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Notes on Approximation Theory

Peter Occil

Some notes that may be useful when finding approximation error bounds that are explicit, with no hidden constants and without introducing transcendental or trigonometric functions.

The notes generally relate to finding bounds on how close a polynomial is to a single-variable function on a closed interval. The mapping from a function to a function (in this case, 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

Notation and Definitions

For definitions of continuous, derivative, convex, concave, bounded, 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.
  • 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 $a$.
  • An operator $L$ is positive if it has the property that, if an allowed function $f$ is nonnegative on its domain, so is $L(f)$.1
  • The operator norm of an operator $L$ is the maximum absolute value of $L(f)$ over all allowed functions $f$ with a maximum absolute value 1 or less. This assumes $L$ maps continuous functions on a closed interval to functions of that kind.
  • 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 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".

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 the real numbers $a_0, ..., a_n$ are the polynomial's Bernstein coefficients.2

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

"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)3.
  • "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)4.
  • $S_{n,r}$: 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.5

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)6. The second row is an improved result of the first, from Molteni (2022)7. The third row follows from Cheng (1983)8. The fourth and fifth rows follow from the first and second as well as that the absolute central moment for odd $r$ can be bounded for every integer $n\ge 2$, using Schwarz's inequality (Weisstein)9 (see also Bojanić and Shisha 197510 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 closed 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)11.) $R_s(f,\lambda)$ 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 $L(f) = LA(f) - LB(f)$.12 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.

In this section, $\Vert g\Vert _\infty$ is the maximum of the absolute value of (the continuous function) $g$ on its domain.

Bounds for General Positive Linear Operators

The following results give bounds that apply to large classes of positive linear operators. 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).
  • $\omega_1(f, \delta)$ is the smallest modulus of continuity of a function $f$ of order 1, with parameter $\delta$.
  • $\tilde\omega_1(f, \delta)$ is the smallest concave modulus of continuity of $f$ of order 1, both with parameter $\delta$.

Lemma 1. Let $f(\lambda)$ be continuous on a closed 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 | $\tilde\omega_1(f, (\sigma_2)^{1/2})$. |

Proof: Inequality 1 follows from a result of Gonska and Meier (1985, theorem 3.1)13. Inequality 2 follows from a result of Shisha and Mond (1968, theorem 1)14; inequality 4 comes from another result in the same paper (see also Mamedov (1959)15); inequality 3 follows from a result of Mond (1978)16; inequality 5, a result of Păltănea (2004, corollary 1.2.2)17; inequality 6, a result of Peetre (1969)18 (also mentioned in Gonska (1998/2023)19, 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 199120, sec. 2.0.2; Gonska 198521. For example, let $f$ be continuous on a closed interval. Then:

  • $\omega_1(f,\delta)\le\tilde\omega_1(f,\delta)$ (Peetre 1969)22.
  • If $f$ is Hölder continuous with Hölder exponent $\alpha$ ($0\lt\alpha\le 1$) and Hölder 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 $f$ is Lipschitz continuous with Lipschitz constant $M$ or less, $\omega_1(f,\delta)\le\tilde\omega_1(f,\delta)\le M\delta$, given that Lipschitz-continuous functions are Hölder continuous with Hölder exponent 1. The same bound holds true if $f$ instead has a continuous derivative with maximum absolute value $M$ or less, since in this case (by a result of Hardy and Littlewood) $f$ is Lipschitz continuous with Lipschitz constant $M$ or less.

□

Example: Let $f$ and $L$ be as in Lemma 1. If $f$ is Lipschitz continuous with Lipschitz constant $M$ or less, or 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 closed 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 $\text{abs}(L(f)(\lambda)-f(\lambda))\le ... $ |

  • | --- | ----- | | 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)13; see also Păltănea and Dimitriu (2016, remark 3)23. 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 closed 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)24; the third bound comes from corollary 3.2 of Dimitriu (2010)25 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)26, where the closed interval assumed was the closed unit interval; see also Gonska (2007)27, Piţul (2007)11. See also Anastassiou (1985, theorem 2.31)28.29□

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

  1. have a Lipschitz-continuous $k$-th derivative on a closed 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. In both, $\Vert L\Vert$ is the operator norm of $L$.

Lemma 4A (special case of Theorem 3.4 in Gonska (1998/2023)19). Let $f(\lambda)$ be continuous on a closed interval or a closed subset thereof, 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:

$$\text{abs}(L(f)(\lambda)-f(\lambda))\le\max(\Vert L\Vert ,L(\text{abs}(e_1-\lambda))(\lambda))\cdot\tilde\omega_1(f,h)$$

$$+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda))$$

$$\le(\Vert L\Vert +L(\text{abs}(e_1-\lambda))(\lambda))\cdot\tilde\omega_(f,h)+\text{abs}(L(e_0)(\lambda)-f(\lambda))\cdot\text{abs}(f(\lambda)),$$

Lemma 4B (special case of Theorem 4.7 in Gonska (1998/2023)19). Let $f(\lambda)$ be continuous on a closed 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:

$$\text{abs}(L(f)(\lambda)-f(\lambda))\le(L(e_0)(\lambda)+L(\text{ceil}((e_0-\lambda)/h-1))(\lambda))\cdot\omega_1(f,h)$$

$$+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda)),$$

$$\text{abs}(L(f)(\lambda)-f(\lambda))\le(L(e_0)(\lambda)+L(\text{abs}(e_0-\lambda))(\lambda)/h)\cdot\omega_1(f,h)$$

$$+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda)).$$

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

Note: Unlike Lemma 4A, Lemma 4B doesn't work for arbitrary closed subsets of $f$'s domain (see Remark 2.5 in Gonska (1998/2023)19.

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

Lemma 5. Let $f(\lambda)$ be continuous on a closed interval, and let $Y$ be a random variable taking only values in that interval. Then Lemmas 1 through 4B 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)30, 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

Roughly speaking, the integral of $f(\lambda)$ on the closed 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—

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

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

Lemmas 7 and 8, which give error bounds for important classes of linear operators (not necessarily positive ones), 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 it31 (for more on this theory, see Brass and Förster 199832; Waldron 199933).

Lemma 7. Let $k$ be zero or a positive integer, let $f(\lambda)$ have a continuous $(k+1)$-th derivative on the closed interval $[a, b]$, let $M$ be its maximum absolute value, and 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.

Then:

$$\text{abs}(LF(f)(\lambda)) = \text{abs}(f(\lambda) - L(f)(\lambda))$$

$$\le \frac{C - c}{2} \frac{1}{k!}\int_a^b \text{abs}\left(LF((e_1-t)_+^k)(\lambda))\right) dt\tag{3}$$

$$= \frac{C - c}{2(k!)} \int_a^b \text{abs}\left((\lambda-t)_+^k-L((e_1-t)_+^k)(\lambda)\right) dt\tag{4}$$

$$\le \frac{M}{k!} \int_a^b \text{abs}\left((\lambda-t)_+^k-L((e_1-t)_+^k)(\lambda)\right) dt,\tag{5}$$

where $LF(f) = f - L(f)$, and the notation $(x)_+^k$ is as follows. If $k\gt 0$, this equals $((x+\text{abs}(x))/2)^k$, or $\max(0, x)^k$, and $k$ 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 on 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)34. Formula (5) is an easy consequence of (4); see also Brass and Förster (1998, theorem 5)32.35

Lemma 8 (see Theorem 4 of Gavrea and Ivan (2015)34). 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:

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

Lemma 9 (special case of Theorem 3.2 in Gonska (1998/2023)19). Let $f(\lambda)$ be continuous on a closed interval or a closed subset thereof, 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:

$$\text{abs}(L(f)(\lambda)-f(\lambda))\le\max((\Vert L\Vert+\alpha)/2, (\gamma(\beta(\lambda)-L(e_0)(\lambda))+\text{abs}(L(\text{abs}(e_1-\lambda))(\lambda)))/h)$$

$$\cdot\tilde\omega_1(f,h)+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda)),$$

where $\alpha$ is the maximum of $\text{abs}(L(e_0))$ over $f$'s domain; $\beta(\lambda)$ is the maximum of $\text{abs}(L(g)(\lambda))$ over all continuous functions $g$ on $f$'s domain with a maximum absolute value of 1 or less; and $\gamma$ 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:

$$\text{abs}(L(f)(\lambda)-f(\lambda))\le\max((1+\Vert L\Vert)/2, (\gamma(\beta(\lambda)-1)+\text{abs}(L(\text{abs}(e_1-\lambda))(\lambda)))/h)\cdot\tilde\omega_1(f,h).$$

Lemma 11 (special case of Theorem 4.4 and Corollary 4.5 in Gonska (1998/2023)19). Let $f(\lambda)$ be continuous on a closed 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:

$$\text{abs}(L(f)(\lambda)-f(\lambda))\le\big((\beta(\lambda)-\text{abs}(L(e_0)(\lambda)))\cdot(1+(b-a)/h)$$

$$+\text{abs}(L(e_0)(\lambda))+\text{abs}(L(\text{abs}(e_1-\lambda))(\lambda))/h\big)\cdot\omega_1(f,h)$$

$$+\text{abs}(L(e_0)(\lambda)-1)\cdot\text{abs}(f(\lambda)),$$

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:

$$\text{abs}(L(f)(\lambda)-f(\lambda))\le\big((\beta(\lambda)-1)\cdot(1+(b-a)/h)$$

$$+1+\text{abs}(L(\text{abs}(e_1-\lambda))(\lambda))/h\big)\cdot\omega_1(f,h)$$

Whitney's Inequality on Polynomial Errors

The following inequality gives a bound on the "best possible" error that a polynomial of degree $n$ can achieve in approximating a function.

Let $n$ be zero or a positive integer, let $f(\lambda)$ be continuous on a closed interval $[a, b]$, and let $P$ be a polynomial of degree $n$ or less with the least maximum absolute difference between $f$ and the polynomial on that interval. Then the error of $P$ in approximating $f$ is bounded as follows (see Babenko and Kryakin 201936):

$$\Vert f-P\Vert _\infty\le W \cdot \omega_{n+1}(f,\frac{b-a}{n+1}),$$

where—

  • $W$ is:
    • 1 if $n\le 7$.
    • $(2+\exp(-2)) (< 2.13534)$ if $n\ge 8$.
    • $3/4$ if $n=1$ and $f$ is convex (Singh Kaire and Prymak 2023/2025)37.
    • $1/2$ if $n=1$, $f$ is convex, and $a=-b$ (Singh Kaire and Prymak 2023/2025)37.
  • $\omega_{n}(f, h)$ is the smallest modulus of continuity of $f$ of order $n$, with parameter $h$.

Using properties of moduli of continuity (see Sevy 199120, sec. 2.0.2; Gonska 198521), if $f$ has a continuous $(n+1)$-th derivative on $[a, b]$:

$$\Vert f-P\Vert _\infty\le W \cdot \left(\frac{b-a}{n+1}\right)^{n+1}\Vert f^{(n+1)}\Vert _\infty,$$

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

$$\Vert f-P\Vert _\infty\le W \cdot \left(\frac{b-a}{n+1}\right)^n\omega_1(f^{(n)}, \frac{b-a}{n+1}).$$

Another Inequality on Polynomial Errors

Like Whitney's inequality, the following gives a bound on the "best possible" error between a polynomial and a function.

Let $n$ be zero or a positive integer, let $f(\lambda)$ have a continuous $(n+1)$-th derivative on the closed interval $[-1, 1]$,38 and let $P$ be a polynomial of degree $n$ or less with the least maximum absolute difference between $f$ and the polynomial on that interval. Then the error of $P$ in approximating $f$ is bounded as follows (Phillips 2003, theorem 2.4.6)39:

$$\Vert f-P\Vert _\infty\le\frac{1}{2^n}\frac{\Vert f^{(n+1)}\Vert _\infty}{(n+1)!}.$$

Lebesgue Inequality for Certain Linear Operators

Let $f(\lambda)$ be a continuous function on a closed interval. For any sequence of linear operators $(L_n)$ that map continuous functions to polynomials and reproduce all polynomials up to degree $m(n)$ (which depends on $n$), the following error bound (also known as Lebesgue's lemma or the Lebesgue inequality) holds true for each $n$:

$$\text{abs}(L_n(f)(x) - f(x))\le(1+\Vert L_n\Vert )\cdot\max_t(\text{abs}(f(t)-P(t))),$$

where $\Vert L_n\Vert$ is the operator norm of $L_n$, and $P$ is a polynomial of degree up to $m(n)$ with the least maximum absolute difference between $f$ and the polynomial (see also DeVore and Lorentz (1993)40, Cheney (1996, chapter 6)41). But this error bound will generally be crude or trivial unless $L_n$ are nonpositive operators. Indeed, the only positive linear operator $L$ that reproduces all polynomials up to degree 2 is the identity operator $L=f$.42

Example: Let $f$ have a continuous third derivative on the closed unit interval. Combining the previous inequality with the Whitney-type inequalities given earlier leads to the following error bound for linear operators $L$ that map continuous functions to polynomials and reproduce all polynomials up to degree 2:

$$\text{abs}(L(f)(x) - f(x))\le(1+\Vert L\Vert )\cdot 1\cdot \left(\frac{1}{3}\right)^{3}\Vert f^{(3)}\Vert _\infty$$

$$ = (1+\Vert L\Vert )\Vert f^{(3)}\Vert _\infty/27.$$

Bounds for Certain Nonlinear Operators

The following comes from a result in Bede and Gal (2010)43; see also Bede et al. (2009)44.

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 $x\ge 0$.

If $L(e_0)=1$, then for every $h>0$:

$$\text{abs}(f(x)-L(f)(x))\le(1+L(\text{abs}(e_0-x))(x)/h)\cdot\omega_1(f, h),$$

provided $L(\text{abs}(e_0-x))(x)$ (the "absolute moment" of $L$) exists (and is finite or infinite).

Notes: 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)45.

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.46
  • $X_k$ is a hypergeometric($2n$, $k$, $n$) random variable.
  • $\sigma_{n,k,i}$ equals ${n\choose i}{n\choose {k-i}}/{2n \choose k}$ and is the probability that $X_k$ equals $i$.
  • $\mathbb{E}[Y]$ is the expected value (or mean or “long-run average”) of the random variable $Y$.

$L_n$ 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)39. That paper, among others (for example, Micchelli 197347), 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)48 and central moments (Weisstein)49 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 $X_k$.

In the following, the notation $\Vert f\Vert$ means $\max_{0\le\lambda\le 1}(\text{abs}(f(\lambda)))$.

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:

$$L_n(f)(\lambda) = L_n(R_3(f, \lambda)) + \sum_{i=2}^3 L_n((e_1-\lambda)^i)(\lambda)\frac{f^{(i)}(\lambda)}{i!},$$

$$\text{abs}(L_n(f)(\lambda)) \le \Vert L_n(R_3(f, \lambda))\Vert + \Vert L_n((e_1-\lambda)^2)\Vert \Vert f^{(2)}\Vert /2$$

$$+ \Vert L_n((e_1-\lambda)^3)\Vert \Vert f^{(3)}\Vert /6.$$

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:

$$\text{abs}(L_n(f)(\lambda)) \le \Vert L_n(R_3(f, \lambda))\Vert + \text{abs}(\frac{3\lambda(\lambda - 1)}{2n(2n-1)})\Vert f^{(2)}\Vert /2$$

$$ + \Vert L_n((e_1-\lambda)^3)\Vert \Vert f^{(3)}\Vert /6$$

$$ \le \Vert L_n(R_3(f, \lambda))\Vert + \frac{3}{8n(2n-1)}\Vert f^{(2)}\Vert /2$$

$$ + \frac{\sqrt{3} (6 n - 5)}{24 n^{2} (2 n - 1)}\Vert f^{(3)}\Vert /6.$$

Meanwhile the remainder is estimated as follows, using the proof of corollary 2.3 of Gonska et al. (2006)3:

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

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 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)50, 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.

(The foregoing sentence would remain true if $B_n$ were replaced with any other operator mapping to and from the same functions.)

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:

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

where $\mu_r$ is as in Proposition 1 and the notation $\Vert f\Vert$ means $\max_{0\le\lambda\le 1}(\text{abs}(f(\lambda)))$.

Alternatively, write:

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

$$=(B_n(f) - f) + (\text{Lag}_m(f) - B_n(\text{Lag}_m(f))),$$

$$\text{abs}((H_{n,m}(f) - f)(\lambda))\le\text{abs}(B_n(f) - f) + \text{abs}(B_n(\text{Lag}_m(f)) - \text{Lag}_m(f)),$$

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:

$$B_n(g)\le M_2(g)/(8n),$$

where $M_i(g)$ is the maximum absolute value of $g$'s $i$-th derivative. (This follows from Lorentz (1963)51 and the well-known fact that $M_2$ is an upper bound of $g$'s first derivative's smallest Lipschitz constant.) Thus what is left is 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:

$$\text{abs}(\text{Lag}_m(f)^{(2)}(\lambda))\le \Vert \text{Lag}_m\Vert M_0(f)\cdot \max(1,m)^2,$$

where $\Vert Lag_m\Vert$ is the operator norm of $Lag_m$, also known as its Lebesgue constant, which will vary depending on the points on the closed unit interval where the polynomial meets (interpolates) $f$. The inequality just shown relies on Bernstein's inequality for the derivatives of polynomials (Weisstein)52.

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

$$\text{abs}((H_{n,m}(f) - f)(\lambda))\le \frac{M_2(f)}{8n} + \frac{\Vert \text{Lag}_m\Vert M_0(f)\cdot \max(1,m)^2}{8n}.$$

Example: If $m$ is 3, and the polynomial generated by $Lag_m$ interpolates $f$ at the points 0, 1/3, 2/3, and 1, the inequality just shown becomes:

$$\text{abs}((H_{n,3}(f) - f)(\lambda))\le \frac{M_2(f)}{8n} + \frac{1.64\cdot M_0(f)\cdot 9}{8n},$$

using an upper bound for $\Vert Lag_3\Vert$.

Probabilistic Interpretations of Linear Operators

The Bernstein polynomials featured in a proof in 1912 of the result that any continuous function on a closed interval can be approximated as well as desired by polynomials (Bernstein 1912)53. 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 199654, 199555).

License

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Notes

Footnotes

  1. A better term for positive operators is probably nonnegativity-preserving operators. ↩

  2. n! = 1*2*3*...*n is also known as n factorial; in this document, (0!) = 1.
    Summation notation, involving the Greek capital sigma (Σ), is a way to write the sum of one or more terms of similar form. For example, $\sum_{k=0}^n g(k)$ means $g(0)+g(1)+...+g(n)$, and $\sum_{k\ge 0} g(k)$ means $g(0)+g(1)+...$. ↩

  3. Gonska, Heiner, Paula Piƫul, and Ioan Raşa. "On differences of positive linear operators." Carpathian Journal of Mathematics (2006): 65-78. ↩ ↩2

  4. Skorski, Maciej. "Handy formulas for binomial moments." Modern Stochastics: Theory and Applications 12.1 (2024): 27-41. ↩

  5. It is also possible to bound the "absolute moment" as $M_{n,r}(p)\le C(r)(\max(1/n, (p(1-p)/n)^{1/2})^r$ or $M_{n,r}(p)\le D(r)(1/n + (p(1-p)/n)^{1/2})^r$ (G.G. Lorentz, "The degree of approximation by polynomials with positive coefficients", 1966), but the constants $C(r)$ and $D(r)$ seem to be higher (and less favorable) than the $E(r)$ in $M_{n,r}(p)\le E(r)/n^{r/2}$. ↩

  6. Adell, J.A., Cárdenas-Morales, D., "Quantitative generalized Voronovskaja’s formulae for Bernstein polynomials", Journal of Approximation Theory 231, July 2018. https://doi.org/10.1016/j.jat.2018.04.007 ↩

  7. Molteni, Giuseppe. "Explicit bounds for even moments of Bernstein’s polynomials." Journal of Approximation Theory 273 (2022): 105658. ↩

  8. Cheng, F., "On the rate of convergence of Bernstein polynomials of functions of bounded variation", Journal of Approximation Theory 39 (1983). ↩

  9. Weisstein, Eric W. "Schwarz's Inequality." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SchwarzsInequality.html ↩

  10. R. Bojanić, O. Shisha, "Degree of $L^1$ approximation to integrable functions by modified Bernstein polynomials", Journal of Approximation Theory 13, 66–72 (1975). ↩

  11. Piţul, P., "Evaluation of the Approximation Order by Positive Linear Operators", dissertation, Universität Duisberg-Essen, 2007. ↩ ↩2

  12. I suspect that, whenever $L$ is a bounded linear operator that maps continuous functions on a closed interval to functions of that kind, $L$ can be written as a difference between two positive linear operators. But I have not seen a proof of that statement; Acu et al. ("Grüss-type and Ostrowski-type inequalities in approximation theory", Ukr Math J 63, 843–864, 2011) give a similar statement but without proof. ↩

  13. Gonska, H.H., Meier, J., "On approximation by Bernstein-type operators: best constants", Studia Sci. Math. Hungar. 22, 1987. ↩ ↩2

  14. Shisha, O., Mond. B, "The degree of convergence of linear positive operators", 1968. ↩

  15. R. G. Mamedov, "On the order of approximation of functions by linear positive operators" (Russian), Dokl. Akad. Nauk SSSR 128 (1959). ↩

  16. Mond, B., "On the degree of approximation by linear positive operators", Journal of Approximation Theory 18 (1976). ↩

  17. Păltănea, R., Approximation Theory Using Positive Linear Operators, Birkhäuser, 2004. https://doi.org/10.1007/978-1-4612-2058-9 ↩

  18. Peetre, J., "On the connection between the theory of interpolation spaces and approximation theory", in Approximation Theory, 1969. ↩

  19. Gonska, Heiner. "The rate of convergence of bounded linear processes on spaces of continuous functions." Journal of Numerical Analysis and Approximation Theory 52.2 (2023): 182-232. https://doi.org/10.33993/jnaat522-1326 ↩ ↩2 ↩3 ↩4 ↩5 ↩6

  20. Sevy, J., "Acceleration of convergence of sequences of simultaneous approximants", dissertation, Drexel University, 1991. https://doi.org/10.17918/00010296 ↩ ↩2

  21. H. H. Gonska, Quantitative Approximation in C(X), Habilitationschrift, Universität Duisburg, 1985. ↩ ↩2

  22. Peetre, J., "Exact interpolation theorems for Lipschitz-continuous functions", Ricerche Mat. 18 (1969). ↩

  23. Păltănea, R, Dimitriu, M.T., "On some second order moduli of smoothness." General Mathematics 24 (2016) ↩

  24. Păltănea, R., Smuc, M. "Sharp Estimates of Asymptotic Error of Approximation by General Positive Linear Operators in Terms of the First and the Second Moduli of Continuity", Results in Mathematics 74, 70 (2019). https://doi.org/10.1007/s00025-019-0997-8 ↩

  25. Dimitriu, M.T., "Estimates with optimal constants using Peetre's K-functionals", Carpathian Journal of Mathematics 26 (2010). ↩

  26. Gonska, Heiner, Paula Piţul, and Ioan Raşa. "On Peano's form of the Taylor remainder, Voronovskaja's theorem and the commutator of positive linear operators". In Proceedings of the International Conference on Numerical Analysis and Approximation Theory, Cluj-Napoca. Romania, July 2006. ↩

  27. Gonska, Heiner. "On the degree of approximation in Voronovskaja's theorem", Studia Univ. Babeş-Bolyai, Math., September 2007. ↩

  28. Anastassiou, George A. "A study of positive linear operators by the method of moments, one-dimensional case." Journal of Approximation Theory 45.3 (1985): 247-270. ↩

  29. The paper Cichoń et al., "On delta-method of moments and probabilistic sums", ANALCO 2013, has very similar results, but they assume the function $f$ has a $k$-th derivative defined on an open interval (say, $0\lt\lambda\lt 1$), rather than a closed one, making those results harder to use if $Y$ is a random variable that can take a value equal to either endpoint of the interval (in this example, 0 or 1). ↩

  30. Frantz, Deborah A. Summability methods, probability distributions, and associated positive linear operators. Lehigh University, 1984. ↩

  31. This kind of estimation is called quadrature or numerical integration, and methods for such estimation, such as the one given in (2A), are called quadrature rules. ↩

  32. Brass, H., Förster, KJ. (1998). On the Application of the Peano Representation of Linear Functionals in Numerical Analysis. In: Milovanović, G.V. (eds) Recent Progress in Inequalities. Mathematics and Its Applications, vol 430. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9086-0_10 ↩ ↩2

  33. Waldron, Shayne. "Refinements of the Peano kernel theorem." Numerical functional analysis and optimization 20.1-2 (1999): 147-161. https://doi.org/10.1080/01630569908816885 ↩

  34. Gavrea, I., Ivan, M., "A sharp estimate for the Peano error representation", Applied Mathematics and Computation 252 (2015). https://doi.org/10.1016/j.amc.2014.12.017 ↩ ↩2

  35. Note that for formulas (3) to (5), $(e_1-t)_+^0$ is discontinuous and so is not accepted by $LF$ and $L$ if they map from only continuous functions; thus the results in this section suppose both operators map from bounded functions for $k=0$. Brass and Förster 1998 adequately provides for the case $k=0$, but not Gavrea and Ivan 2015, unfortunately. ↩

  36. Babenko, Alexander G., and Yuriy V. Kryakin. "Special difference operators and the constants in the classical Jackson-type theorems." Topics in Classical and Modern Analysis: In Memory of Yingkang Hu. Cham: Springer International Publishing, 2019. 35-46. ↩

  37. Jaskaran Singh Kaire and Andriy Prymak, "Whitney-type estimates for convex functions", arXiv:2311.00912 (2023). ↩ ↩2

  38. It would be interesting to find a version of this inequality that works for any closed interval $[a, b]$. ↩

  39. Güntürk, C. Sinan, and Weilin Li. "Approximation with one-bit polynomials in Bernstein form", arXiv:2112.09183 (2021); Constr Approx 57, 601–630 (2023). https://doi.org/10.1007/s00365-022-09608-y ↩ ↩2

  40. R.A. DeVore and G.G. Lorentz, Constructive Approximation, 1993. https://link.springer.com/book/9783540506270 ↩

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