1.8 Numerical Determination of Fourier Coefficients

1.8 Numerical Determination of Fourier Coefficients

• Most of the series disappears.

• The sum term by term is what Lemma 1 needs to be verified.

• The integrals involved in many functions are not known in terms of easily evaluated functions.
• It is possible that a function can be found at some point.
• If a Fourier series is to be found for the function, some numerical technique must be used to approximate the integrals that give the coefficients.
• One of the crudest numerical integration techniques is the best.

• We want to find its coefficients numerically.

• The trapezoidal rule is used to approximate the integral.

• The two terms can be combined.

• The usual coefficients name is used to designate approximations.

• The other coefficients are approximated in the same way.

• Calculating efficients can be done.

• Table 3 has the numerical information.

• Table 4 contains the results of the calculation.
• The approx imate coefficients are on the left.

• This is not a consideration when the calculation is done with a digital computer.

• The first month of the year represents the depth of the water in Lake Ontario.
• The mean level and fluctuations of period 12 months, 6 months, 4 months, and so forth are identified.

• The table shows the monthly precipi tation in Lake Placid, NY, from 1950 to 1959.