1.3 Convergence of Fourier Series

1.3 Convergence of Fourier Series

  • Consider the integral as a sum of signed areas.
  • Justify or prove the statements.
  • It is an even function.

  • In this section we will state that there are some theorems that answer the question without proof.
    • We need some definitions about limits and continuity.

  • The ordinary limit is equal to the one-handed limits if both left- and right-hand limits are equal.
    • There is a chance that the left and right-handed limits are different.
  • The right-hand limit is +2.
  • It is1-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-6556 The function will become continuous if the value at the troublesome point is equal to the limit.
    • We can assume that the discontinuities have been removed from any function.
  • Other discontinuities are more serious.
    • One or both of the one-handed limits fail to exist.

  • If a function is continuous on every interval, it is sectionally continuous.
    • If a periodic function is sectionally continuous on any interval whose length is more than one period, then it is sectionally continuous.

  • Some facts about the meaning of sectional con tinuity are clarified by the examples.
    • Sectionally continuous functions must not blow up at any point in an interval.
    • It is not necessary for a function to be defined at every point in order to be sectionally continuous.
  • The graph of a smooth function has a finite number of jumps, corners and discontinuities.
    • The derivative is infinite and no vertical tangents are allowed.

  • The wave is smooth but not continuous.
  • Sectionally smooth functions are used in mathematical modeling.
    • We can give a positive statement about the functions in the Fourier series.

  • The answer to the question is given at the beginning of the section.
  • A sectionally smooth function only has a finite number of jumps and no bad discontinuities.

  • It is periodic with period 2.
  • This shows that the conditions are too strong.

If the function is not sectionally smooth on the interval, then why not?

  • Check to see if the function described is section ally smooth.

  • Give an example.
  • From half-range expansions, the state convergence theorems for the Fourier series arise.