Is the Set {cos x, cos 2x, cos 3x, ...} Orthogonal Using Integral Products?

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Discussion Overview

The discussion revolves around the orthogonality of the set {cos x, cos 2x, cos 3x, ...} using integral products. Participants explore the mathematical definitions and properties related to orthogonality and integral products over a specified interval.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant asks how to prove the orthogonality of the set using the integral product.
  • Another participant suggests defining the integral product and the concept of an orthogonal set before applying these definitions to the problem.
  • A participant claims that over the interval -pi to pi, the integral of cos(mx)cos(nx)dx is zero for integer values of m and n, implying that any pair from the set satisfies the orthogonality condition.
  • A later reply reiterates the previous claim about the integral being zero and suggests using integration by parts for those unsure of how to begin the proof.

Areas of Agreement / Disagreement

Participants generally agree on the mathematical property that the integral of the product of cos(mx) and cos(nx) is zero for distinct integers m and n, supporting the notion of orthogonality. However, the discussion does not reach a consensus on the overall proof structure or methodology.

Contextual Notes

The discussion does not clarify the specific conditions under which the integral is evaluated or the assumptions regarding the definitions of orthogonality and integral products.

Who May Find This Useful

Readers interested in mathematical proofs, particularly in the context of orthogonality in function spaces, may find this discussion relevant.

roto25
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How would you prove, using the integral product, that the set of {cos x, cos 2x, cos 3x, cos 4x, ...} is an orthogonal set?
 
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welcome to pf!

hi roto25! welcome to pf! :wink:

i] define the integral product

ii] define orthogonal set

iii] apply i and ii …

what do you get? :smile:​
 
over the interval -pi to pi, the integral of cos(mx)cos(nx)dx is zero, as long as m and n are integers. Therefore, if you select ANY pair of elements from the set, the 'integral of their product' will be zero, thereby satisfying the condition of orthogonality.
 
Bavid said:
over the interval -pi to pi, the integral of cos(mx)cos(nx)dx is zero, as long as m and n are integers. Therefore, if you select ANY pair of elements from the set, the 'integral of their product' will be zero, thereby satisfying the condition of orthogonality.

On top of what Bavid said if you don't know where to start set up the integral and use integration by parts.
 

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