Is there a function for a sinc curve with increasing frequency at the tails?

Question

The idea is to simulate a signal with lower amplitude and higher frequency as we move away from zero in the positive and negative directions:

I am trying to find, perhaps, a modification of $\frac{\sin(x)}{x}.$ The function needs to be even.

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Thank you very much for the suggestions in the comments, but when I plot $\frac{\sin(x^3)}{x^3}$ I get some ugly flattened wave in the middle:

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I am looking for something like a modified sinc function as suggested in the comments, combined with a Gaussian, but without the big peak in the middle. See for instance

$$\sin(x^2)/x^2 + \exp(-x^2) $$

Another - not completely satisfactory - equation $(-\frac{\sin(x^2)}{x^2} + 2\frac{\cos(x^2)}{x^2})$:

Answer 1

Here is the plot of $\cos(12x^2) \cdot e^{-x^2}$:

Instead of dividing by $x$, which causes a lot of unnecessary issues, just use $\cos$ to start. Then multiply it by whatever envelope you'd like.

Answer 2

It could work by changing the input of an even oscillating function (cosine) by including a monotonously increasing function such as a logarithmic function truncated below $1.$ This would increase the frequency progressively. An exponential envelop would take care of the tapering:

$$f(x)=\begin{cases} 3\cos(8\log_2(\vert x\vert+1)x)e^{-\frac x 2}\quad x>0\\[2ex] 3\cos(8\log_2(\vert x\vert+1)x)e^{\frac x 2}\quad x<0 \end{cases}$$