Delta-shaped kernels (classical approximate identities)

Let $f(x)$, $0 \leq x \leq 1$, be a real valued continuous function. Show that

Prove the equalities:

if the integral is absolutely convergent, and $f(x+0)$ and $f(x-0)$ are the right and left limits of $f(\xi )$ as $\xi \rightarrow x$.

Prove that as $h \rightarrow 0$ the equality holds

if $f(x)$ is a continuous function for which $\int_{-\infty }^{\infty } \left|f(t)\right| e^{-\frac{t^{2}}{h^{2}}}\, dt$ makes sense for sufficiently small $\left|h\right|$.

A sequence $\left\{ f_k\right\}$ of functions on a topological space $X$ with a Borel measure $\mu$ is said to be $\delta$-shaped for a point $a\in X$ if:

1. $f_k(x)\geq 0$ on $X$;

2. $\int_X f_k(x)\, d\mu (x)=1$;

3. $\int_{X\setminus U}f_k(x)\, d\mu (x)\to 0$ for any neighborhood $U$ of $a$ as $k\to \infty$. Prove that the following sequences are $\delta$-shaped for the point $0\in \mathbb {R}^{n}$:

Let $G$ be a topological abelian group with invariant measure $\mu$, $\left\{ f_k\right\}$ a $\delta$-shaped sequence for the point $a\in G$ (see Problem 510), and $L$ a Banach space of functions on $G$ satisfying the conditions of Problem 509. Prove that $S(f_k)\to T(a)$ as $k\to \infty$.

Construct a $\delta$-shaped sequence consisting of trigonometric polynomials on the torus $\mathrm{T}^n$.

Let the function $f(x)$ be continuous and take positive values on the segment $[0 ; 1]$. Prove that the function

is discontinuous at $\alpha = 0$.

Let $\left\{ \varphi_{n}(x)\right\}$ be a sequence of functions, Riemann integrable on the segment $[a ; b]$ and taking nonnegative values on this segment, converging uniformly to zero on the set $E = \left\{ x: 0 < \delta \leqslant \left|x\right| \leqslant 1\right\}$ for every $\delta > 0$, and moreover

Prove that for any function $f(x)$ continuous on the segment $[-1 ; 1]$, the equality holds

Prove that if the function $f$ is continuous and bounded on the interval $[0 ;+\infty )$, then

If the functions

are continuous on the interval $\left[a,b\right]$ and if they satisfy the conditions

then the terms of the sequence

are between the minimum and the maximum of the continuous function $f(t)$. (Cf. Problems 65, 79, 83 of Part I.)

Let $x$ be a fixed point of the interval $\left[a,b\right]$ considered in Problem 128. In order that

holds for all functions $f(t)$ continuous on $\left[a,b\right]$ it is necessary and sufficient that

for all positive values of $\varepsilon$ for which $a < x-\varepsilon < x+\varepsilon < b$ (if $x=a$ or $x=b$ the first or second integral resp. under the limit sign has to be omitted.) (Cf. Problems 66, 80, 84 of Part I.)

We assume that

are continuous functions of $x$ and $t$, $a \leq x, t \leq b$, and that for each $n$

Let $f(t)$ denote a continuous function. The functions

lie between the minimum and the maximum of $f(t)$ on $\left[a,b\right]$ for any $x$, $a \leq x \leq b$; i.e. $\min_{a \leq x \leq b} f(x) \leq f_{n}(x) \leq \max_{a \leq x \leq b} f(x)$. Furthermore

provided

uniformly for $a+\varepsilon \leq x \leq b-\varepsilon$, $\varepsilon$ fixed and positive. The convergence is uniform on any closed subinterval of $(a,b)$.