Emily Jennings. Georgia Institute of Technology. Nebraska Conference for Undergraduate Women in Mathematics, 2012

Transcription

1 δ 2 Transform and Fourier Series of Functions with Multiple Jumps Georgia Institute of Technology Nebraska Conference for Undergraduate Women in Mathematics, 2012 Work performed at Kansas State University s REU with students Daniel Muñiz (University of Florida) and Ashley Toth (Rollins College) under Dr. Charles Moore.

2 Fourier Series Background Previous Work Fourier series are used to approximate a periodic function using a combination of sines and cosines. Heat equation and Joseph Fourier Similarly used to find eigensolutions/simple solutions to PDEs.

3 Fourier Series Background Previous Work Fourier series are used to approximate a periodic function using a combination of sines and cosines. Heat equation and Joseph Fourier Similarly used to find eigensolutions/simple solutions to PDEs.

4 Fourier Series Background Previous Work Fourier series are used to approximate a periodic function using a combination of sines and cosines. Heat equation and Joseph Fourier Similarly used to find eigensolutions/simple solutions to PDEs.

5 Fourier Series Background Previous Work For a function f integrable on [ π, π], we define the Fourier coefficients by for each integer n 1. a 0 := 1 2π π π π f (x)dx a n := 1 f (x) cos nxdx π π π b n := 1 f (x) sin nxdx π π

6 Fourier Series Background Previous Work The N th partial sum of the Fourier series is N S N f (x) := a 0 + a k cos kx + b k sin kx, k=1 where N is a positive integer and x [ π, π]. Complex exponentials: Euler s formula e inx = cos nx + i sin nx

7 Fourier Series Background Previous Work The N th partial sum of the Fourier series is N S N f (x) := a 0 + a k cos kx + b k sin kx, k=1 where N is a positive integer and x [ π, π]. Complex exponentials: Euler s formula e inx = cos nx + i sin nx

8 Fourier Series Background Previous Work Using complex exponentials, we define the Fourier coefficients for a function f integrable on [ π, π] by, for each integer n. 1 ˆf (n) := 2π π π f (x)e inx dx The N th partial sum of the Fourier series is S N f (x) := N k= N ˆf (k)e ikx, where N is a positive integer and x [ π, π].

9 Fourier Series Background Previous Work Using complex exponentials, we define the Fourier coefficients for a function f integrable on [ π, π] by, for each integer n. 1 ˆf (n) := 2π π π f (x)e inx dx The N th partial sum of the Fourier series is S N f (x) := N k= N ˆf (k)e ikx, where N is a positive integer and x [ π, π].

10 Accelerating Convergence Background Previous Work The Fourier series for any function in L 2 converges. Problem: Convergence of Fourier series of certain functions can be relatively slow. Example: Functions with jump discontinuities. If (s n ) s, we say that a transformation t n accelerates the convergence of (s n ) if there exists a k > 0 such that each t n depends only on s 1, s 2,..., s n+k and (t n ) converges to s faster than (s n ).

11 Accelerating Convergence Background Previous Work The Fourier series for any function in L 2 converges. Problem: Convergence of Fourier series of certain functions can be relatively slow. Example: Functions with jump discontinuities. If (s n ) s, we say that a transformation t n accelerates the convergence of (s n ) if there exists a k > 0 such that each t n depends only on s 1, s 2,..., s n+k and (t n ) converges to s faster than (s n ).

12 Accelerating Convergence Background Previous Work The Fourier series for any function in L 2 converges. Problem: Convergence of Fourier series of certain functions can be relatively slow. Example: Functions with jump discontinuities. If (s n ) s, we say that a transformation t n accelerates the convergence of (s n ) if there exists a k > 0 such that each t n depends only on s 1, s 2,..., s n+k and (t n ) converges to s faster than (s n ).

13 Accelerating Convergence Background Previous Work The Fourier series for any function in L 2 converges. Problem: Convergence of Fourier series of certain functions can be relatively slow. Example: Functions with jump discontinuities. If (s n ) s, we say that a transformation t n accelerates the convergence of (s n ) if there exists a k > 0 such that each t n depends only on s 1, s 2,..., s n+k and (t n ) converges to s faster than (s n ).

14 The δ 2 Transformation Background Previous Work What happens when we apply the δ 2 transform, a convergence acceleration method, to the partial sums of the Fourier series? For the sequence (S N f (x)), the δ 2 transform results in a sequence with terms as follows: t N := S N (S N+1 S N )(S N S N 1 ) (S N+1 S N ) (S N S N 1 ), where we set t N = S N if the denominator of the fraction is zero.

15 The δ 2 Transformation Background Previous Work What happens when we apply the δ 2 transform, a convergence acceleration method, to the partial sums of the Fourier series? For the sequence (S N f (x)), the δ 2 transform results in a sequence with terms as follows: t N := S N (S N+1 S N )(S N S N 1 ) (S N+1 S N ) (S N S N 1 ), where we set t N = S N if the denominator of the fraction is zero.

16 Background Previous Work Some General Previous If a complex series and its δ 2 transform converge, then their sums are equal (Tucker, 1967). Ciszweski, Gregory, Moore and West (2011) found continuous functions for which the transformed Fourier series fails to converge.

17 Background Previous Work Some General Previous If a complex series and its δ 2 transform converge, then their sums are equal (Tucker, 1967). Ciszweski, Gregory, Moore and West (2011) found continuous functions for which the transformed Fourier series fails to converge.

18 Some Notation Overview of Examples Throughout this presentation, f is a piecewise smooth function integrable on [ π, π] having a finite number of jump discontinuities at a 1, a 2,..., a m ( π, π). Let d j = f (a + j ) f (a j ) and dj = f (a j ) f (a + j ) for all j {1,..., m}. (Assume the derivatives f (a j ) and f (a + j ) exist and are finite.)

19 Some Notation Overview of Examples Throughout this presentation, f is a piecewise smooth function integrable on [ π, π] having a finite number of jump discontinuities at a 1, a 2,..., a m ( π, π). Let d j = f (a + j ) f (a j ) and dj = f (a j ) f (a + j ) for all j {1,..., m}. (Assume the derivatives f (a j ) and f (a + j ) exist and are finite.)

20 Partial Sum of Fourier Series Overview of Examples After some computation, the N th partial sum of the Fourier series is S N f (x) = ˆf N (0) + 1 m [ dj sin k(x a j ) ] + ɛ k, kπ where ɛ k = 1 k 2 π m k=1 [ ] dj f cos k(x a j ) (k)e ikx + f ( k)e ikx k 2.

21 Partial Sum of Fourier Series Overview of Examples After some computation, the N th partial sum of the Fourier series is S N f (x) = ˆf N (0) + 1 m [ dj sin k(x a j ) ] + ɛ k, kπ where ɛ k = 1 k 2 π m k=1 [ ] dj f cos k(x a j ) (k)e ikx + f ( k)e ikx k 2.

22 Transformed Series Overview of Examples We note that ɛ k O( 1 k 2 ) and call this the error term. Applying the δ 2 process and simplifying gives a sequence of transformed partial sums t N f (x) S N f (x) ( m [ dj sin (N + 1)(x a j ) ]) ( m [ dj sin N(x a j ) ]) Nπ m [d j sin (N + 1)(x a j )] (N + 1)π m [d j sin N(x a j )]. (1)

23 Transformed Series Overview of Examples We note that ɛ k O( 1 k 2 ) and call this the error term. Applying the δ 2 process and simplifying gives a sequence of transformed partial sums t N f (x) S N f (x) ( m [ dj sin (N + 1)(x a j ) ]) ( m [ dj sin N(x a j ) ]) Nπ m [d j sin (N + 1)(x a j )] (N + 1)π m [d j sin N(x a j )]. (1)

24 Transformed Series Overview of Examples We note that ɛ k O( 1 k 2 ) and call this the error term. Applying the δ 2 process and simplifying gives a sequence of transformed partial sums t N f (x) S N f (x) ( m [ dj sin (N + 1)(x a j ) ]) ( m [ dj sin N(x a j ) ]) Nπ m [d j sin (N + 1)(x a j )] (N + 1)π m [d j sin N(x a j )]. (1)

25 Two Lemmas Overview of Examples Lemma 1: Denominator is bounded above (small). For a fixed real number x 0, the function g(x) = m [ d j sin (N + 1)(x aj ) sin N(x a j ) ] is bounded above [ by 2C N, where ] C is some constant, on a subinterval of x 0, x 0 +. π N+1 Lemma 2: Numerator is bounded below (not zero). If the points of discontinuity of f are restricted to rational multiples of π, then the numerator of fraction (1) is bounded below by a positive constant at the points where the denominator is close to zero.

26 Two Lemmas Overview of Examples Lemma 1: Denominator is bounded above (small). For a fixed real number x 0, the function g(x) = m [ d j sin (N + 1)(x aj ) sin N(x a j ) ] is bounded above [ by 2C N, where ] C is some constant, on a subinterval of x 0, x 0 +. π N+1 Lemma 2: Numerator is bounded below (not zero). If the points of discontinuity of f are restricted to rational multiples of π, then the numerator of fraction (1) is bounded below by a positive constant at the points where the denominator is close to zero.

27 Main Theorem Overview of Examples Theorem: (t n ) does not converge to f. If f fits our constraints described and if there is at least one value of N for which the numerator of (1) is not 0, then for each N N, there exist intervals for which the fraction in (1) is greater than or equal to M R.

28 Overview of Examples Rearranging the numerator of (1), define r N := m [ ] m cos Naj + i sin Na j = d j e ina j. We can show that if for any N, r N = 0 or r N+1 = 0, then the numerator of (1) is 0 for all N N. (The theorem does not apply.)

29 Overview of Examples Rearranging the numerator of (1), define r N := m [ ] m cos Naj + i sin Na j = d j e ina j. We can show that if for any N, r N = 0 or r N+1 = 0, then the numerator of (1) is 0 for all N N. (The theorem does not apply.)

30 Example 1 Overview of Examples Example 1. Consider a function f with jump discontinuities at 3π 4, π 4, π 4, and 3π 4 such that d 1 = d 3 and d 2 = d 4. Then r N = 0 for N odd, so the numerator of (1) is 0 for all N.

31 Example 1 Overview of Examples Example 1. Consider a function f with jump discontinuities at 3π 4, π 4, π 4, and 3π 4 such that d 1 = d 3 and d 2 = d 4. Then r N = 0 for N odd, so the numerator of (1) is 0 for all N.

32 Example 1 Overview of Examples 4 20 terms of series 4 transformed 3 20 terms of series 3 transformed terms of series of a function with equal jumps 1 1 (d 1 = d 0 3 = 2, d 2 = d 0 4 = 1) transformed Figure: δ 2 applied to partial sums terms of series transformed 3 3 Figure: 2 δ 2 applied to 2partial sums 1 1 of a function with unequal jumps 0 0 (d 1 = 1 d 3 = 2, d 2 = 1, d 4 = 2)

33 Example 2 Overview of Examples Example 2. Consider a function f with jump discontinuities at 2π 3, 0, and 2π 3 such that d 1 = d 2 = d 3. Then r N = 0 for N not a multiple of 3, so the numerator of (1) is 0 for all N.

34 Example 2 Overview of Examples Example 2. Consider a function f with jump discontinuities at 2π 3, 0, and 2π 3 such that d 1 = d 2 = d 3. Then r N = 0 for N not a multiple of 3, so the numerator of (1) is 0 for all N.

35 Example 2 20 terms of series 3 3 transformed Overview of Examples 4 20 terms of series 4 transformed terms of series 3 transformed 4 30 terms of series 4 transformed Figure: δ 2 applied to partial sums of a function with equal jumps (d 1 = d 2 = d 3 = 1). Figure: δ 2 applied to partial sums of a function with unequal jumps (d 1 = 3,d 2 = 2, d 3 = 1).

36 Directions for Investigate: Functions with a finite number of jumps anywhere in the interval [ π, π]. Functions with an infinite number of jumps. Other non-linear transforms applied to Fourier series.

37 Directions for Investigate: Functions with a finite number of jumps anywhere in the interval [ π, π]. Functions with an infinite number of jumps. Other non-linear transforms applied to Fourier series.

38 Directions for Investigate: Functions with a finite number of jumps anywhere in the interval [ π, π]. Functions with an infinite number of jumps. Other non-linear transforms applied to Fourier series.

39 Appendix References References I E. Abebe, J. Graber, and C. N. Moore, Fourier Series and the δ 2 Process, J. Computational and Applied Math. 224 (2009), no. 1, C. N. Moore, Acceleration of Fourier Series, J. Analysis, 17 (2009), D. Shanks, Non-linear transformations of divergent and slowly convergent sequences, J. Math. Phys. 34 (1955), R. Tucker, The δ 2 -Process and related topics,pacific Journal of Mathematics, 22 (1967), no. 2. J. Wimp, Sequence Transformations and Their Applications, Academic Press, New York