Fix an integer $n\ge 2$. Let $[a,b]$ be an interval and $f: [a,b]\to \mathbb R$ be a continuous function and for $x_1,...,x_n$ being the Gaussian Quadrature nodes in $[a,b]$, and Gaussian Quadrature wights $w_1,...,w_n$ in $[a,b]$ , (everything being calculated with weight function $\omega(x)=1$ ) (see https://en.wikipedia.org/wiki/Gaussian_quadrature for details ), let $T_{n,[a,b]} (f) := \sum_{i=1}^n w_if(x_i)$.

Now, for every $f\in C[0,1]$, let $T_n(f)(0)=0$ and $T_n(f)(x)=T_{n,[0,x]}(f), \forall x \in (0,1]$. Let ${\mathbb R}^{[0,1]}$ be the set of all functions from $[0,1]$ to $\mathbb R$.

Then of-course $T_n$ is a linear map from $C([0,1])$ to ${\mathbb R}^{[0,1]}$. My questions are the following :

(1) For which $n$, is it true that $T_n (C[0,1]) \subseteq C[0,1]$ ?

(2) Let $\mathcal B([0,1])$ be the set of all bounded real-valued functions on $[0,1]$. For which $n$, is it true that $T_n(C[0,1]) \subseteq \mathcal B([0,1])$ ?

(3) For which $n$, is $T_n$ an injective function ?

(4) If $n\ge 2$ is an integer for which $(2) $ (or (1)) holds, then is it true that $\sup_{x\in [0,1]} |T_n (f)(x)| \le M_n \sup_{x\in [0,1]} |f(x)|, \forall f \in C([0,1]) $, where the constant $M_n$ only depends on $n$ ?

(5) Is it true that $T_n (f) \to T(f), \forall f \in C[0,1]$ ? Or at least does a subsequence of $\{T_n\}$ converge to $T$ pointwise on $C[0,1]$ ? Where $T$ is the operator $T:C[0,1] \to C[0,1]$ , defined as $T(f)(x):=\int_0^x f(t) dt$

up vote 2 down vote accepted

The key here is the simple change-of-interval/rescaling formula, found e.g. at the link in the OP, according to which \begin{equation} T_n(f)(x)=T_{n,[0,x]}(f)=x\sum_1^n w_i f(xx_i), \tag{*} \end{equation} where the $w_i$'s and $x_i$'s are such that $T_n(f)(1)=T_{n,[0,1]}(f)=\sum_1^n w_i f(x_i)$ for all $f$.

From here, one immediately has the affirmative answer to Questions (1), (2), and (4). In particular, in (4), one may take $M_n=\sum_1^n|w_i|=\sum_1^n w_i =1$.

The answer to Question (5) is also positive. Indeed, recall that for any polynomial $p$ of degree $\le2n-1$ we have $T_n(p)(1)=\int_0^1 p$ and hence, by the rescaling, $T_n(p)(x)=\int_0^x p$ for all $x\in[0,1]$. Take now any $f\in C[0,1]$ and any real $\delta>0$ and let $p$ be a polynomial, of degree $\le2n-1$ for some natural $n$, such that $p-\delta\le f\le p+\delta$ on $[0,1]$. The operator $T_n$ is positive, since $w_i\ge0$ for all $i$. So, for $x\in[0,1]$ \begin{multline} T(p)(x)-\delta\cdot x=\int_0^x (p-\delta)=T_n(p-\delta)(x)\le T_n(f)(x) \\ \le T_n(p+\delta)(x)=\int_0^x (p+\delta) =T(p)(x)+\delta\cdot x, \end{multline} whence \begin{equation} |T_n(f)(x)-T(f)(x)|\le\delta\cdot x\le\delta, \end{equation} so that $T_n(f)\to T(f)$ uniformly on $[0,1]$, for any $f\in C[0,1]$.

Moreover, if two functions, say $f$ and $g$ in $C[0,1]$, are uniformly close to each other, than any polynomial $p$ uniformly close to $f$ will also be uniformly close to $g$. Hence, it follows from the above reasoning that $T_n\to T$ in the $(\|\cdot\|_\infty,\|\cdot\|_\infty)$ operator norm.

The answer to Question (3) is this: the linear operator $T_n$ is not injective for any natural $n$. Indeed, letting, as usual, the nodes $x_i$ be increasing in $i$, we have $x_n<1$. On the other hand, by (*), $T_n(f)=0$ for any $f\in C[0,1]$ such that $f=0$ on $[0,x_n]$. So, letting e.g. $f(x):=\max(0,x-x_n)$ for $x\in[0,1]$, we have $f\in C[0,1]$ and $f\ne0$, whereas $T_n(f)=T_n(0)$.


Added in response to a comment by the OP: No, the operator $T_n$ is not compact for any natural $n$. Indeed, take any compact intervals $J$ and $K$ such that $\emptyset\ne J^\circ\subset J\subset K^\circ\subset K\subset (x_{n-1}/x_n,1)$, where ${}^\circ$ denotes the interior and $x_{n-1}:=0$ for $n=1$. Take any $h\in C(J)$. Then $h$ equals $g|_J$, the restriction to $J$ of some function $g\in C[0,1/x_n]$ such that the support $\text{supp}\,g$ of $g$, defined as the closure of the set $\{x\colon g(x)\ne0\}$, is contained in $K$ and $\|g\|_{[0,1/x_n]}=\|h\|_J$, where $\|u\|_I:=\max_{x\in I}|u(x)|$ for any interval $I$ and any $u\in C(I)$. Let now \begin{equation} f(y):=\frac{x_n}{w_n y}\,g\Big(\frac y{x_n}\Big) \end{equation} for $y\in(0,1]$, with $f(0):=0$. Then $\text{supp}f\subseteq x_n\,K\subset(x_{n-1},x_n)\subset(0,1)$, whence $y_*:=\min\text{supp}f\ge\min(x_n K)=x_n\min K>0$, $f\in C[0,1]$, and \begin{equation} \|f\|_{[0,1]}\le M\|g\|_{[0,1/x_n]}=M\|h\|_J, \end{equation} where $M:=\frac1{w_ny_*}<\infty$. Next, for $x\in(0,1]$, \begin{equation} xw_i f(xx_i)=xw_i \frac{x_n}{w_n xx_i}\,g\Big(\frac{xx_i}{x_n}\Big)= \left\{ \begin{aligned} 0&\text{ if }i\le n-1,\\ g(x)&\text{ if }i=n, \end{aligned} \right. \end{equation} because $\text{supp}\,g\subseteq K\subset (x_{n-1}/x_n,1)$ and hence $g=0$ on $[0,x_{n-1}/x_n]$. So, $T_n(f)=g$ on $[0,1]$ and hence $T_n(f)=h$ on $J$, that is, $(R_JT_n)(f)=h$, where $R_J\psi:=\psi|_J$, the restriction of $\psi$ to $J$. So, $(R_JT_n)(MB_{[0,1]})\supseteq B_J$, where $B_I$ denotes the unit ball in $C(I)$. Since $B_J$ is not compact, it follows that the operator $R_JT_n$ is not compact. Therefore and because the operator $R_J$ is continuous, it follows that the operator $T_n$ is indeed not compact.

  • Thank you very much for your answer. Say ... do you think the $T_n$'s are compact operators ? – user521337 Nov 18 at 0:02
  • @user521337 : I have added a proof that $T_n$ is not compact. – Iosif Pinelis Nov 18 at 15:45
  • thanks ... btw do you have any idea about the injectivity of the $T_n$ s that I also asked in the original question ? – user521337 Nov 19 at 2:00
  • No, at this point I don't have a good idea about the injectivity. – Iosif Pinelis Nov 19 at 12:32
  • D you have any idea about whether any $T_n$ is bijective or not ? – user521337 Nov 25 at 0:37

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