linear Operators
  • “Moments” of Bernstein Polynomials
• Taylor Expansion of Linear Operators
• Results on Error Bounds
  • Bounds for General Positive Linear Operators
  • Bounds for Remainder of Bernstein Polynomials
  • Bounds for General Linear Operators
  • Smoothing Error Bounds
  • Lebesgue Inequality for Certain Linear Operators
  • Bounds for Certain Nonlinear Operators
  • Lipschitz-Continuous Operators
• Example
• Example: An Interesting Linear Operator
• Example: The Lorentz Operators
• Probabilistic Interpretations of Linear Operators
• Conclusion and Ways to Improve This Article
• License
• Notes

About This Document

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Notation and Definitions

For definitions of continuous, derivative, convex, concave, Hölder continuous, and Lipschitz continuous, see the definitions section in “Supplemental Notes for Bernoulli Factory Algorithms”.

• The closed unit interval (written as [0, 1]) means the set consisting of 0, 1, and every real number in between.
• A compact interval (written as $[a, b]$), means a set of real numbers consisting of $a$, $b$, and every real number in between, where $a$ is less than or equal to $b$.¹ The closed unit interval is one example of a compact interval.
• A function $f(\lambda)$ is bounded if there are two real numbers $a$ and $b$ such that $a\le f(\lambda) \le b$ over the domain of $f$.
• The least upper bound of a function $f(x)$ is the smallest number $b$ such that $f(x) \le b$ over the domain of $f$. If $f$ is continuous, the least upper bound is the same as the maximum.
• An operator is a mapping from a function to a function.
  • An operator $L$ is linear if it satisfies $L(af)=aL(f)$ and $L(f+g)=L(f)+L(g)$ for all allowed functions $f$ and $g$ and every number $a$.².
  • An operator $L$ is positive if it has the property that, if an allowed function $f$ is nonnegative on its domain, so is $L(f)$.³
  • The operator norm of an operator $L$ is the least upper bound of the following value over all allowed functions $f$ other than 0: the “norm” of $L(f)$ divided by the “norm” of $f$ (De Villiers 2012, (5.2.2))⁴.⁵ The “norm” of $L(f)$ and that of $f$ depend on the spaces of functions that $L$ maps to and from, respectively.
  • An operator is bounded if its operator norm is finite.
• The expected value (or mean or “long-run average”) of a random variable $Y$ is denoted $\mathbb{E}[Y]$.
• A modulus of continuity of order 1 of a function f, denoted $\omega_1(f, \delta)$, means a nonnegative and nowhere decreasing function where, for each $\delta\ge 0$, $\text{abs}(f(x)-f(y))\le\omega_1(f, \delta)$ whenever $x$ and $y$ are in $f$’s domain and no more than $\delta$ apart. Loosely speaking, $\omega_1(f, \delta)$ gives how much $f$ can vary when $f$ is restricted to a window of size $\delta$ or less. The modulus of continuity reflects the “regularity” of $f$; generally, the smaller it is, the more “regular”.
• In this document:
  • $e_i$ is a function such that $e_i(t) = t^i$, so that $e_0(t) = 1$ and $e_1(t) = t$; as an example, if $L(f) = f(0) + f(1)$, then $L(e_1 - x)$ = $(e_1(0) - x) + (e_1(1) - x)$ = $(0-x)+(1-x)=1-2x$.
  • The notation $\Vert f\Vert$, where $f$ is a function, means the function’s “norm”, which is the maximum absolute value over its domain unless the context indicates otherwise.
  • The notation $\Vert L\Vert_{op}$, where $L$ is an operator, means its operator norm; the “norm” used to define the operator norm is the maximum absolute value unless noted otherwise.

Bernstein Form and Bernstein Polynomials

Among the best known examples of linear operators are the Bernstein polynomials.

In this document, a polynomial $P(x)$ is written in Bernstein form of degree $n$ if it is written as—

\[P(x)=\sum_{k=0}^n a_k \frac{n!}{(k!)((n-k)!)} x^k (1-x)^{n-k},\]

where $0\le x\le 1$ and the real numbers $a_0, …, a_n$ are the polynomial’s Bernstein coefficients.⁶

The degree-$n$ Bernstein polynomial of a function $f(x)$, denoted $B_n(f)$ in this document, has Bernstein coefficients $a_k = f(k/n)$. In general, this Bernstein polynomial differs from $f$ even if $f$ is a polynomial.

$B_n(f)$ is a positive linear operator. It maps a function like $f(x)$ on the closed unit interval to a polynomial of degree $n$ or less on that interval.

This page’s emphasis is on methods that produce polynomials in Bernstein form, ratios of such polynomials, or functions like $f(g(x))$ where $f$ and $g$ are such polynomials or ratios.

“Moments” of Linear Operators

To examine the approximation behavior of linear operators, it is helpful to find the so-called “moments” of those operators, that is, the functions they map certain functions to.

For a linear operator $L$, they are:

• “Raw moments”: The values of $L(e_i)$ for each integer $i\ge 0$.
• “Central moments”: The values of $L((e_1-x)^i)$ for each integer $i\ge 0$. If the “raw moments” $L(e_0), …, L(e_j)$ are known, then $L((e_1-x)^j)$ is also known, thanks to proposition 5.6 of Gonska et al. (2006)⁷.
• “Absolute moments”: The values of $L(\text{abs}(e_1-x)^i)(x)$ for each integer $i\ge 0$. When $i$ is even, $L(\text{abs}(e_1-x)^i)$ = $L((e_1-x)^i)$.

Because $L$ is linear, if $L(e_i) = e_i$ for each $i$ from 0 through $j$ ($j$ is zero or a positive integer), then:

• $L$ reproduces all polynomials up to degree $j$ (that is, $L(f) = f$ whenever $f$ is a polynomial of degree $j$ or less).
• The $0$-th “central moment” is $L(e_0)$ = $L((e_1-x)^0)$ = 1, and for each $i$ from 1 through $j$, $L((e_1-x)^j) = 0$.

Also, because $L$ is linear, the “moments” of degree up to $m$, say, lead to easy ways to find the mapping by $L$ of any polynomial of degree up to $m$, when the polynomial is written in “power” form.

Example: Let $f(x)$ be the polynomial $4x^3 - 6x^2 + 8x^1 - 10$. Then:

\[L(f) = 4L(e_3) - 6L(e_2) + 8L(e_1) - 10L(e_0).\]

“Moments” of Bernstein Polynomials

The following results deal with useful quantities when discussing the error in approximating a function by Bernstein polynomials.

Suppose a coin shows heads with probability $p$, and $n$ independent tosses of the coin are made, where $n$ is 1 or greater. Then the total number of heads $X$ follows a binomial distribution. The following are useful quantities of this distribution.

• $T_{n,r}$: The central moment (moment about the mean) of $X$ is denoted $T_{n,r}(p)$ = $\mathbb{E}[(X-\mathbb{E}[X])^r]$ = $B_n((e_1-p)^r)(p)\cdot n^r$. Formulas for computing this central moment are given in Skorski (2024)⁸.
• $S_{n,r}$: Traditionally, the central moment of $X/n$ or the ratio of heads to tosses is denoted $S_{n,r}(p)$ = $T_{n,r}(p)/n^r$ = $\mathbb{E}[(X/n-\mathbb{E}[X/n])^r]$ = $B_n((e_1-p)^r)(p)$. ($T$ and $S$ are notations of S.N. Bernstein, known for Bernstein polynomials.) $S_{n,r}$ is thus the $r$-th “central moment” of degree-$n$ Bernstein polynomials.
• $M_{n,r}$: The $r$-th central absolute moment of $X/n$ is denoted $M_{n,r}(p)$ = $\mathbb{E}[\text{abs}(X/n-\mathbb{E}[X/n])^r]$ = $B_n(\text{abs}(e_1-p)^r)(p)$. If $r$ is even, $M_{n,r}(p) = S_{n,r}(p)$. $M_{n,r}$ is thus the $r$-th “absolute moment” of degree-$n$ Bernstein polynomials.

The following gives bounds on $M_{n,r}$; some results in approximation theory rely on bounds like these.⁹

Proposition 1: Let $r\ge 0$, and let $\sigma(r,t) = (r!)/(((r/2)!)t^{r/2})$. Then for real numbers $r$ and integers $n$ described in the following table, $M_{n, r}(p)\le \mu_{n,r}/n^{r/2}$, where $\mu_{n,r}$ is as given in the table.

If $r$…	Then $\mu_{n,r}$ is…
Is an even integer.	$\sigma(r,6)$, for positive $n$.
Is an even integer, but not greater than 44.	$\sigma(r,8)$, for positive $n$.
Is 1.	$1/2$, for positive $n$.
Is odd, and $3\le r\le 43$.	$\sqrt{\sigma(r-1,8)\sigma(r+1,8)} = r^{1/2}(r-1)! / (2\cdot 8^{(r-1)/2}((r-1)/2)!)$, for $n\ge 2$.
Is odd and greater than 43.	$\sqrt{\sigma(r-1,6)\sigma(r+1,6)}$, for $n\ge 2$.

Proof: The first row comes from a result of Adell and Cárdenas-Morales (2018)¹⁰. The second row is an improved result of the first, from Molteni (2022)¹¹. The third row follows from Cheng (1983)¹². The fourth and fifth rows follow from the first and second as well as that the absolute central moment for odd $r$ can be overestimated for every integer $n\ge 2$, using Schwarz’s inequality (Weisstein)¹³ (see also Bojanić and Shisha 1975¹⁴ for the case $r=4$). □

Taylor Expansion of Linear Operators

Continuous functions can be “unwrapped” into a Taylor expansion. The linear mapping of those functions also has a Taylor expansion of sorts, which is described next.

Let $f(\lambda)$ have a continuous $s$-th derivative on a compact interval, where $s$ is zero or a positive integer, and let $L(f)$ be a linear operator that maps continuous functions on that interval to functions of that kind. Then:

\[L(f)(\lambda) = L(R_s(f, \lambda)) + \sum_{i=0}^s L((e_1-\lambda)^i)(\lambda)\frac{f^{(i)}(\lambda)}{i!}, \tag{1}\]

where $R_s(f,\lambda)$ is the “remainder” after subtracting from $f$ the degree-$s$ Taylor polynomial of $f$ centered at $\lambda$. (See also Piţul (2007, proof of theorem 5.8)¹⁵.) $R_s(f,\lambda)$ is 0 if $f$ is a polynomial of degree $s$ or less.

If $L$ reproduces constants, so that $L(e_0)=1$, this becomes:

\[L(f)(\lambda) - f(\lambda) = L(R_s(f, \lambda)) + \sum_{i=1}^s L((e_1-\lambda)^i)(\lambda)\frac{f^{(i)}(\lambda)}{i!}.\tag{2}\]

If $L$ reproduces polynomials up to degree $s$, this even reduces to $L(f)(\lambda) - f(\lambda) = L(R_s(f, \lambda))$.

It can be seen from the expansions just given that finding upper bounds for $L_n(f)(\lambda)$ involves:

• Finding upper bounds for $L$’s “central moments” up to the $s$-th order.
• Finding upper bounds for $L(R_s(f,\lambda))$. If $L$ is positive linear, such bounds are given in the section “Bounds for General Positive Linear Operators”. If $L$ is nonpositive linear, bounds are given in the section “Bounds for General Linear Operators”, and this can be helped if $L$ can be written as a difference between two positive linear operators $LA$ and $LB$, so that $L(f) = LA(f) - LB(f)$.¹⁶ See the “Example” section later in this document.

Meanwhile, bounds for the derivatives of $f$ (here, $f^{(i)}$) are often assumed to be known beforehand.

Results on Error Bounds

Some results on error bounds for certain classes of operators.

Bounds for General Positive Linear Operators

The results in this section give bounds that apply to large classes of positive linear operators.

But many classes of positive linear operators, including Bernstein polynomials, generally do not approximate functions with more than two continuous derivatives “better” than functions with only two (Voronovskaya 1932)¹⁷, (DeVore 1972/2006)¹⁸, (Cárdenas-Morales et al. 2012)¹⁹.

In this section:

• $\sigma_i = L((e_i-\lambda)^i)(\lambda)$ (the $i$-th “central moment” of the linear operator $L$ in question).
• $\tau_i = L(\text{abs}(e_i-\lambda)^i)(\lambda)$ (the $i$-th “absolute moment” of the linear operator $L$ in question).
• $\omega_1(f, \cdot)$ is the smallest modulus of continuity of a function $f$ of order 1.
• $\tilde\omega_1(f, \cdot)$ is the smallest concave modulus of continuity of $f$ of order 1.

Lemma 1. Let $f(\lambda)$ be continuous on a compact interval, and let $L$ be a positive linear operator that maps continuous functions on that interval to functions of that kind and reproduces all constants (so that $L(e_0) = 1$ ). Then:

No.	$\text{abs}(L(f)(\lambda)-f(\lambda))\le …$
1	$\tilde\omega_1(f, \tau_1)$.
2	$2 \omega_1(f, (\sigma_2)^{1/2})$.
3	$(1 + (\sigma_2)^{1/2}/h) \omega_1(f, h)$.
4	$(1 + (\sigma_2)/h^2) \omega_1(f, h)$.
5	(Use ineq. 3 if $h<(\sigma_2)^{1/2}$, or ineq. 4 otherwise.)
6	$\tilde\omega_1(f, (\sigma_2)^{1/2})$.
7	$(1 + 1/h^2) \omega_1(f, h\cdot\text{Max}(\sigma_2)^{1/2})$.

In the table, $\text{Max}(\sigma_2)$ is the maximum of $\sigma_2$ over all values of $\lambda$ in the compact interval.

Proof: Inequality 1 follows from a result of Gonska and Meier (1985, theorem 3.1)²⁰. Inequality 2 follows from a result of Shisha and Mond (1968, theorem 1)²¹; inequality 4 comes from another result in the same paper (see also Mamedov (1959)²²); inequality 7 follows from a result of Mond (1978)²³; inequality 3 is a special case of Remark 1.2.5 of Păltănea (2004)²⁴; inequality 5 is from corollary 1.2.2 of the same book; inequality 6, a result of Peetre (1969)²⁵ (also mentioned in Gonska (1998/2023)²⁶, which has an extensive discussion on error bounds for linear operators). □

Remark 1: The moduli of continuity $\omega_1(f, \delta)$ and $\tilde\omega_1(f, \delta)$ offer concise ways to express different error bounds depending on how “regular” $f$ is. Properties of these moduli are given in Sevy 1991²⁷, sec. 2.0.2; Gonska 1985²⁸. For example, let $f$ be continuous on a compact interval. Then:

• $\omega_1(f,\delta)\le\tilde\omega_1(f,\delta)$ (Peetre 1969)²⁹.
• If $f$ is Hölder continuous with exponent $\alpha$ ($0\lt\alpha\le 1$) and constant $M$ or less, $\omega_1(f,\delta)\le\tilde\omega_1(f,\delta)\le M\delta^\alpha$. Indeed, in this case, $f$ admits the continuous and concave modulus of continuity $\omega_1(\delta)=M\delta^\alpha$, where $\delta>0$.
• If $f$:
  1. is Lipschitz continuous with Lipschitz constant $M$ or less, or
  2. has a continuous derivative with maximum absolute value $M$ or less,

  then $\omega_1(f,\delta)\le\tilde\omega_1(f,\delta)\le M\delta$; in case (1) because $f$ is Hölder continuous with Hölder exponent 1, and in case (2) because of a result of Hardy and Littlewood.

Example: Let $f$ and $L$ be as in Lemma 1. If $f$ has a continuous derivative with maximum absolute value $M$ or less, $\text{abs}(L(f)(\lambda)-f(\lambda))\le M (\sigma_2)^{1/2}$; this follows from the combination of Remark 1 and inequality 6 of Lemma 1.

Lemma 2. Let $f(\lambda)$ be continuous on a compact interval, and let $L$ be a positive linear operator that maps continuous functions on that interval to functions of that kind and reproduces all polynomials up to degree 1 (constants and linear functions). Let $h>0$ be a real number. Then:

No.	If $f$ …	Then abs($(L(f)(\lambda)-f(\lambda))$ ≤ …
1	Has a continuous derivative.	$((h+2)^2/(8h))\cdot \omega_1(f^{(1)}, h\cdot\sqrt{\sigma_2}) \cdot\sqrt{\sigma_2}$.
2	Has a continuous derivative.	$\frac{1}{2}(\sigma_2)^{1/2} \tilde\omega_1(f^{(1)}, (\sigma_2)^{1/2})$.
3	Has a Hölder-continuous derivative with Hölder exponent $\alpha$ ($0\lt\alpha\le 1$) and Hölder constant $M$ or less.	$\frac{M}{2}(\sigma_2)^{(1+\alpha)/2}$.
4	Has a Lipschitz-continuous derivative with Lipschitz constant $M$ or less, or has a continuous second derivative with maximum absolute value $M$ or less.	$\frac{M}{2} (\sigma_2)$.

Proof: Inequality 1 is a special case of Theorem 2.19 (in conjunction with Remark 2.21) of Anastassiou (1985), with the interval $[a, b]$, $m=1$ (since the function is defined on all of $[a, b]$), $r=h$, and $x_0$ equal to $\lambda$. Inequality 2 follows from a result of Gonska and Meier (1985, theorem 4.1)²⁰; see also Păltănea and Dimitriu (2016, remark 3)³⁰. Inequalities 3 and 4 follow from inequality 2 because of Remark 1. □

Lemma 3. Let $f(\lambda)$ have a continuous $k$-th derivative on a compact interval, and let $L$ be a positive linear operator that maps continuous functions on that interval to functions of that kind. Let $h>0$ be a real number. Then $L(f)(\lambda) = L(Q_k(f,\lambda))(\lambda) + L(R_k(f,\lambda))(\lambda)$, where:

\[\text{abs}(L(Q_k(f,\lambda)) - f(\lambda))\le \left(\sum_{i=0}^k \frac{\max(\text{abs}(f^{(i)})) \text{abs}(\sigma_i)}{i!}\right),\] \[\text{abs}(L(R_k(f,\lambda)))\le\left(\frac{\tau_k}{k!}+\frac{\tau_{k+1}}{(k+1)!\cdot h}\right)\cdot\omega_1(f^{(k)}, h),\] \[\text{and }\text{abs}(L(R_k(f,\lambda)))\le\max\left(\frac{\tau_k}{k!}, \frac{\tau_{k+1}}{(k+1)!\cdot 2h}\right)\cdot\tilde\omega_1(f^{(k)}, 2h),\] \[\text{and }\text{abs}(L(R_k(f,\lambda)))\le\frac{\tau_k}{k!}\cdot\tilde\omega_1(f^{(k)}, \frac{\tau_{k+1}}{(k+1)\tau_k}),\]

and where:

• $Q_k(f,\lambda)=$ $\sum_{i=0}^k f^{(i)}(\lambda)\cdot(e_0-\lambda)^i/(i!)$ is the degree-$k$ Taylor polynomial of $f$ centered at $\lambda$.
• $R_k(f,\lambda)$ is the Taylor remainder that results from subtracting $Q(f,\lambda)$ from $f$.

Proof: The second to fourth bounds given relate to the Taylor remainder. The second bound comes from Păltănea and Smuc (2019, Theorem 1)³¹; the third bound comes from corollary 3.2 of Dimitriu (2010)³² and Brudnyĭ’s lemma; and the fourth bound follows from the second with $h=\tau_{k+1}/(2(k+1)\tau_k)$ and comes from Gonska et al. (2006)³³, where the compact interval assumed was the closed unit interval; see also Gonska (2007)³⁴, Piţul (2007)¹⁵. See also Anastassiou (1985, theorem 2.31)³⁵.³⁶□

Lemma 4. Let $k$ be zero or a positive integer. Let $f(\lambda)$—

1. have a Lipschitz-continuous $k$-th derivative on a compact interval, with Lipschitz constant $M$ or less, or
2. have a continuous $(k+1)$-th derivative on that interval, with maximum absolute value $M$ or less,

and let $L$ be a positive linear operator that maps continuous functions on that interval to functions of that kind. Then $\text{abs}(L(R_k(f,\lambda)))\le M \tau_{k+1}/((k+1)!)$, where $R_k(f,\lambda)$ is as in Lemma 3.

Proof: Follows from the third bound for $L(R_k(f,\lambda))$ in Lemma 3 in the same manner as inequality 10 of Lemma 2, using Remark 1. □

The following two lemmas are more general, but not as easy to use.

Lemma 4A (special case of Theorem 3.4 in Gonska (1998/2023)²⁶). Let $f(\lambda)$ be continuous on a union of nonempty compact intervals, and let $L$ be a positive linear operator that maps continuous functions on $f$’s domain to bounded functions on that domain. Let $h>0$ be a real number. Then:

\[\begin{multline}\text{abs}(L(f)(\lambda)-f(\lambda))\le\max(\text{Max}(L(e_0)) ,L(\text{abs}(e_1-\lambda))(\lambda))\cdot\tilde\omega_1(f,h)\\\\+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda))\\\\\le(\text{Max}(L(e_0)) +L(\text{abs}(e_1-\lambda))(\lambda))\cdot\tilde\omega_1(f,h)+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda)),\end{multline}\]

where $\text{Max}(L(e_0))$ is the maximum of $L(e_0)$ over $f$’s domain.

Lemma 4B (special case of Theorem 4.7 in Gonska (1998/2023)²⁶). Let $f(\lambda)$ be continuous on a compact interval, and let $L$ be a positive linear operator that maps bounded functions on $f$’s domain to bounded functions on that domain. Let $h>0$ be a real number. Then:

\[\begin{multline}\text{abs}(L(f)(\lambda)-f(\lambda))\le(L(e_0)(\lambda)\\\\+L(\text{ceil}((e_1-\lambda)/h-1))(\lambda))\cdot\omega_1(f,h)\\\\+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda)),\end{multline}\] \[\begin{multline}\text{abs}(L(f)(\lambda)-f(\lambda))\le(L(e_0)(\lambda)\\\\+L(\text{abs}(e_1-\lambda))(\lambda)/h)\cdot\omega_1(f,h)\\\\+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda)).\end{multline}\]

The second inequality also works if $L$ maps from continuous functions instead of from bounded functions.

Notes:

1. Using Lemma 4A requires calculating $L(e_0)$, $\text{Max}(L(e_0))$, and $L(\text{abs}(e_1-\lambda))$, or finding upper bounds for these.
2. Using Lemma 4B requires calculating $L(e_0)$ and either $L(\text{abs}(e_1-\lambda))$ or $L(\text{ceil}((e_1-\lambda)/h-1))$, or finding upper bounds for these values.
3. Unlike Lemma 4A, Lemma 4B is not guaranteed to work if $f$’s domain is not a single compact interval (see Remark 2.5 in Gonska (1998/2023)²⁶).

The following lemma adapts the previous lemmas to the setting of random variables.

Lemma 5. Let $f(\lambda)$ be continuous on a compact interval, and let $Y$ be a random variable taking only values in that interval. Then Lemmas 1 through 4A apply as appropriate to $f$ meeting their conditions, with $L(f)=\mathbb{E}[f(Y)]$ and $\lambda =\mathbb{E}[Y]$.

Proof: With these assumptions there is a positive linear operator $L(f) = \mathbb{E}[f(Y)]$ for $Y$ and $f$, according to Theorem 3.1.1 of Frantz (1984)³⁷, letting $x_o = \lambda$. Then $L(e_0)$ = $\mathbb{E}[e_0(Y)]$ = $\mathbb{E}[1]$ = 1 regardless of $Y$, and $L(e_1)$ = $\mathbb{E}[e_1(Y)]$ = $\mathbb{E}[Y]$ = $\lambda$, so $L$ reproduces all polynomials of degree up to 1. □

Bounds for Remainder of Bernstein Polynomials

The following results specialize the previous ones to the case of Bernstein polynomials $B_n$. They apply to the Bernstein polynomial of the result of subtracting a Taylor polynomial from a function, and are useful when a linear operator contains $B_n(f)$ in its definition and reproduces all polynomials of degree $r$ or less.

Lemma 6: Let $k$ be zero or a positive integer. Let $f(\lambda)$—

1. have a Lipschitz-continuous $k$-th derivative on the closed unit interval, with Lipschitz constant $M$ or less, or
2. have a continuous $(k+1)$-th derivative on that interval, with maximum absolute value $M$ or less.

Then the following bound holds true: $\text{abs}(B_n(R_k(f, \lambda)) \le (M \mu_{k+1})/ ( ((k+1)!) n^{(k+1)/2})$ for every integer $n\ge 2$ (and also for $n=1$ if $k$ is odd), where $\mu_k$ is as defined in Proposition 1.

Proof: Follows from Lemma 4, with $L(f)=B_n(f)$, and from Proposition 1. □

Corollary 1: Let $f(\lambda)$, $k$, and $M$ be as in Lemma 6. Then, for every $0\le\lambda\le 1$:

If $k$ is:	Then $\text{abs}(B_n(R_k(f, \lambda))) \le$ …
0.	$M(1/2)/n^{1/2}$ for every integer $n\ge 1$.
1.	$M(1/8)/n = 0.125M/n$ for every integer $n\ge 1$.
2.	$M(\sqrt{3}/48)/n^{3/2} < 0.3609M/n^{3/2}$ for every integer $n\ge 2$.
3.	$M(1/128)/n^{2} = 0.0078125M/n^{2}$ for every integer $n\ge 1$.
4.	$M(\sqrt{5}/1280)/n^{5/2} < 0.001747/n^{5/2}$ for every integer $n\ge 2$.
5.	$M(1/3072)/n^{3} < 0.0003256/n^{3}$ for every integer $n\ge 1$.

Bounds for General Linear Operators

The results in this section give error bounds for important classes of linear operators (not necessarily positive ones). But, in general, they are harder to use than the ones for positive linear operators, because more has to be computed for nonpositive operators than just the “moments”.

Roughly speaking, the integral of $f(\lambda)$ on the compact interval $[a,b]$ is the “area under the graph” of that function when the function is restricted to that interval. If $f$ is continuous there, this is the value that—

\[\frac{b-a}{n} \sum_{i=1}^n f\left(a+(b-a)(i-\frac{1}{2})/n\right),\tag{2A}\]

approaches as $n$ gets larger and larger.³⁸ The integral of $f(\lambda)$ on $[a,b]$ is denoted $\int_a^b f(\lambda) d\lambda$.

The next two lemmas rely on the so-called Peano kernel theorem, which was originally developed to assess the error in estimating the integral of a function from samples of it³⁹ (for more on this theory, see Brass and Förster 1998⁴⁰; Waldron 1999⁴¹).

Lemma 7. Let $k$ be zero or a positive integer, let $f(\lambda)$ have a continuous $(k+1)$-th derivative on the compact interval $[a, b]$, where that derivative has maximum absolute value $M$ or less. Let $C$ and $c$ be real numbers such that $c\le f^{(k+1)}\le C$ over that interval. Let $L$ be a bounded linear operator that—

• reproduces all polynomials of degree $k$ or less, and
• maps continuous functions (or, if $k=0$, bounded functions) on the interval $[a, b]$ to continuous functions on that interval.

Let $LF(f) = f - L(f)$. Then, whenever $a\le\lambda\le b$:

\[\begin{multline}\text{abs}(LF(f)(\lambda)) = \text{abs}(f(\lambda) - L(f)(\lambda))\\\\\le \frac{C - c}{2}\int_a^b \text{abs}\left(LF(\lgroup e_1-t\rgroup_+^k)(\lambda)/(k!)\right) dt\quad\quad\mbox{(3)}\\\\= \frac{C - c}{2(k!)} \int_a^b \text{abs}\left(LF(\lgroup e_1-t\rgroup_+^k)(\lambda)\right) dt\quad\quad\mbox{(4)}\\\\= \frac{C - c}{2(k!)} \left\lgroup\int_a^\lambda \text{abs}\left(LF(\lgroup e_1-t\rgroup_+^k)(\lambda)\right) dt +\\\\\int_\lambda^b \text{abs}\left(LF(\lgroup e_1-t\rgroup_+^k)(\lambda)\right) dt\right\rgroup\\\\= \frac{C - c}{2(k!)} \left\lgroup\int_a^\lambda \text{abs}\left((\lambda-t)^k - L(\lgroup e_1-t\rgroup_+^k)(\lambda)\right) dt +\\\\ \int_\lambda^b \text{abs}\left(L((e_1-t)_+^k)(\lambda)\right) dt\right\rgroup,\\\\\end{multline}\] \[\begin{multline} \text{abs}(LF(f)(\lambda))\le \frac{M}{k!} \int_a^b \text{abs}\left(LF(\lgroup e_1-t\rgroup_+^k)(\lambda))\right) dt,\quad\quad\mbox{(5)} \end{multline}\]

where the notation $\lgroup x\rgroup^k_+$ is as follows. If $k\gt 0$, this equals $((x+\text{abs}(x))/2)^k$, or $\max(0, x)^k$, and if $k$ is 0, this equals either 1 if $x\ge 0$ or 0 otherwise.

Formulas (3) and (4) are because, in this case, the operator $LF$ equals 0 for every polynomial of degree $k$ or less, so that $LF(e_i)=0$ whenever $0\le i\le k$, so that $LF$ satisfies theorem 3 of Gavrea and Ivan (2015)⁴². Formula (5) is an easy consequence of (4); see also Brass and Förster (1998, theorem 5)⁴⁰.⁴³

Note: The operator—

\[\frac{LF(\lgroup e_1-t\rgroup_+^k)}{k!} = \frac{\lgroup e_1-t\rgroup_+^k - L(\lgroup e_1-t\rgroup_+^k)}{k!},\]

where $k\ge 0$, is called the Peano kernel of order $k+1$ of $LF$ (Brass and Förster 1998)⁴⁰), with a fixed value of $t$ such that $a\le t\le b$. But finding a “closed form” of Peano kernels is relatively hard compared to “raw moments” and “central moments”. Luckily, only an upper bound of the Peano kernel is needed to use the formulas in this lemma.

Lemma 8 (see Theorem 4 of Gavrea and Ivan (2015)⁴²). With the assumptions in Lemma 7, if $LF$ is the difference of two positive linear operators $LA$ and $LB$, so that $LF(f)=LA(f)-LB(f)$ (or $L(f)=f-LA(f)+LB(f)$), and $LA$ and $LB$ both map continuous functions on that interval to functions of that kind, then:

\[\text{abs}(L(f)(\lambda) - f(\lambda))\le \frac{C - c}{(k+1)!} \text{abs}(LA(e_{k+1})(\lambda)),\] \[\text{abs}(L(f)(\lambda) - f(\lambda))\le \frac{2M}{(k+1)!} \text{abs}(LA(e_{k+1})(\lambda)).\]

Lemma 9 (special case of Theorem 3.2 in Gonska (1998/2023)²⁶). Let $f(\lambda)$ be continuous on a union of nonempty compact intervals, and let $L$ be a bounded linear operator that maps continuous functions on $f$’s domain to bounded functions on that domain. Let $h>0$ be a real number. Then for each $\lambda$ in $f$’s domain:

\[\begin{multline}\text{abs}(L(f)(\lambda)-f(\lambda))\le\max((\Vert L\Vert_{op}+\alpha)/2, (\gamma(\beta(\lambda)-L(e_0)(\lambda))+\\\\\text{abs}(L(\text{abs}(e_1-\lambda))(\lambda)))/h)\\\\\cdot\tilde\omega_1(f,h)+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda)),\end{multline}\]

where:

• $\alpha$ is the maximum of $\text{abs}(L(e_0))$ over $f$’s domain.
• $\Vert L\Vert_{op}$ is the operator norm of $L$, where the “norm” used to define the operator norm is the least upper bound on the absolute value.⁴⁴
• $\beta(\lambda)$ is the least upper bound of $\text{abs}(L(g)(\lambda))$ over all continuous functions $g$ on $f$’s domain with a maximum absolute value of 1 or less.
• $\gamma$ is the difference between the highest and lowest value of $\lambda$ in $f$’s domain.

Lemma 10. With the assumptions in Lemma 9, if $L$ reproduces constants, so that $L(e_0)=1$, the inequality in that lemma becomes:

\[\begin{multline}\text{abs}(L(f)(\lambda)-f(\lambda))\le\max((1+\Vert L\Vert_{op})/2, (\gamma(\beta(\lambda)-1)\\\\+\text{abs}(L(\text{abs}(e_1-\lambda))(\lambda)))/h)\cdot\tilde\omega_1(f,h).\end{multline}\]

Lemma 11 (special case of Theorem 4.4 and Corollary 4.5 in Gonska (1998/2023)²⁶). Let $f(\lambda)$ be continuous on a compact interval $[a, b]$, and let $L$ be a bounded linear operator that maps continuous functions on $f$’s domain to bounded functions on that domain. Let $h>0$ be a real number. Then for each $\lambda$ in $f$’s domain:

\[\begin{multline}\text{abs}(L(f)(\lambda)-f(\lambda))\le\big((\beta(\lambda)-\text{abs}(L(e_0)(\lambda)))\cdot(1+(b-a)/h)\\\\+\text{abs}(L(e_0)(\lambda))+\text{abs}(L(\text{abs}(e_1-\lambda))(\lambda))/h\big)\cdot\omega_1(f,h)\\\\+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda)),\end{multline}\]

where $\beta(\lambda)$ is as in Lemma 9.

Lemma 12. With the assumptions in Lemma 11, if $L$ reproduces constants, so that $L(e_0)=1$, the inequality in that lemma becomes:

\[\begin{multline}\text{abs}(L(f)(\lambda)-f(\lambda))\le\big((\beta(\lambda)-1)\cdot(1+(b-a)/h)\\\\+1+\text{abs}(L(\text{abs}(e_1-\lambda))(\lambda))/h\big)\cdot\omega_1(f,h)\end{multline}\]

 

Note: Lemmas 11 and 12 are not guaranteed to work if $f$’s domain is not a single compact interval, unlike Lemmas 9 and 10 (see Remark 2.5 in Gonska (1998/2023)²⁶).

Smoothing Error Bounds

The following results give bounds on the “best possible” error in approximating a function with a “smoother” function, such as a polynomial of degree $n$.

Lemma 13. Let $f(\lambda)$ be continuous on the compact interval $[a, b]$. Then there is at least one polynomial $P$ on that interval of degree up to $n$ (where $n$ is zero or a positive integer) such that the error of $P$ in approximating $f$ is bounded as follows (see Babenko and Kryakin 2019⁴⁵, Babenko and Kryakin 2018⁴⁶, and references therein):

\[\text{abs}(f(\lambda)-P(\lambda))\le\begin{cases} W \cdot \omega_{n+1}(f,\frac{b-a}{n+1}) & \textrm{(Wh)}\\\\ J\cdot\omega_k(f,\frac{\alpha\pi}{n+1}), & \textrm{(JS)}\end{cases}\]

where:

• $k$ is a positive integer.
• $W$ is:
  • 1/2 if $n$ is 0 or 1 (Whitney 1957)⁴⁷.
  • 1 if $2\le n\le 7$.
  • 2 if $8\le n\lt 81999$.
  • $(2+\exp(-2)) (< 2.13534)$ if $n\ge 81999$.
  • $3/4$ if $n=1$ and $f$ is convex (Singh Kaire and Prymak 2023/2025)⁴⁸.
  • $1/2$ if $n=1$, $f$ is convex, and $a=-b$ (Singh Kaire and Prymak 2023/2025)⁴⁸.
• $J$ is (Babenko and Kryakin 2018⁴⁶):
  • 1 if $k=1$ and $\alpha=1$.
  • $3/4+3/(16\alpha^2)$ if $k=2$ and $n\ge 1$.
  • 4.38 if $k=4$, $\alpha=1$, and $n\ge 11$.
  • 2.59 if $k=4$, $\alpha=2$, and $n\ge 11$.
  • 8.9 if $n\ge k(k-1)-1$, $k$ is 2, 4, 6, or 8, and $\alpha$ is 1 or 2.
  • 9.74 if $n\ge k(k-1)-1$, $k\ge 10$, $k$ is even, and $\alpha=2$.
• $\alpha\gt 0$.
• $\omega_{n}(f, h)$ is the smallest modulus of continuity of $f$ of order $n$, with parameter $h$.
• The inequality (JS) is valid only for the compact interval $[-1,1]$.

Note: The inequality (Wh) is also known as Whitney’s inequality; the inequality (JS), the Jackson–Stechkin inequality.

Lemma 13A. With the setting in Lemma 13, if $f$ has a continuous $(n+1)$-th derivative whose maximum absolute value is $M$ or less, then:

\[\text{abs}(f(\lambda)-P(\lambda))\le \frac{M (b-a)^{n+1}}{((n+1)!)\cdot 2^{2n+1}}. \tag{Ph}\]

 

Note: The following are references on the inequality (Ph): Brass and Petras (2011, example 3.1.2)⁴⁹; Phillips (2003, theorem 2.4.6)⁵⁰.

Using properties of moduli of continuity (see Sevy 1991²⁷, sec. 2.0.2; Gonska 1985²⁸), if $f$ has a continuous $(n+1)$-th derivative on $[a, b]$:

\[\text{abs}(f(\lambda)-P(\lambda))\le W \cdot \left(\frac{b-a}{n+1}\right)^{n+1}\max(\text{abs}(f^{(n+1)})),\]

and if $f$ has a continuous $n$-th derivative on that interval:

\[\text{abs}(f(\lambda)-P(\lambda))\le W \cdot \left(\frac{b-a}{n+1}\right)^n\omega_1(f^{(n)}, \frac{b-a}{n+1}).\]

 

Note: Upper bounds similar to those in this section can be found by considering an operator $L$ that maps continuous functions to a family of “smoother” functions (such as polynomials of degree $n$ or less, as in this section); in that case $P$ would be a function in the family with the “best approximation” to $f$. As one example, Marsden (1972)⁵¹ (see also Beutel et al. 2002⁵²) proved an upper bound of this kind using an operator that maps to a family of splines, that is, continuous functions whose pieces are polynomials. So did De Boor (1968)⁵³. Another example is in the result given next.

Lemma 13B. Let $f(\lambda)$ be continuous on the compact interval $[a, b]$, and let $0\lt h\le (b-a)/2$. Then there is at least one function $P$ with a continuous second derivative on that interval such that the error of $P$ in approximating $f$ is bounded as follows:

\[\text{abs}(f(\lambda)-P(\lambda))\le (3/4)\omega_2(f, h),\]

and such that $P$’s first derivative’s absolute value is no more than $(1/2)\omega_1(f,h)$ $+ (3/8)\omega_2(f,h)$ $\le (5/h)\omega_1(f,h)$ and $P$’s second derivative’s absolute value is no more than $(3/(2h^2))\omega_2(f,h)$.

The proof comes from lemmas 2.1 and 2.4 and corollary 2.5 of Gonska and Kovacheva (1994/2024)⁵⁴, where, as mentioned in that paper, lemma 2.1 is a result of Zhuk (1989)⁵⁵.

Example: Suppose it is known that an approximation error is no more than:

\[3\Vert f - P\Vert + 4 \Vert P^{(1)}\Vert + 5\Vert P^{(2)}\Vert,\]

and that this is true for every function $P$ with a continuous second derivative on $[a,b]$ and every continuous function $f$ on that interval. Then with Lemma 13B, this bound can be transformed to the following:

\[\begin{multline}3((3/4)\omega_2(f, h))+4((5/h)\omega_1(f,h))\\\\+5((3/(2h^2))\omega_2(f,h)).\end{multline}\]

Lebesgue Inequality for Certain Linear Operators

For certain operators, the approximation error of a function under that operator can be related to the “best approximation” error between that function and every function the operator maps to.

Lemma 14 (Lebesgue’s lemma or Lebesgue inequality): Suppose the following:

• $S$ is the space of continuous functions on a compact interval, and its “norm” ($\Vert\cdot\Vert$) is the maximum absolute value.
• $f$ is a function in $S$.
• $T$ is a subspace of $S$ that has a “best approximation” to $f$, so that there is a function $g$ in $T$ such that $\Vert f - g\Vert\le\Vert f - h\Vert$ for every function $h$ in $T$ (De Villiers 2012, theorem 5.3.2)⁴.⁵⁶ Also, $T$ admits the same “norm” as $S$.
• $L$ is a linear operator that maps functions in $S$ to functions in $T$. Also, $L$ is idempotent, that is, applying the linear operator twice or more is the same as applying it once, so that $L(L(f))=L(f)$ for every allowed function $f$ (for example, $L$ reproduces all functions in $T$).
• $I$ is the identity operator $I(f)=f$.

Then for every function $P$ in $T$:

\[\begin{multline}\Vert f - L(f)\Vert\le(\Vert I - L\Vert_{op})\cdot\Vert f-P\Vert\quad\text{(Leb2)}\\\\ \le(\Vert I\Vert_{op} + \Vert L\Vert_{op}) \cdot\Vert f-P\Vert\\\\ \le(1 + \Vert L\Vert_{op}) \cdot\Vert f-P\Vert\quad\text{(Leb)}\end{multline}\]

(see also DeVore and Lorentz (1993, p. 30; ch. 5)⁵⁷, Powell (1981, theorem 3.1)⁵⁸, De Villiers (2012, theorem 5.3.2)⁴; for (Leb2) see De Boor (1982, chapter 2)⁵⁹). ⁶⁰

Examples:

1. Let $f$ have a continuous third derivative on the closed unit interval. Combining the inequalities (Leb) and (Wh) leads to the following error bound if $T$ is the set of polynomials up to degree 2:

   \[\begin{multline}\Vert f - L(f)\Vert\le(1+\Vert L\Vert_{op})\cdot 1\cdot \left(\frac{1}{3}\right)^{3}\Vert f^{(3)}\Vert\\\\= (1+\Vert L\Vert_{op})\Vert f^{(3)}\Vert /27.\end{multline}\]

2. The Bernstein polynomial $B_n$ maps continuous functions to polynomials up to degree $n$, but if $n$ is 2 or greater it is not idempotent because it does not reproduce all polynomials up to degree $n$. For example, for $e_2(x)=x^2$, a degree-2 polynomial, $B_n(e_2)=x^2+x(1-x)/n\ne x^2$. Thus, the inequalities (Leb) and (Leb2) do not apply to Bernstein polynomials of degree 2 or greater.

3. The identity operator $I(f)=f$ has operator norm 1 and maps every function to itself. For this operator, the inequality (Leb2) becomes:

   \[\Vert f - I(f)\Vert\le\Vert I - I\Vert_{op}\cdot 0 = 0.\]

Notes:

1. The only positive linear operator that maps continuous to continuous functions and reproduces all polynomials up to degree 2 (constants, linear functions, and quadratic functions) is the identity operator (Păltănea 2004, corollary 1.1.2)²⁴.⁶¹
2. Inequalities similar to (Leb) and (Leb2) may apply to spline operators that map to a continuous function (a spline) that equals a polynomial at subintervals of its domain (for example, Sablonnière 2007⁶²). These operators may have a local approximation property, such that $L(f)(\lambda)$ depends only on the behavior of $f$ near $\lambda$. But it’s not clear to me when inequalities similar to (Leb) and (Leb2) apply to those cases.⁶³

Bounds for Certain Nonlinear Operators

Not all mappings from functions to functions are linear transformations. The results in this section provide approximation errors for a broader class of transformations, including nonlinear ones.

Lemma 15 (from a result in Bede and Gal (2010)⁶⁴; see also Bede et al. (2009)⁶⁵): Let $f(\lambda)$ be continuous, bounded, and nonnegative on an interval. Let $L$ be an operator that maps functions of that kind to functions of that kind and also has the following properties:

1. (Monotone.) For every pair of allowed functions $g$ and $h$, if $g\le h$, then $L(g)\le L(h)$.
2. (Subadditive.) For every pair of allowed functions $g$ and $h$, $L(g+h)\le L(g)+L(h)$.
3. (Positively homogeneous.) $xL(g)=L(xg)$ for every allowed function $g$ and every $x\ge 0$.

Finally, let $h>0$ be a real number. Then the first inequality applies, and the second one also applies if $L(e_0)=1$:

\[\begin{multline} \text{abs}(f(x)-L(f)(x))\le(L(e_0)(x)+L(\text{abs}(e_1-x))(x)/h)\cdot\omega_1(f, h)\\\\+f(x)\cdot\text{abs}(L(e_0)(x)-1),\end{multline}\] \[\text{abs}(f(x)-L(f)(x))\le(1+L(\text{abs}(e_1-x))(x)/h)\cdot\omega_1(f, h),\]

provided that, in either case, $L(\text{abs}(e_1-x))(x)$ (the “absolute moment” of $L$) exists (and is finite or infinite).

Note: An operator meeting conditions 2 and 3 is also called a sublinear operator. Every linear operator is also sublinear. A linear operator is monotone if and only if it is positive. For more on nonlinear operators, see Gal and Niculescu (2023)⁶⁶; on nonlinear approximation, see DeVore (1998)⁶⁷.)

Example: Take the operator $L(f) = (f)^3$. Then $L$ is not a linear operator because $2\cdot L(e_2) = 2e_6$, yet $L(2 e_2) = 8e_6$. For the same reason $L$ is not positively homogeneous (and thus not sublinear). And $L$ is unbounded because $L(z)/z=z^2$ grows without bound as $z$ increases. But $L$ is positive: a nonnegative number raised to a positive power (such as 3) is itself nonnegative.

The following result allows for getting a more refined estimate on a mapping’s upper bounds, using the first-and second order moduli of continuity ($\omega_1$ and $\omega_2$).

Lemma 16 (Gonska and Kovacheva [1994/2024, theorem 4.2]⁵⁴): Let $H$ be an operator that maps continuous functions on a compact interval $[a,b]$ to functions in a Banach space equipped with the “norm” $\Vert\cdot\Vert_B$.⁶⁸ Let $\Vert\cdot\Vert_C$ be the maximum absolute value. If—

• (Boundedness of $H$:) there is $\alpha\ge 0$ such that $\Vert H(f)\Vert_B\le\alpha\Vert f\Vert_C$ whenever $f$ is continuous on the interval,
• there is $\gamma\ge 0$ such that $\Vert H(f+g)\Vert_B\le$ $\gamma(\Vert H(f)\Vert_B$ + $\Vert H(g)\Vert_B)$ whenever $f$ and $g$ are continuous on the interval (or $H$ is linear, so that $\gamma=1$), and
• there are nonnegative numbers $\beta_0$, $\beta_1$, and $\beta_2$ such that $\Vert H(f)\Vert_B\le\beta_0\Vert f\Vert_C+$ $\beta_1\Vert f^{(1)}\Vert_C+\beta_2\Vert f^{(2)}\Vert_C$ whenever $f$ has a continuous second derivative on the interval,

then for every positive $h$ with $h\le (b-a)/2$ and every continuous function $f$ on the interval, the following bound applies:

\[\begin{multline}\Vert H(f)\Vert_B\le\gamma(\beta_0\Vert f\Vert_C+2\beta_1\omega_1(f,h)/h+\\\\(3/4)(\alpha+\beta_0+2\beta_1/h+2\beta_2/h^2)\omega_2(f,h)).\end{multline}\]

Lipschitz-Continuous Operators

An operator $L$ is Lipschitz continuous if it satisfies $\Vert L(f)-L(g)\Vert\le M\Vert f-g\Vert$ for some $M\ge 0$ and all allowed functions $f$ and $g$. (The left-hand side is a “norm” that depends on the space of functions $L$ maps to; the right-hand “norm”, on the functions $L$ maps from.) When this is the case, $M$ is a Lipschitz constant of that operator.

Every bounded linear operator $L$ (with the same “norm” on both sides) is Lipschitz continuous with $M$ equal to its operator norm (because $\Vert L(f)-L(g)\Vert$ = $\Vert L(f-g)\Vert$ $\le\Vert L\Vert_{op}\cdot\Vert f-g\Vert$), and so is every monotone and sublinear operator (Gal and Niculescu 2023)⁶⁶.

The definition of Lipschitz continuity easily leads to…

Lemma 17: If $L$ is a Lipschitz-continuous operator, then for every allowed function $f$:

\[\Vert L(f)\Vert\le M\Vert f-P\Vert,\]

where $P$ is any function such that $L(P)=0$.

Example

This example shows how to find a linear operator’s bounds.

Let $L_n(f)$ be a linear operator inspired by a conjecture I have on polynomial approximation. It is described as follows:

\[L_n(f)(\lambda) = \sum_{i=0}^n \left( W_{2n}\left(f\right)\left(\frac{k}{2n}\right) - W_n\left(f\right)\left(\frac{i}{n}\right)\right)\sigma_{n,k,i}\] \[=\mathbb{E}\left[W_{2n}\left(f\right)\left(\frac{k}{2n}\right) - W_n\left(f\right)\left(\frac{X_k}{n}\right)\right],\]

where:

• $k = 2n\lambda$, where $0\le\lambda\le 1$.
• $W_n(f)$ is a linear operator that approaches $f$ as $n$ increases.⁶⁹
• $X_k$ is a hypergeometric($2n$, $k$, $n$) random variable. This is the number of “good” balls out of $k$ balls taken uniformly at random, all at once, from a bag containing $2n$ balls, $n$ of which are “good”.
• $\sigma_{n,k,i}$ equals ${n\choose i}{n\choose {k-i}}/{2n \choose k}$ and is the probability that $X_k$ equals $i$.

$L_n$ and $W_n$ are generally nonpositive operators. As an example, take $W_n=2f-B_n(f)$. Then $B_n(W_n(f))$ is a linear operator that is the iterated Boolean sum of degree-$n$ Bernstein polynomials, with one iteration; see Güntürk and Li (2021a, Theorem 5)⁷⁰. That paper, among others (for example, Micchelli 1973⁷¹), showed that this operator approaches $f$ at the rate $O(1/n^{3/2})$ if $f$ has a continuous third derivative. (“$O(1/n^{3/2})$” means the error is no greater than a constant times $1/n^{3/2}$ for all values of $n$.)

With this choice of $W_n$, $L_n$ becomes:

\[L_n(f)(\lambda) = \sum_{i=0}^n\left((2f\left(\frac{k}{2n}\right) - B_{2n}(f)\left(\frac{k}{2n}\right)) - (2f\left(\frac{i}{n}\right) - B_n(f)\left(\frac{i}{n}\right))\right) \sigma_{n,k,i}\] \[= \mathbb{E}\left[(2f\left(\frac{k}{2n}\right) - B_{2n}(f)\left(\frac{k}{2n}\right)) - (2f\left(\frac{X_k}{n}\right) - B_n(f)\left(\frac{X_k}{n}\right))\right]\] \[= \sum_{i=0}^n\left((2f\left(\frac{k}{2n}\right) + B_{n}(f)\left(\frac{i}{n}\right))\right)\sigma_{n,k,i} - \sum_{i=0}^n \left((2f\left(\frac{i}{n}\right) + B_{2n}(f)\left(\frac{k}{n}\right))\right) \sigma_{n,k,i}\] \[= LA_n(f)(\lambda) - LB_n(f)(\lambda).\]

Here, $LA_n$ and $LB_n$ are positive linear operators, making it easier to assess their approximation properties.

It will be shown that, if $f$ has a continuous third derivative, the rate of $L_n$ towards zero is $O(M/n^{3/2})$, where $M$ is the maximum absolute value of $f$ and its derivatives up to the third derivative. The proof of this relies on exact expressions of $L_n$’s “raw moments” and “central moments”, and those for the combined operator $(LA_n+LB_n)$.

The following are some of these values and those for related operators:

• $L_n(e_0)(x) = L_n((e_1-x)^0)(x) = 0$.
• $L_n(e_1)(x) = L_n((e_1-x)^1)(x) = 0$.
• $L_n(e_2)(x) = L_n((e_1-x)^2)(x)$ = $3x(x - 1)/(2n(2n-1))$ = $O(1/n^2)$.
• $L_n(e_3)(x)$ = $n^3 x^2(2nx - 4x + 3)/(2n - 1)$.
• $LA_n((e_1-x)^2)(x)$ = $-x(3n - 2)\cdot(x - 1)/(n(2n-1))$ = $O(1/n)$.
• $LB_n((e_1-x)^2)(x)$ = $-x(6n - 1)\cdot(x - 1)/(2n(2n-1))$ = $O(1/n)$.
• $(LA_n+LB_n)((e_1-x)^2)(x)$ = $LA_n(\text{abs}(e_1-x)^2)(x) + LB_n(\text{abs}(e_1-x)^2)(x)$ = $-x(12n - 5)\cdot(x - 1)/(2n(2n - 1)) = O(1/n)$.

To find values like those just listed, it is useful to calculate raw moments (Wang et al. 2023)⁷² and central moments (Weisstein)⁷³ of hypergeometric random variables (such as $X_k$). Indeed, if $g(y)=W_{2n}(e_r;k/(2n))-W_n(e_r;y)$ is a polynomial in $y$ of degree $r$ or less, then $L_n(e_r)$ can be found using a Taylor expansion, namely as—

\[L_n(e_r) = \sum_{i=0}^r \mathbb{E}[(X_k/n-\mathbb{E}[X_k/n])^i]\frac{g^{(i)}(\mathbb{E}[X_k/n])}{i!}\] \[= \sum_{i=0}^r \frac{\mathbb{E}[(X_k-\mathbb{E}[X_k])^i]}{n^i}\frac{g^{(i)}(k/(2n))}{i!},\]

where the derivatives are taken with respect to $y$, and where $\mathbb{E}[(X_k-\mathbb{E}[X_k])^i]$ is the $i$-th central moment of $X_k$.

The first step is to find the Taylor expansion of $L_n(f)(\lambda)$. Given that $L_n((e_1-x)^0)(x)$ = $L_n((e_1-x)^1)(x)$ = 0, this becomes:

\[L_n(f)(\lambda) = L_n(R_3(f, \lambda)) + \sum_{i=2}^3 L_n((e_1-\lambda)^i)(\lambda)\frac{f^{(i)}(\lambda)}{i!},\] \[\text{abs}(L_n(f)(\lambda)) \le \Vert L_n(R_3(f, \lambda))\Vert + \Vert L_n((e_1-\lambda)^2)\Vert\cdot\Vert f^{(2)}\Vert /2\] \[+ \Vert L_n((e_1-\lambda)^3)\Vert\cdot\Vert f^{(3)}\Vert /6.\]

The function $\text{abs}(L_n((e_1-x)^3)(x))$ has its maximum at $x=1/2-\sqrt{3}/6$; and $\text{abs}(L_n((e_1-x)^2)(x))$ has its maximum at $x=1/2$, so:

\[\text{abs}(L_n(f)(\lambda)) \le \Vert L_n(R_3(f, \lambda))\Vert + \text{abs}(\frac{3\lambda(\lambda - 1)}{2n(2n-1)})\Vert f^{(2)}\Vert /2\] \[+ \Vert L_n((e_1-\lambda)^3)\Vert\cdot\Vert f^{(3)}\Vert /6\] \[\le \Vert L_n(R_3(f, \lambda))\Vert + \frac{3}{8n(2n-1)}\Vert f^{(2)}\Vert /2\] \[+ \frac{\sqrt{3} (6 n - 5)}{24 n^{2} (2 n - 1)}\Vert f^{(3)}\Vert /6.\]

Meanwhile the remainder is estimated as follows, using the proof of corollary 2.3 of Gonska et al. (2006)⁷:

\[\Vert L_n(R(f, \lambda))\Vert \le \frac{1}{6} \Vert f^{(3)}\Vert\cdot\Vert (LA_n+LB_n)(\text{abs}(e_1-\lambda)^3)\Vert .\]

In turn, using Schwarz’s inequality (see proof of the same paper’s corollary 2.1):

\[\Vert (LA_n+LB_n)(\text{abs}(e_1-\lambda)^3)\Vert \le (\Vert (LA_n+LB_n)((e_1-\lambda)^4)\Vert )^{1/2}\] \[\times (\Vert (LA_n+LB_n)((e_1-\lambda)^2)\Vert )^{1/2} \le \frac{3\sqrt{3}}{8n^{3/2}}.\]

(The expression in the middle takes its maximum at $\lambda = 1/2$; the right-hand side is an upper bound of that expression for all positive integers $n$.) Altogether:

\[\Vert L_n(f)\Vert \le \frac{3}{8n(2n-1)}\frac{1}{2}\Vert f^{(2)}\Vert\] \[+ \left(\frac{3\sqrt{3}}{8n^{3/2}} + \frac{\sqrt{3} (6 n - 5)}{24 n^{2} (2 n - 1)}\right)\frac{1}{6}\Vert f^{(3)}\Vert = LC_n(f)\] \[\le 0.1875 \frac{\Vert f^{(2)}\Vert }{n^{3/2}} + \frac{5\sqrt{3}}{72} \frac{\Vert f^{(3)}\Vert }{n^{3/2}} \le \frac{0.3078 M}{n^{3/2}} = O(1/n^{3/2}).\]

If $n\ge 2$ is an integer, $LC_n(f)\le 0.2165 M/n^{3/2}$.

Example: An Interesting Linear Operator

For a continuous function $f$ on the closed unit interval and for nonnegative integers $m$ and $n$, let $H_{n,m}$ be a linear operator as follows:

\[H_{n,m}(f)=B_n(f) + \text{Lag}_m(f) - B_n(\text{Lag}_m(f)),\]

where $B_n$ is the degree-$n$ Bernstein polynomial (see “Bernstein Form and Bernstein Polynomials”, earlier) and $\text{Lag}_m$ is the polynomial of degree up to $m$ that equals $f$ at “$m+1$ distinct points on” the closed unit interval. This operator was mentioned in Remark 2 of Gavrea and Ivan (2018)⁷⁴, but appears not to have been studied elsewhere.

It is known that $Lag_m$ is a linear operator and reproduces all polynomials of degree $m$ or less, so that $Lag_m(e_i) = e_i$ whenever $0\le i\le m$ is an integer. Thus, if $f$ is such a polynomial, $B_n(f)=B_n(Lag_m(f))$ and therefore $H_{n,m}(f)$ = $Lag_m(f)=f$, and therefore $H_{n,m}(e_i)=e_i$ whenever $0\le i\le m$ is an integer.

Because $H_{n,m}$ is linear and reproduces all polynomials up to degree $m$, the following holds if $f$ has a continuous $m$-th derivative:

\[H_{n,m}(f)(\lambda) - f(\lambda) = H_{n,m}(R_m(f, \lambda))(\lambda)\] \[=B_n(R_m(f,\lambda)) + \text{Lag}_m(R_m(f,\lambda)) - B_n(\text{Lag}_m(R_m(f,\lambda))).\]

With the help of Lemma 6, the following holds if $n$ is also 2 or greater and $m$ is a positive integer:

\[\Vert H_{n,m}(f)(\lambda)\Vert \le \frac{\Vert f^{(m)}\Vert \mu_{m}}{ (m!) n^{m/2}} + \Vert \text{Lag}_m(R_m(f,\lambda)) - B_n(\text{Lag}_m(R_m(f,\lambda)))\Vert\]

where $\mu_r$ is as in Proposition 1.

It will be helpful to estimate the second derivative of $\text{Lag}_m(f)$.

Given that that function is a polynomial of degree $m$ or less, this can be estimated as:

\[\text{abs}(\text{Lag}_m(f)^{(2)}(\lambda))\le \Vert \text{Lag}_m\Vert_{op}\cdot\Vert f\Vert\cdot M(m),\tag{E}\]

where:

• $\Vert Lag_m\Vert_{op}$ is the operator norm of $Lag_m$, which in this case equals its Lebesgue constant, which will vary depending on the points on the closed unit interval where the polynomial meets (interpolates) $f$ (Ibrahimoglu 2016)⁷⁵.
• $M(m) = (4/3)\cdot \max(2,m)^2(\max(2,m)^2-1)$. This is an upper bound on the maximum absolute value of a polynomial’s second derivative (on the closed unit interval) when that polynomial has a maximum absolute value of 1 (on that interval). This uses the following lemma based on one proved for the interval $[-1,1]$ by V. Markov in 1892 (see also Schaeffer and Duffin 1938 ⁷⁶).

Lemma: Let $p(\lambda)=c_0 \lambda^0 + … + c_n \lambda^n$ be a polynomial on the interval $[a,b]$, where $c_0$, …, $c_n$ are real numbers and $c_n$ is not zero. If $\Vert p\Vert\le 1$, then $\Vert p^{(2)}\Vert\le (b-a)^2 n^2 (n^2-1)/3$.

An error bound for $H_{n,m}$ can also be written as:

\[\begin{multline}H_{n,m}(f) - f=B_n(f) + \text{Lag}_m(f) - B_n(\text{Lag}_m(f)) - f\\\\=(B_n(f) - f) + (\text{Lag}_m(f) - B_n(\text{Lag}_m(f))),\end{multline}\] \[\text{abs}((H_{n,m}(f) - f)(\lambda))\le\text{abs}(B_n(f) - f) + \text{abs}(B_n(\text{Lag}_m(f)) - \text{Lag}_m(f)),\]

so now there are two error bounds to find: one for $f$ and the other for $\text{Lag}_m(f)$. And, if $f$ has a continuous second derivative, both have the same form:

\[B_n(g)\le \Vert g^{(2)}\Vert/(8n).\]

(This follows from Lorentz (1964)⁷⁷ and the well-known fact that $\Vert g^{(2)}\Vert$, the maximum absolute value of $g$’s second derivative, is an upper bound of $g$’s first derivative’s smallest Lipschitz constant.)

Altogether, if $f$ has a continuous second derivative and $m$ is fixed:

\[\text{abs}((H_{n,m}(f) - f)(\lambda))\le \frac{\Vert f^{(2)}\Vert}{8n} + \frac{\Vert \text{Lag}_m\Vert_{op}\cdot\Vert f\Vert\cdot M(m)}{8n}.\]

It is suspected that, using $(E)$, the following is true if $f$ has a continuous $m$-th derivative and $m$ is a positive integer:

\[H_{n,m}(f)(\lambda) - f(\lambda) = B_n(R_m(f,\lambda)) + \lgroup\text{Lag}_m(f) - B_n(\text{Lag}_m(f))\rgroup,\]

so that, using Lemma 6 again, there is a better error bound if $n$ is 2 or greater:

\[\text{abs}((H_{n,m}(f) - f)(\lambda))\le \frac{\Vert f^{(m)}\Vert \mu_{m}}{ (m!) n^{m/2}} + (\Vert Lag_m\Vert_{op}\cdot\Vert f\Vert\cdot M(m)) \frac{\mu_{m}}{ (m!) n^{m/2}}.\]

Example: The Lorentz Operators

The Lorentz operators were introduced by Lorentz (1963)⁷⁸ and studied by Holtz et al. (2011)⁷⁹.

This section touches on the Lorentz operator of order 2, defined as—

\[Q_{n,2}(f)(\lambda)=B_n(f)(\lambda)-\frac{\lambda(1-\lambda)}{2n} B_n(f^{(2)})(\lambda)\] \[=B_n\left(f(e_1)-\frac{\lambda(1-\lambda)}{2n}f^{(2)}(e_1)\right)(\lambda),\]

where $0\le\lambda\le 1$.⁸⁰

This operator is a nonpositive linear operator.

• Unlike in the previous examples, this operator takes in only continuous functions with a second derivative.
• The operator maps those functions to polynomials of degree up to $n+2$.

Because $Q_{n,2}(e_i) = e_i$ if $i$ is 0, 1, or 2, the operator reproduces all polynomials of degree 2 or less (for another proof, see Lemma 14 of Holtz et al. 2011⁷⁹). (The Lorentz operators of order 0 and 1 are simply the Bernstein polynomials.)

$Q_{n,2}$ can be bounded as follows:

\[\Vert Q_{n,2}(f)\Vert\le\Vert B\Vert_{op}\cdot\Vert f\Vert+\frac{1}{8n}\Vert B\Vert_{op}\cdot\Vert f^{(2)}\Vert = \Vert f\Vert+\Vert f^{(2)}\Vert/(8n),\]

where $\Vert B\Vert_{op} = 1$ (De Villiers 2012, Theorem 5.2.2)⁴ is the operator norm of the Bernstein polynomials.

Some of the “moments” of this operator are:

• $Q_{n,2}(e_3)=\frac{x \left(n^{2} x^{2} + 2 x^{2} - 3 x + 1\right)}{n^{2}}.$
• $Q_{n,2}((e_1-\lambda)^3)=\frac{x \left(2 x^{2} - 3 x + 1\right)}{n^{2}}.$

Let $LF = f - Q_{n,2}(f)$ be the error in approximating $f$ with $Q_{n,2}(f)$.

It is suspected that—

\[\Vert LF(f)\Vert \le \frac{C \Vert f^{(3)}\Vert}{n^{3/2}},\]

for some $C>0$. Specifically, using Lemma 6 and Corollary 1, it is suspected that—

\[\Vert LF(f)\Vert \le \frac{\mu_3\Vert f^{(3)}\Vert}{6\cdot n^{3/2}}+\frac{1}{8n}\frac{\mu_1\Vert f^{(3)}\Vert}{1\cdot n^{1/2}}\le(0.06015+0.0625)\frac{\Vert f^{(3)}\Vert}{n^{3/2}}.\]

The operator norm for $Q_{n,2}$ requires some care to define. Because $Q_{n,2}$ doesn’t map from all continuous functions, a different kind of “norm” than the maximum absolute value is needed to define this operator norm. Denote $C^k$, where $k\ge 0$, as the space of functions with a continuous $k$-th derivative on the closed unit interval. The following “norm” gives $C^k$ the property that a sum of functions in that space converges whenever the sum of “norms” of those functions is finite⁸¹:

\[\Vert f\Vert_{C^k} = \begin{cases}\Vert f\Vert + \Vert f^{(k)}\Vert & \text{if }k\gt 0,\\\\\Vert f\Vert & \text{if }k=0.\end{cases}\]

Using the $C^2$ “norm” to define the operator norm of $Q_{n,2}$, that operator can be bounded as follows:

\[\Vert Q_{n,2}(f)\Vert\le\Vert f\Vert+\Vert f^{(2)}\Vert/(8n)\le\Vert f\Vert_{C^2}+\Vert f\Vert_{C^2}/(8n)=\Vert f\Vert_{C^2}\frac{9}{8n},\]

so the operator norm satisfies $\Vert Q_{n,2}\Vert_{op}\le 9/(8n)$.

Probabilistic Interpretations of Linear Operators

The Bernstein polynomials featured in a proof in 1912 of the result that any continuous function on a compact interval can be approximated as well as desired by polynomials (Bernstein 1912)⁸². That proof used probability theory. In a series of papers, Adell and De la Cal use probability theory to interpret a number of linear operators in addition to those polynomials (Adell and De la Cal 1996⁸³, 1995⁸⁴).

Conclusion and Ways to Improve This Article

Many academic papers and books present results in approximation theory, especially error bounds, with hidden constants and without giving upper bounds for such constants. For example, a result may say the following about an operator (approximation scheme) $L$ and a function $f(\lambda)$. If certain conditions are met, then:

\[\text{abs}(f(\lambda) - L(\lambda)) \le C\omega_1(f, 1/n^{1/2}),\text{ or }\] \[\text{abs}(f(\lambda) - L(\lambda)) \le D\Vert f^{(3)}\Vert/n^{3/2},\]

where $C$ and $D$ are unspecified constants with no upper bounds given. Or:

\[\text{abs}(f(\lambda) - L(\lambda)) = O(1/n^{1/2}),\]

where $O(1/n^{1/2})$ is a function whose absolute value is no more than an unspecified constant times $1/n^{1/2}$. (For example, compare Sevy 1991²⁷ with Gonska and Zhou 1994⁸⁵ and Holtz et al. 2011⁷⁹.)

It was a goal of this article to catalog general-purpose error bounds without such hidden constants.

To improve this article, explicit error bounds (with no hidden constants) of the following kinds are sought:

• Inequalities similar to the Lebesgue inequality for certain operators (not necessarily linear ones) where the normal Lebesgue inequality doesn’t apply (say, spline operators with local approximation, or numerical integration rules). For example, inequalities of the following forms, where $C$ is an explicitly given positive constant:
  • $\Vert f-L(f)\Vert$ is no more than $C$ times the smallest $\Vert f-P\Vert$ over all functions $P$ mapped to by $L$.
  • $\Vert f-L(f)\Vert \le (C \Vert I - L\Vert_{op})$ times the smallest $\Vert f-P\Vert$ over a subset of functions $P$ mapped to by $L$.
  • $\Vert f-L(f)\Vert \le (C\Vert I - L\Vert_{op})$ times the smallest $\Vert f^{(k)}-P^{(k)}\Vert$ over a subset of functions $P$ mapped to by $L$, where $k\ge 1$ (for example, De Boor 1975⁸⁶).
  • Inequalities of the foregoing kinds where $L$ and $f$ are restricted to some compact subinterval.
• General-purpose upper bounds on the error when approximating a function with:
  1. Polynomials, especially polynomials in Bernstein form with nonnegative coefficients.
  2. Ratios of polynomials described in (1).
  3. Convex combinations of functions described in (1) or (2). A convex combination has the form $c_0 f_0(\lambda) + c_1 f_1(\lambda) + …$ where $c_i$ are nonnegative and sum to 1.
  4. Compositions of functions described in (1), (2), or (3) (for example, Yeon 2025)⁸⁷. A composition of functions $f$ and $g$ is a function like $f(g(\lambda))$.
• Inequalities similar to Lemmas 13 and 13B that give the “best possible” error when approximating functions with “smoother” functions. For example, results like: If $f$ is continuous, there is at least one function $P$ in a subset of continuous functions such that $\Vert f-P\Vert\le …$ and $\Vert P^{(1)}\Vert\le …$, and so on up to $\Vert P^{(k)}\Vert\le …$, for some value of $k$.
• Easy-to-use upper bounds for estimating Peano kernels, such as the Peano kernels corresponding to the linear Lorentz operators.
• Additional error bounds for nonlinear operators.
• Error bounds that are sharper than those in Lemmas 9 to 12 for functions with a continuous $k$-th derivative for some $k\ge 2$.

In addition, the following will be helpful.

• Results on ways to rewrite a nonpositive linear operator into a difference of two positive linear operators, as used in Lemma 8 and the first Example section.
• Does Lemma 7 also work for linear operators, such as the Lorentz operator, whose norm is not based on the maximum absolute value?
• In which cases are the Lebesgue inequalities (Leb) and (Leb2) true even if the set of functions $L$ maps to is not a (linear) subspace?
• Given the work by Rohwer (2005)⁸⁸, are the inequalities (Leb) and (Leb2) true when—
  • $L$ is nonlinear,
  • $L$ is Lipschitz continuous, replacing $\Vert L\Vert_{op}$ with the Lipschitz constant of $L$, and
  • both $L$ and $I-L$ are idempotent?

Also:

• Is it helpful to introduce the concept of functions of bounded variation in this article?

Those are not all the possible ways this article can be improved.

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Notes