A Question in Fourier Analysis proposing a conjecture

Question

Let $f$ be a $2\pi$ periodic BV function whose derivative is also BV.Let the amount of jump at a point $x$ is denoted as $\lfloor f \rfloor (x) = f(x+0)-f(x-0)$ Define function $J:\mathbb{R} \to\mathbb{R}$, such that $J(x) = \lfloor f \rfloor (x)$. Define the set $L = \{x/x\in(0,2\pi)\wedge J(x)\neq 0\}$. Let there not be any jumps at $0, \pi, 2\pi$. and all jumps at rational multiples of $2\pi$.

Let the Fourier series of $f$ be defined as \begin{equation} S[f] = \sum_{\nu=-\infty}^{\nu=\infty}c_{\nu}e^{i\nu x} %a=x^2 \end{equation} and the conjugate Fourier series as \begin{equation} \tilde{S}[f] = -i\sum_{\nu=-\infty}^{\nu=\infty}sign(\nu)c_{\nu}e^{i\nu x} \end{equation} Let the Fourier partial sum of $f$ be defined as \begin{equation} S_n(x;f) = \sum_{\nu=-n}^{\nu=n}c_{\nu}e^{i\nu x} \end{equation} and the conjugate partial sum of $f$ as \begin{equation} \tilde{S}_n(x;f) = -i\sum_{\nu=-n}^{\nu=n}sign(\nu)c_{\nu}e^{i\nu x} \end{equation}

Define $$G(n) = \sum_{i=0}^{n}\left|c_i\right|$$

The Dirichlet kernel $D_n(x)$ and conjugate Dirichlet kernel $\tilde{D}_n(x)$ are defined as \begin{equation} D_n(x) = \frac{1}{2} + \sum_{\nu=1}^{n}\cos(\nu x) = \frac{\sin(n+\frac{1}{2})x}{2\sin(\frac{1}{2}x)} \end{equation}

\begin{equation} \tilde{D}_n(x) = \sum_{\nu=1}^{n}\sin(\nu x) = \frac{\cos(\frac{x}{2})-\cos(n+\frac{1}{2})x}{2\sin(\frac{1}{2}x)} \end{equation}

Let \begin{equation} Y_n(x) = -\frac{1}{\log(n) G(n)}\sum_{k=1}^{n}\tilde{S}_k(x;f)\left|c_k\right| \end{equation} Conjectures

show that \begin{equation} \lim_{n\to\infty} Y_n(x) = \frac{J(x)}{2\pi} \end{equation}

Given any $(a,b)\subseteq (0,2\pi)$, show that \begin{equation} V_a^b(Y_n) \sim \frac{1}{\pi}\sum_{x\in L\cap (a,b)}\left|J(x)\right| \end{equation}

PS: $V_a^b(f)$ is the variation of $f$ in $(a,b)$.

Answer 1

These are proofs for first and second conjectures.

This is a proof to the second conjecture.

Let $$V_n = V_a^b(Y_n) = \int_a^b\left|\frac{1}{\log(n) G(n)}\sum_{k=1}^{n}\tilde{S}'_k(x;f)\left|c_k\right| \right|dx$$

$$V_n \leq \frac{1}{\log(n) G(n)}\sum_{k=1}^{n}\int_a^b\left|\tilde{S}'_k(x;f)\left|c_k\right| \right|dx$$

using a lemma from Trigonometric Series, Volume 1 - Antoni Zygmund

$$V_n \leq \frac{1}{\log(n) G(n)}\sum_{k=1}^{n}\int_a^b\left|\tilde{S}_k(x;f')-\sum_{y\in L}\frac{J(y)}{\pi}\tilde{D}_k(x-y))\left|c_k\right| \right|dx$$

$$V_n \leq \frac{1}{\log(n) G(n)}\sum_{k=1}^{n}\int_a^b\left|\tilde{S}_k(x;f')-\sum_{y\in L}\frac{J(y)}{\pi}\tilde{D}_k(x-y))\left|c_k\right| \right|dx$$

$$V_n \leq \frac{1}{\log(n) G(n)}(\sum_{k=1}^{n}\int_a^b\left|\tilde{S}_k(x;f')\left|c_k\right|\right|dx-\sum_{k=1}^{n}\int_a^b\left|\sum_{y\in L}\frac{J(y)}{\pi}\tilde{D}_k(x-y))\left|c_k\right| \right|dx)$$

Since $f'$ is BV, with a suitable reference, $\int_a^b\left|\tilde{S}_k(x;f')\right|dx \sim O(1)$, So $\sum_{k=1}^{n}\int_a^b\left|\tilde{S}_k(x;f')\left|c_k\right|\right|dx \sim \sum_{k=1}^{n}(O(1))\frac{V}{k} \sim O(\log(n))$

$$\sum_{k=1}^{n}\int_a^b\left|\sum_{y\in L}\frac{J(y)}{\pi}\tilde{D}_k(x-y))\left|c_k\right| \right|dx \leq \sum_{k=1}^{n}\sum_{y\in L}\frac{J(y)}{\pi}\left|c_k\right|\int_a^b\left|\tilde{D}_k(x-y)\right|dx$$

$$ \sum_{k=1}^{n}\sum_{y\in L}\frac{J(y)}{\pi}\left|c_k\right|\int_a^b\left|\tilde{D}_k(x-y)\right|dx = \sum_{k=1}^{n}\sum_{y\in L\setminus(a,b)}\frac{J(y)}{\pi}\left|c_k\right|\int_a^b\left|\tilde{D}_k(x-y)\right|dx + \sum_{k=1}^{n}\sum_{y\in L\cap(a,b)}\frac{J(y)}{\pi}\left|c_k\right|\int_a^b\left|\tilde{D}_k(x-y)\right|dx $$

$\int_a^b\left|\tilde{D}_k(x-y)\right|dx \sim 2\log(k)$ if $y \in (a,b)$ and $\int_a^b\left|\tilde{D}_k(x-y)\right|dx \sim O(1)$, if $y \notin (a,b)$

$V_n \leq \frac{1}{\log(n)G(n)}(\sum_{k=1}^{n}\int_a^b\left|\tilde{S}_k(x;f')\left|c_k\right|\right|dx-\sum_{k=1}^{n}\sum_{y\in L\setminus(a,b)}\frac{J(y)}{\pi}\left|c_k\right|\int_a^b\left|\tilde{D}_k(x-y)\right|dx + \sum_{k=1}^{n}\sum_{y\in L\cap(a,b)}\frac{J(y)}{\pi}\left|c_k\right|\int_a^b\left|\tilde{D}_k(x-y)\right|dx$

Therefore

In the first term on the RHS $\int_a^b\left|\tilde{S}_k(x;f')\right|dx \sim O(1)$ and hence first term goes to zero.

Coming to second term, $y\notin(a,b)$, so $\int_a^b\left|\tilde{D}_k(x-y)\right|dx \sim O(1)$ and hence second term also vanishes.

In the third term $ y \in (a,b)$ and hence $\int_a^b\left|\tilde{D}_k(x-y)\right|dx = 2log(k)$,also $|c_k|\sim \frac{1}{k}$, so after summation via integration and knowing $G(n)\sim \log(n)$, we get the result, but the equality hold only up to a constant factor.

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This is a proof to first conjecture.

Case 1 Points of no jump.

$\tilde{S_n}(x;f)\sim O(1)$ and hence $Y_n(x)\sim 0$, as $J(x) = 0$ for no jump the result is proved.

Case 2 Points of jump.

By Lukacs theorem, at points of jump, $\tilde{S_n}(x;f)\sim \frac{J(x)}{\pi}\log(n)$, So $$Y_n(x) = \frac{1}{\log(n)G(n)}\sum\limits_{k=1}^n\frac{J(x)}{\pi}\log(k)|c_k|$$

Noting $|c_k|\sim\frac{1}{k}$ (as we are assuming points of jump),

$$Y_n(x) = \frac{1}{\log(n)K\log(n)}\sum\limits_{k=1}^n\frac{J(x)}{\pi}\log(k)K^2\frac{1}{k}$$ where $K$ is a constant.

on taking summation as integration, we get $$ \frac{1}{K(\log(n))^2}K\frac{J(x)}{\pi}\int_0^n\log(k)K\frac{1}{k}dk$$

Substituting $\log(k) = t$ in the integral, we get $$ \frac{1}{(\log(n))^2}\frac{J(x)}{\pi}\int_0^t Ktdt$$

Hence we get $$\lim_{n\to\infty}Y_n(x) = K \frac{J(x)}{2\pi}$$ equality in conjecture holding only upto a constant factor.