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Let ${𝑒}_{𝜉}(𝑥)={𝑒}^{𝑖𝑥\cdot 𝜉}$, $𝑥,𝜉\in {ℝ}^{𝑛}$. The Fourier transform on tempered distributions ${𝑆}^{\prime }={𝑆}^{\prime }({ℝ}^{𝑛})$ is defined by duality with Schwartz functions $𝑆=𝑆({ℝ}^{𝑛})$: $(\mathrm{ℱ︀}𝑢)(𝜑)=𝑢(\mathrm{ℱ︀}𝜑)$ for $𝑢\in {𝑆}^{\prime }$, $𝜑\in 𝑆$, where $\mathrm{ℱ︀}𝜑(𝜉)={\int }_{ℝ}{𝑒}_{-𝜉}(𝑥)𝜑(𝑥)d𝑥$, $𝜑\in 𝑆$.

For compactly supported distributions, i.e. distributions $𝑢$ for which $𝑢=𝜒𝑢$ for some $𝜒\in {𝐶}_{𝑐}^{\infty }({ℝ}^{𝑛})$, $𝑢({𝑒}_{-𝜉})=𝑢(𝜒{𝑒}_{-𝜉})$ is directly defined.

Show that for such distributions, $\mathrm{ℱ︀}𝑢$ is the tempered distribution given by the polynomially bounded function $𝜉\mapsto 𝑢(𝜒{𝑒}_{-𝜉})$ (include a proof of this polynomially bounded statement: there exist $𝐶>0$, $𝑁\in \mathbb{N}$ such that $|𝑢(𝜒{𝑒}_{-𝜉})|\le {𝐶(1+|𝜉|)}^{𝑁}$).

Let $𝐾\subseteq {ℝ}^{𝑛}$ be a compact set, and $𝑓:𝐾\to ℝ$ be continuous. Show that there exists a continuous function such that for all $𝑥\in 𝐾$.

Suppose that $𝑋,𝑌,𝑍$ are Banach spaces, $𝑃:𝑋\to 𝑌$, $𝑋\subseteq 𝑍$ with the inclusion map compact. Suppose also that there is a constant $𝐶>0$ such that for all $𝑥\in 𝑋$, ${\Vert 𝑥\Vert }_{𝑋}\le 𝐶({\Vert 𝑃𝑥\Vert }_{𝑌}+{\Vert 𝑥\Vert }_{𝑍})$.

(a) If $𝑃$ is injective, show that there is another constant ${𝐶}^{\prime }>0$ such that for all $𝑥\in 𝑋$, ${\Vert 𝑥\Vert }_{𝑋}\le {𝐶}^{\prime }{\Vert 𝑃𝑥\Vert }_{𝑌}$.

(b) Without assuming that $𝑃$ is injective show that there is a complementary subspace $𝑊$ of $Ker𝑃$, i.e. a closed subspace $𝑊$ of $𝑋$ with $𝑊\oplus Ker𝑃=𝑋$, such that there is another constant ${𝐶}^{\prime }>0$ such that for all $𝑥\in 𝑊$, ${\Vert 𝑥\Vert }_{𝑋}\le {𝐶}^{\prime }{\Vert 𝑃𝑥\Vert }_{𝑌}$.

Let $\widehat{𝑓}(𝜉)={\int }_{ℝ}{𝑒}^{-𝑖𝑥𝜉}𝑓(𝑥)d𝑥$ be the Fourier transform on Schwartz functions $𝑓\in 𝑆(ℝ)$.

Suppose $𝑓\in 𝑆(ℝ)$ satisfies $𝑓(2𝜋𝑛)=0$ and $\widehat{𝑓}(𝑛)=0$ for all integers $𝑛$. Prove or disprove that $𝑓$ must be the zero function.

Let $𝑄$ be the unit cube in ${ℝ}^{𝑛}$, and let ${𝐻}_0^2(𝑄)$ be the closure of ${𝐶}^{\infty }$ functions supported in $𝑄$ with respect to the norm whose square is ${\Vert 𝑓\Vert }_{{𝐻}_0^2(𝑄)}^2={\int }_{𝑄}{|𝑓|}^2+{|\nabla 𝑓|}^2+{|{\nabla }^2𝑓|}^2$.

(a) Prove that for every $𝜀>0$, there exists a constant ${𝐶}_{𝜀}$ so that for all $𝑓\in {𝐻}_0^2(𝑄)$,

(b) Suppose that $𝑢$ is in the Hölder space ${𝐶}^{1,𝛼}(𝐼)$, $𝛼\in (0,1)$, $𝐼=[𝑎,𝑏]$. Prove that for every $𝜀>0$ there exists ${𝐶}_{𝜀}$ such that

(c) Let , , with ${𝐶}^2$ coefficients. Suppose that for every $𝑢\in {𝐻}_0^2(𝑄)$, ${\Vert 𝑢\Vert }_{{𝐻}_0^2(𝑄)}^2\le 𝐶({\Vert {𝐿}_0𝑢\Vert }_{{𝐿}^2(𝑄)}^2+{\Vert 𝑢\Vert }_{{𝐿}^2(𝑄)}^2)$. Prove, using (a), that a similar estimate holds if ${𝐿}_0$ is replaced by $𝐿$.

For $𝑢\in {𝑆}^{\prime }(ℝ)$ let $𝐷𝑢$ denote the distributional derivative of $𝑢$, and let $𝑀𝑢$ be the distribution $𝑥𝑢$ (i.e. $(𝑀𝑢)(𝜑)=𝑢(𝑥𝜑)$, $𝜑\in 𝑆(ℝ)$). Let ${𝐻}_1=\{𝑢\in {𝐿}^2(ℝ):𝐷𝑢\in {𝐿}^2,𝑀𝑢\in {𝐿}^2\}$, equipped with the norm ${\Vert 𝑢\Vert }_{{𝐻}_1}^2={\Vert 𝑢\Vert }_{{𝐿}^2}^2+{\Vert 𝐷𝑢\Vert }_{{𝐿}^2}^2+{\Vert 𝑀𝑢\Vert }_{{𝐿}^2}^2$, and let ${𝐻}_2=\{𝑢\in {𝑆}^{\prime }(ℝ):\exists {𝑢}_0,{𝑢}_1,{𝑢}_2\in {𝐿}^2(ℝ):𝑢={𝑢}_0+𝐷{𝑢}_1+𝑀{𝑢}_2\}$ with norm ${\Vert 𝑢\Vert }_{{𝐻}_2}=inf\{{({\Vert {𝑢}_0\Vert }_{{𝐿}^2}^2+{\Vert {𝑢}_1\Vert }_{{𝐿}^2}^2+{\Vert {𝑢}_2\Vert }_{{𝐿}^2}^2)}^{1/2}:𝑢={𝑢}_0+𝐷{𝑢}_1+𝑀{𝑢}_2\}$.

(a) Show that ${𝐻}_1$ and ${𝐻}_2$ are Banach spaces.

(b) Show that the map $𝑗:{𝐻}_2\to {𝐻}_1^{\ast }$, , $𝑣={𝑣}_0+𝐷{𝑣}_1+𝑀{𝑣}_2$, ${𝑣}_{𝑗}\in {𝐿}^2(ℝ)$, is well defined, and is an invertible (conjugate) linear map between Banach spaces.

Note: You may use without proof that $𝑆(ℝ)$ is dense in ${𝐻}_1$.

Two short problems:

(a) Define the function $𝜈$ from subsets of $ℝ$ to $[0,\infty ]$ as follows. If $𝐴\subseteq ℝ$, let $𝜈(𝐴)=\infty$ if $0$ is in the closure of $𝐴$, and let $𝜈(𝐴)=0$ otherwise. Show that $𝜈$ is finitely additive but it is not countably additive.

(b) Let $𝑚$ be the Lebesgue measure on $ℝ$. Suppose that $𝑓:ℝ\to ℝ$ is measurable and let $𝑆=\{𝑥\in ℝ:𝑚({𝑓}^{-1}(\{𝑥\}))>0\}$. Show that $𝑆$ is countable, and give an example to show that $𝑆$ can be dense in $ℝ$.

Two short problems:

(a) Show that a Hilbert space is separable as a metric space (i.e. has a countable dense subset) if and only if it has a countable complete orthonormal set (basis).

(b) Show that in a separable Hilbert space the closed unit ball is weakly sequentially compact, i.e. any sequence ${\{{𝑥}_{𝑛}\}}_{𝑛=1}^{\infty }$ in it has a weakly convergent subsequence.

Given a Schwartz function $𝑓\in \mathrm{𝒮︀}(ℝ)$, define the ${𝐻}^{𝑠(ℝ)}$ norm by ${\Vert 𝑓\Vert }_{{𝐻}^{𝑠(ℝ)}}≔{\Vert {(1+{|𝜉|}^2)}^{𝑠/2}\widehat{𝑓}(𝜉)\Vert }_{{𝐿}^2(ℝ)}$, where $\widehat{𝑓}$ denotes the Fourier transform of $𝑓$ and $𝑠\in ℝ$.

(a) Prove that for every $𝑠>1/2$, there exists a constant $𝐶>0$ (depending only on $𝑠$) such that for every $𝑓\in \mathrm{𝒮︀}$,

(b) Let $𝐾=\left\{𝑓\in \mathrm{𝒮︀}(ℝ):\exists 𝑑>0s.t.\widehat{𝑓}(𝜉)\ne 0⟹𝑑\le |𝜉|\le 2𝑑\right\}$. Prove that there exists a constant $𝐶>0$ (independent of $𝑑$) such that for every $𝑓\in 𝐾$,

Suppose $𝑋,𝑌$ are Banach spaces, ${𝐴}_{𝑗},𝐴\in \mathrm{ℒ︀}(𝑋,𝑌)$, $𝑗\in \mathbb{N}$, and $𝐴$ is invertible. (Here $\mathrm{ℒ︀}(𝑋,𝑌)$ is the set of bounded linear operators $𝑋\to 𝑌$.)

(a) Suppose that ${𝐴}_{𝑗}\to 𝐴$ in norm in $\mathrm{ℒ︀}(𝑋,𝑌)$. Show that there exists $𝑁$ such that for $𝑗\ge 𝑁$, ${𝐴}_{𝑗}$ is also invertible.

(b) Suppose now that ${𝐴}_{𝑗}\to 𝐴$ in the strong operator topology on $\mathrm{ℒ︀}(𝑋,𝑌)$. Does it follow that ${𝐴}_{𝑗}$ is invertible for sufficiently large $𝑗$? Prove it, or give a counterexample.