Max of Sum of Sines: Find the Max Value for Even n

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In summary, the "Max of Sum of Sines" problem is a mathematical optimization problem where the maximum value of a sum of sine functions is calculated. The maximum value is found by first determining the local maximum values for each individual sine function and then adding them together. The number of sine functions involved must be even for accurate calculation. This problem cannot be solved analytically and has applications in various fields such as signal processing, image processing, and physics.
  • #1
ekkilop
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Hi!

Consider the function

[itex] \frac{d^n}{dx^n} \sum_{k=1}^m \sin{kx}, \quad 0 \leq x \leq \pi/2 [/itex].

If [itex] n [/itex] is odd this function attains its largest value, [itex] \sum_{k=1}^m k^n [/itex] at [itex] x=0 [/itex]. But what about if [itex] n [/itex] is even? Where does the maximum occur and what value does it take?

Any help is much appreciated. Thank you!
 
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  • #2
Please make some attempt at the problem when you ask for help.
i.e. how would you normally go about finding the maximum?
Can you prove the statement about when m is odd?
 

1. What is the "Max of Sum of Sines" problem?

The "Max of Sum of Sines" problem is a mathematical optimization problem that involves finding the maximum possible value of a sum of sine functions. The problem is typically defined for an even number of sine functions, with each function having a different amplitude and frequency.

2. How is the maximum value for "Max of Sum of Sines" calculated?

The maximum value for "Max of Sum of Sines" is calculated by first finding the local maximum values for each individual sine function. These local maximum values are then added together to determine the overall maximum value for the sum of sines. This process is repeated for different combinations of amplitudes and frequencies until the global maximum value is found.

3. What is the significance of "even n" in the problem?

The "even n" in the problem refers to the number of sine functions involved. This number must be even because the sum of sines will have a symmetric pattern, with half of the functions producing positive values and the other half producing negative values. This symmetry is necessary for the maximum value to be accurately calculated.

4. Can the "Max of Sum of Sines" problem be solved analytically?

No, the "Max of Sum of Sines" problem cannot be solved analytically because it involves a complex combination of sine functions. The maximum value can only be found through numerical methods, such as using a computer algorithm to test different combinations of amplitudes and frequencies.

5. What are the real-world applications of the "Max of Sum of Sines" problem?

The "Max of Sum of Sines" problem has applications in fields such as signal processing, image processing, and data compression. It is also used in physics and engineering to model the behavior of waves and vibrations. Additionally, the problem has been studied in mathematics as an interesting and challenging optimization problem.

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