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Define the function $$\mathbb{X} \times \mathbb{X} \ni (x,y) \mapsto f(x,y) := \begin{cases}\frac{A}{\pi}\frac{\sin{A(x-y)}}{A(x-y)}\,\, if \,\,x \neq y,\\ \frac{A}{\pi} \,\, \hspace{16.5mm}if \,\,x =y.\end{cases}$$ Here we can assume that $\mathbb{X} \subset \mathbb{R}$ is a compact set and $A \in \mathbb{R}$ is a constant. I want to find a Lipschitz constant $M$ for $f$ ? Recall $M := \sup_{x \neq y \in \mathbb{X} } ||\nabla f(x,y)||$.

The above query is a small part of a bigger problem (which I cannot reveal). Any ideas and thoughts on how to proceed will be appreciated, as I have been stuck at this point for quite some time.

Edits: What I have done:
Observe that $$\sup_{x \neq y \in \mathbb{X} } ||\nabla f(x,y)|| \leq \frac{\sqrt{2}A}{\pi}\sup_{x \neq y \in \mathbb{X}}\Bigg|\frac{cos{A(x-y)}}{x-y}\Bigg| + \frac{\sqrt{2}}{\pi}\sup_{x \neq y \in \mathbb{X}}\Bigg|\frac{sin{A(x-y)}}{(x-y)^2}\Bigg|,$$ after calculating the gradient $\nabla f(x,y)$ of the function. From the preceding inequality I can get an estimate that depends on $x$ and $y$ which in turn will depend on $\mathbb{X}.$ I am asking whether it is possible to somehow solve the optimization problem $M := \sup_{x \neq y \in \mathbb{X} } ||\nabla f(x,y)||$.
Thank you for the help.

souvikd
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  • Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or closed. To prevent that, please [edit] the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Jun 15 '21 at 08:34
  • What is $\mathbb{X} $ ? – Fred Jun 15 '21 at 08:39
  • You can assume $\mathbb{X}$ to be a compact subset of $\mathbb{R}$. – souvikd Jun 15 '21 at 10:52
  • @JoséCarlosSantos Point taken. Thanks a lot. – souvikd Jun 15 '21 at 10:53
  • First recap that there is not THE Lipschitz constant but A Lipschitz constant. Any constant $L$ for which it holds $$|f(x_1,y_1) - f(x_2,y_2)| \le L|(x_1,y_1) - (x_2,y_2)|$$ is called Lipschitz constant. What's your problem of calculating $$\sup_{x \neq y \in \mathbb{X} } ||\nabla f(x,y)||$$ on your own? – Gono Jun 15 '21 at 12:33
  • @Gono Well I can only estimate the value of $\sup_{x \neq y \in \mathbb{X}}||\nabla f(x,y)||$. Is it somehow possible to find the exact value? – souvikd Jun 15 '21 at 14:19
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    Letting $$ g(x,y) := \frac{\sin(A(x-y))}{A(x-y)} $$ it follows that $f(x,y) := \frac{A}{\pi}g(x,y)$. So $$ |\nabla g(x,y)| = \frac{\sin(A(x-y)) - A(x-y)\cos(A(x-y))}{A(x-y)^2}. $$ L:et $h:=x-y$. Then, we are interested in determining the supremum of $$ h \mapsto \frac{\sin(Ah) - Ah\cos(Ah)}{Ah^2} \quad \text{for } h \neq 0, h \in \mathbb{X}. $$ Then to determine the supremum, one could proceed in the following two steps:
    1. Determine supremum of the function dependent on $h$
    2. Show that the supremum of the norm of the gradient of $g$ lies on the diagonal.
     – spaceman Jun 15 '21 at 15:37
  • @souvikd Why haven't you shown what you got, yet? Then it's easier to see where you're stuck. $M$ will depend on $\Bbb X$ somehow. Additionally your function is not well defined. What happens for $x=y$? – Gono Jun 15 '21 at 16:37
  • @Gono I have added some more remarks. – souvikd Jun 15 '21 at 20:44
  • @Spaceman Thanks for the reply. I can infer point (2). However, your point (1) is where I got stuck. – souvikd Jun 15 '21 at 20:44
  • @souvikd I see, and this is precisely the hard step. I'm not sure if is possible to find the explicit roots of the derivative of the $h$ function since this would rely on solving trig/poly problems, see https://math.stackexchange.com/questions/2082103/trigonometric-polynomial-equations-and-the-algebraic-nature-of-trig-functions. Although, that doesn't mean that there is no hope. Best thing I can think of would be to represent the $h$ function as a series/product series and try to obtain a sharp upper bound (easier said than done) – spaceman Jun 15 '21 at 21:39
  • @souvikd Alternatively, it may be useful to approach differently. For example, how about considering the Fourier transform of $g(h) = sin(h)/h$, and obtaining bounds for that, and then using some bound comparison between Fourier and real space to conclude – spaceman Jun 15 '21 at 21:44
  • @Spaceman Thanks for the reply. I have tried something similar by representing the gradients by a polynomial, not very useful. Your second approach seems interesting. Let me try and let you know. – souvikd Jun 16 '21 at 05:17

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