Completeness of systems of functions

Prove that the system of functions $\left\{ t^{100 k}: k \geq k_{0}\right\} , k_{0} \in \mathbb {N}$, is total in $L_{p}(\left[a, b\right]), 1 \leq p < +\infty$ for $a \geq 0$.

Let a function $f \in L_{2}(\mathbb {R})$ be almost everywhere nonzero and suppose there exists a positive number $a$ such that $e^{a\left|x\right|} \cdot f \in L_{2}(\mathbb {R})$. Prove that the system of functions $\left\{ x^{n} f\right\}_{n = 0}^{\infty }$ is complete in $L_{2}(\mathbb {R})$.

Prove the stated assertion. The system of powers $1, t, t^{2}, \ldots , t^{n}, \ldots$ is complete in the space $C[a ; b]$ (see problem 8).

Prove the stated assertion.

Prove the stated assertion. The system of Legendre polynomials (see problem 28) is complete in the spaces $C[a ; b]$ (see problem 8) and $C L_{p}[a ; b], 1 \leqslant p < +\infty$ (see problem 66).

Prove the stated assertion.

Let $\mathcal{H}$ be Hilbert space with an orthonormal basis $\left\{ e_{n}\right\}$, and let

Prove that $\left\{ f_{n}\right\}$ is an orthonormal basis for $\mathcal{H}$.

Let $\lambda_{1}, \lambda_{2}, \lambda_{3}, \ldots , \lambda_{n}, \ldots$ be distinct positive numbers such that the series

is divergent. If a function $h(t)$ which is properly integrable over the interval $0\leq t\leq 1$ satisfies the condition

then $h(t)=0$ at every point of continuity $t$. (A generalization of Problem 139 of Part II.)

Let $g(z)$ be a transcendental entire function of order $\lambda <\tfrac {1}{2}$. Let its coefficients be different from $0$ and its zeros $w_{1}, w_{2}, w_{3}, \ldots , w_{n}, \ldots$ be distinct, $w_{k}\neq w_{l}$ for $k\neq l$. If a function $h(t)$ which is defined and properly integrable over the interval $0\leq t\leq 1$ satisfies the condition

then $h(t)=0$ at every point of continuity $t$. (See Problem 139 of Part II.)

Let $g(z)$ be a transcendental entire function of genus $0$ with all its zeros $w_{1}, w_{2}, w_{3}, \ldots , w_{n}, \ldots$, real and distinct. If a function $h(t)$ defined and properly integrable in the interval $0\leq t\leq 1$ satisfies the condition

then $h(t)=0$ at every point of continuity $t$. (We may, for example, set $g(z)=J_{0}(\sqrt{z})$ or $\cos \pi \sqrt{z}$.)

Let the numbers $a, b, m_{1}, m_{2}, m_{3}, \ldots$ be real, and satisfy

Furthermore, let $f(x)$ be continuous in the interval $[a,b]$. Then it follows from

for $n=1,2,3,\ldots$, that $f(x)$ vanishes identically. (A fraction of the whole system of Fourier functions is closed over the corresponding fraction of the whole interval.)