The expression \( \text{sinc}(w) \cdot \delta(w) \) involves the multiplication of the sinc function and the Dirac delta function. Here, \(\text{sinc}(w)\) is defined as:

\[ \text{sinc}(w) = \frac{\sin(\pi w)}{\pi w} \]

The Dirac delta function, \(\delta(w)\), is a generalized function or distribution that is zero everywhere except at \( w = 0 \), where it is infinite, and it integrates to one over the entire real line.

When multiplying \(\text{sinc}(w)\) by \(\delta(w)\), the result leverages the sifting property of the delta function, which extracts the value of the function it multiplies at the location where the delta function is centered. Mathematically, this is expressed as:

\[ \text{sinc}(w) \cdot \delta(w) = \text{sinc}(0) \cdot \delta(w) \]

Since \(\text{sinc}(0) = \frac{\sin(0)}{0} = 1\), the result simplifies to:

\[ \text{sinc}(w) \cdot \delta(w) = \delta(w) \]

Thus, the product of \(\text{sinc}(w)\) and \(\delta(w)\) is simply \(\delta(w)\).

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. What are the properties of the Dirac delta function?
2. How does the Dirac delta function interact with other common functions?
3. What is the Fourier transform of the sinc function?
4. How is the sinc function used in signal processing?
5. Can you explain the sifting property of the Dirac delta function?
6. How does the convolution of a function with the Dirac delta function work?
7. What is the inverse Fourier transform of the delta function?
8. How is the delta function used in solving differential equations?

**Tip:** When dealing with the Dirac delta function, remember that it is a distribution, not a traditional function, which means it is best understood through its effect on integrals.

Explore the mathematical concept of multiplying the sinc function and the Dirac delta function. Learn about the properties of the sinc function and the sifting property of the Dirac delta function. Suitable for advanced mathematics enthusiasts.

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