Acceleration of Fourier Series Charles Moore Kansas State - - PowerPoint PPT Presentation

Acceleration of Fourier Series

Charles Moore

Kansas State University, U.S.A.

International Conference on Scientific Computing October 13, 2011

Introduction Functions with multiple jumps Comments, further results

Outline

Introduction Fourier series Acceleration Previous work Functions with multiple jumps Notation and lemmas Main results Some examples Comments, further results

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

Fourier Series

For a function f integrable on [−π, π] and n ∈ Z, we define the nth Fourier coefficient by ˆ f(n) := 1 2π

π

• −π

f(x)e−inxdx.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

Fourier Series

For a function f integrable on [−π, π] and n ∈ Z, we define the nth Fourier coefficient by ˆ f(n) := 1 2π

π

• −π

f(x)e−inxdx. For n = 0, 1, 2, . . . and x ∈ [−π, π], define the nth partial sum of the Fourier series: Snf(x) :=

n

• k=−n

ˆ f(k)eikx.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

Accelerating Convergence

The convergence of Fourier series can be quite slow and often depends on subtle cancellation.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

Accelerating Convergence

The convergence of Fourier series can be quite slow and often depends on subtle cancellation. This is especially true for functions with jump discontinuities.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

Accelerating Convergence

The convergence of Fourier series can be quite slow and often depends on subtle cancellation. This is especially true for functions with jump discontinuities. For a sequence {sn} that converges to s, we say that a transformation {tn} accelerates the convergence of {sn} if there exists a positive integer k such that each tn depends only

• n s1, s2, ..., sn+k and {tn} converges to s faster than {sn}, that

is lim

n→∞

tn − s sn − s = 0

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

The δ2 Transformation

Given a numerical sequence sn, the δ2 process transforms it to the sequence tn := sn − (sn+1 − sn)(sn − sn−1) (sn+1 − sn) − (sn − sn−1), where we set tn = sn if the denominator of the fraction is zero.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

The δ2 Transformation

Given a numerical sequence sn, the δ2 process transforms it to the sequence tn := sn − (sn+1 − sn)(sn − sn−1) (sn+1 − sn) − (sn − sn−1), where we set tn = sn if the denominator of the fraction is zero. If a complex series and its δ2 transform converge, then their sums are equal (Tucker, 1967).

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

The δ2 Transformation

Given a numerical sequence sn, the δ2 process transforms it to the sequence tn := sn − (sn+1 − sn)(sn − sn−1) (sn+1 − sn) − (sn − sn−1), where we set tn = sn if the denominator of the fraction is zero. If a complex series and its δ2 transform converge, then their sums are equal (Tucker, 1967). It is well known that the δ2 process behaves badly for many sequences; it can turn a convergent sequence into one with multiple convergent subsequences. It can also turn divergent sequences into convergent sequences.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

Applying this to Fourier Series

We consider the possibility of applying the δ2 transform pointwise to the partial sums Snf(x) of an integrable function f

• n [−π, π]. This results in the sequence of functions Tnf(x)

given by: Tnf(x) := Snf(x) − (Sn+1f(x) − Snf(x))(Snf(x) − Sn−1f(x)) (Sn+1f(x) − Snf(x)) − (Snf(x) − Sn−1f(x)),

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

Applying this to Fourier Series

We consider the possibility of applying the δ2 transform pointwise to the partial sums Snf(x) of an integrable function f

• n [−π, π]. This results in the sequence of functions Tnf(x)

given by: Tnf(x) := Snf(x) − (Sn+1f(x) − Snf(x))(Snf(x) − Sn−1f(x)) (Sn+1f(x) − Snf(x)) − (Snf(x) − Sn−1f(x)), Various authors have done numerical experiments: Smith and Ford, Drummond.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

Applying this to Fourier Series

We consider the possibility of applying the δ2 transform pointwise to the partial sums Snf(x) of an integrable function f

• n [−π, π]. This results in the sequence of functions Tnf(x)

given by: Tnf(x) := Snf(x) − (Sn+1f(x) − Snf(x))(Snf(x) − Sn−1f(x)) (Sn+1f(x) − Snf(x)) − (Snf(x) − Sn−1f(x)), Various authors have done numerical experiments: Smith and Ford, Drummond. If f(x) = −1 on [−π, 0) and f(x) = 1 on [0, π], then Snf( π

2) gives

the partial sums for the Leibnitz series 1 = f( π

2) = 4 π(1 − 1 3 + 1 5 − 1 7 + . . . ).

The convergence of this series is extremely slow but is dramatically accelerated by applying the δ2 process.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

The Transform for Functions With One Jump

Theorem (joint with E. Abebe and P . Graber, 2007)

(a) Suppose f ∈ C2([−π, π]) and that f(−π) = f(π). Then the transformed sequence of partial sums Tnf(x) diverges at every x of the form x = 2aπ, where a ∈ [−.5, .5] is irrational.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

The Transform for Functions With One Jump

Theorem (joint with E. Abebe and P . Graber, 2007)

(a) Suppose f ∈ C2([−π, π]) and that f(−π) = f(π). Then the transformed sequence of partial sums Tnf(x) diverges at every x of the form x = 2aπ, where a ∈ [−.5, .5] is irrational. (b) Suppose f is as above and let x := 2jπ

k where j k is in lowest

terms and k is odd. Then Tnf(x) has three limit points, f(x) and f(x) ±

α2 sin2 (x/2) α sin (x/2)+2β cos (x/2), where α = [f(π) − f(−π)]/π and

β = [f ′(π) − f ′(−π)]/π.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

Fourier series and transformed Fourier series for f(x) = x

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 3.5 4

Figure: S100f for f(x) = x.

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 3.5 4

Figure: T100f

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

The δ2 process is same as ε(n)

2

in family of transforms ε(n)

k .

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

The δ2 process is same as ε(n)

2

in family of transforms ε(n)

k .

Brezinski (see also Wynn): To Snf add conjugate function Snf; create analytic function Gnf(z). Apply epsilon algorithm to Gnf(eiθ); take real part.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

The δ2 process is same as ε(n)

2

in family of transforms ε(n)

k .

Brezinski (see also Wynn): To Snf add conjugate function Snf; create analytic function Gnf(z). Apply epsilon algorithm to Gnf(eiθ); take real part. Brezinski gave numerical experiments to demonstrate that this procedure reduces the Gibbs phenomenon.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

The δ2 process is same as ε(n)

2

in family of transforms ε(n)

k .

Brezinski (see also Wynn): To Snf add conjugate function Snf; create analytic function Gnf(z). Apply epsilon algorithm to Gnf(eiθ); take real part. Brezinski gave numerical experiments to demonstrate that this procedure reduces the Gibbs phenomenon. Beckermann, Matos, and Wielonsky show that this method accelerates convergence for functions of the form f = f1 + f2, where f1 has prescribed discontinuities but is smooth elsewhere, f2 has quickly decaying Fourier coefficients, and G(f1) = limn→∞ Gn(f1) is a certain type of hypergeometric function.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

Consider the analytic function

∞

• k=1

eick log k eikx k

1 2 +α

α ∈ R and c > 0. This was studied by Hardy and Littlewood.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

Consider the analytic function

∞

• k=1

eick log k eikx k

1 2 +α

α ∈ R and c > 0. This was studied by Hardy and Littlewood. If α > 0 then the partial sums are uniformly convergent; consequently, it is the Fourier series of a continuous function.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Fourier series Acceleration Previous work

Consider the analytic function

∞

• k=1

eick log k eikx k

1 2 +α

α ∈ R and c > 0. This was studied by Hardy and Littlewood. If α > 0 then the partial sums are uniformly convergent; consequently, it is the Fourier series of a continuous function.

Theorem

Suppose c > 0, 0 < α ≤ 1

• 2. Consider the partial sums

Sn(x) =

n

• k=1

eick log k eikx k

1 2 +α

and let Tn(x) be the sequence of functions which results from applying the δ2 process to the sequence Sn(x). Then at every x, Tn(x) fails to converge to the same limit as Sn(x). In fact, if 0 < α < 1

2, {Tn(x)} has subsequences which become

unbounded.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Some Notation

Throughout this section, f is a piecewise smooth function integrable on [−π, π] having a finite number of jump discontinuities at a1, a2, ..., am ∈ (−π, π).

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Some Notation

Throughout this section, f is a piecewise smooth function integrable on [−π, π] having a finite number of jump discontinuities at a1, a2, ..., am ∈ (−π, π). Suppose that for each j, f(aj±) = limx→aj ± f(x) and f ′(aj±) = limx→aj ± f ′(x) exist and are finite.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Some Notation

Throughout this section, f is a piecewise smooth function integrable on [−π, π] having a finite number of jump discontinuities at a1, a2, ..., am ∈ (−π, π). Suppose that for each j, f(aj±) = limx→aj ± f(x) and f ′(aj±) = limx→aj ± f ′(x) exist and are finite. Set dj = f(a+

j ) − f(a− j ) and d∗ j = f ′(a− j ) − f ′(a+ j ) for all

j ∈ {1, ..., m}.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Partial Sums of the Fourier Series

After some computation, the Nth partial sum of the Fourier series is

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Partial Sums of the Fourier Series

After some computation, the Nth partial sum of the Fourier series is SNf(x) = ˆ f(0) +

N

• k=1

  1 kπ

m

• j=1
• dj sin k(x − aj)
• + ǫk

  , where

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Partial Sums of the Fourier Series

After some computation, the Nth partial sum of the Fourier series is SNf(x) = ˆ f(0) +

N

• k=1

  1 kπ

m

• j=1
• dj sin k(x − aj)
• + ǫk

  , where ǫk = 1 k2π

m

• j=1
• d∗

j cos k(x − aj)

• −
• f ′′(k)eikx +

f ′′(−k)e−ikx k2 . We note that ǫN = O( 1

N2 ).

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Transformed Series

Applying the δ2 process and simplifying gives SNf(x) − TNf(x) ≈

• m
• j=1
• dj sin (N + 1)(x − aj)
• m
• j=1
• dj sin N(x − aj)
• Nπ

m

• j=1

[dj sin (N + 1)(x − aj)] − (N + 1)π

m

• j=1

[dj sin N(x − aj)] . Investigate the behavior of both the numerator and denominator

• f the large fraction on the right.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Denominator estimates

Lemma

For any fixed real number x0, on a subinterval of

• x0, x0 +

π N+1

• ,

g(x) =

m

• j=1

dj[sin (N + 1)(x − aj) − sin N(x − aj)] is bounded above by C

N .

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Denominator estimates

Lemma

For any fixed real number x0, on a subinterval of

• x0, x0 +

π N+1

• ,

g(x) =

m

• j=1

dj[sin (N + 1)(x − aj) − sin N(x − aj)] is bounded above by C

N .

proof: Roughly g(x0) and g(x0 +

π N+1) have opposite sign.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Denominator estimates

Lemma

For any fixed real number x0, on a subinterval of

• x0, x0 +

π N+1

• ,

g(x) =

m

• j=1

dj[sin (N + 1)(x − aj) − sin N(x − aj)] is bounded above by C

N .

proof: Roughly g(x0) and g(x0 +

π N+1) have opposite sign.

Thus, on 2(N + 1) subintervals of [−π, π] we estimate

• Nπ

m

• j=1

[dj sin (N + 1)(x − aj)] − (N + 1)π

m

• j=1

[dj sin N(x − aj)]

• ≤ Nπ|g(x)| + π

m

• j=1

|dj| ≤ C

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Set αN =

m

• j=1

dj cos Naj, βN =

m

• j=1

dj sin Naj, AN =

• αN2 + βN2,

φN = arctan βN αN

• , so that with this notation

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Set αN =

m

• j=1

dj cos Naj, βN =

m

• j=1

dj sin Naj, AN =

• αN2 + βN2,

φN = arctan βN αN

• , so that with this notation

m

• j=1

dj sin N(x − aj) =

m

• j=1

dj(sin Nx cos Naj − cos Nx sin Naj) = αN sin Nx − βN cos Nx = AN sin(Nx − φN).

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Set αN =

m

• j=1

dj cos Naj, βN =

m

• j=1

dj sin Naj, AN =

• αN2 + βN2,

φN = arctan βN αN

• , so that with this notation

m

• j=1

dj sin N(x − aj) =

m

• j=1

dj(sin Nx cos Naj − cos Nx sin Naj) = αN sin Nx − βN cos Nx = AN sin(Nx − φN). Note that if aj = pjπ

qj in lowest terms, AN is periodic of period

L = 2 lcm{qj}.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Another denominator estimate

m

• j=1

dj[ sin (N + 1)(x − aj) − sin N(x − aj)] = AN+1 sin ((N + 1)x − φN+1) − AN sin (Nx − φN)

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Another denominator estimate

m

• j=1

dj[ sin (N + 1)(x − aj) − sin N(x − aj)] = AN+1 sin ((N + 1)x − φN+1) − AN sin (Nx − φN)

Lemma

Suppose each aj is a rational multiple of π, aj = pj

qj π, in lowest

• terms. Suppose x = 2aπ, where a ∈ [−.5, .5] is irrational. Fix

J ∈ {1, ..., L}, L = 2 lcm{qj}, which has AJAJ+1 = 0. Then there are an infinite number of integers N = kL + J such that |AN+1 sin ((N + 1)x − φN+1) − AN sin (Nx − φN)| ≤ max{AJ, AJ+1}24L2π N .

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Another denominator estimate

m

• j=1

dj[ sin (N + 1)(x − aj) − sin N(x − aj)] = AN+1 sin ((N + 1)x − φN+1) − AN sin (Nx − φN)

Lemma

Suppose each aj is a rational multiple of π, aj = pj

qj π, in lowest

• terms. Suppose x = 2aπ, where a ∈ [−.5, .5] is irrational. Fix

J ∈ {1, ..., L}, L = 2 lcm{qj}, which has AJAJ+1 = 0. Then there are an infinite number of integers N = kL + J such that |AN+1 sin ((N + 1)x − φN+1) − AN sin (Nx − φN)| ≤ max{AJ, AJ+1}24L2π N . proof: Chebyshev’s theorem: Given a irrational, θ real, |na − m − θ| < 3/n, infinite # of m, n

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Numerator estimates

We want to estimate  

m

• j=1
• dj sin (N + 1)(x − aj)
• 

  

m

• j=1
• dj sin N(x − aj)
• 

 = AN+1 sin((N + 1)x − φN+1) AN sin(Nx − φN)

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Numerator estimates

We want to estimate  

m

• j=1
• dj sin (N + 1)(x − aj)
• 

  

m

• j=1
• dj sin N(x − aj)
• 

 = AN+1 sin((N + 1)x − φN+1) AN sin(Nx − φN) Suppose x is a point at which |AN+1 sin((N + 1)x − φN+1) − AN sin(Nx − φN)| ≤ C N

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Numerator estimates

We want to estimate  

m

• j=1
• dj sin (N + 1)(x − aj)
• 

  

m

• j=1
• dj sin N(x − aj)
• 

 = AN+1 sin((N + 1)x − φN+1) AN sin(Nx − φN) Suppose x is a point at which |AN+1 sin((N + 1)x − φN+1) − AN sin(Nx − φN)| ≤ C N Squaring each side and rearranging gives

ANAN+1 sin ((N + 1)x − φN+1) sin (Nx − φN) ≥ 1 2 min{AN, AN+1}2 sin2 ((N + 1)x − φN+1) + sin2 (Nx − φN)

• −

C N 2 . Except for a small set, the term in brackets is bounded below.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Theorems

Theorem (joint with E. Jennings, D. Muñiz, A. Toth)

Let f be a piecewise C2 function on [−π, π] having a finite number of jump discontinuities at a1 < a2 < ... < am ∈ (−π, π), and suppose f(aj±) and f ′(aj±) exist and are finite.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Theorems

Theorem (joint with E. Jennings, D. Muñiz, A. Toth)

Let f be a piecewise C2 function on [−π, π] having a finite number of jump discontinuities at a1 < a2 < ... < am ∈ (−π, π), and suppose f(aj±) and f ′(aj±) exist and are finite. (a) Suppose N is a sufficiently large positive integer with ANAN+1 = 0. Then there exists intervals on which |TNf(x) − SNf(x)| ≥ m min{AN, AN+1}2 − C

N , where m and C

are constants which do not depend on N or x.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Theorems

Theorem (joint with E. Jennings, D. Muñiz, A. Toth)

Let f be a piecewise C2 function on [−π, π] having a finite number of jump discontinuities at a1 < a2 < ... < am ∈ (−π, π), and suppose f(aj±) and f ′(aj±) exist and are finite. (a) Suppose N is a sufficiently large positive integer with ANAN+1 = 0. Then there exists intervals on which |TNf(x) − SNf(x)| ≥ m min{AN, AN+1}2 − C

N , where m and C

are constants which do not depend on N or x. Proof: For (a) there are 2(N + 1) intervals of length C

N2 ,

uniformly spaced, where the denominator of TNf(x) − SNf(x) is

• bounded. The numerator is estimated using the previous

lemma.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Theorem (continued)

Let f be a piecewise C2 function on [−π, π] having a finite number of jump discontinuities at a1 < a2 < ... < am ∈ (−π, π), and suppose f(aj±) and f ′(aj±) exist and are finite.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Theorem (continued)

Let f be a piecewise C2 function on [−π, π] having a finite number of jump discontinuities at a1 < a2 < ... < am ∈ (−π, π), and suppose f(aj±) and f ′(aj±) exist and are finite. (b) Suppose that the aj are rational multiples of π, aj = pj

qj π (in

lowest terms), so that the sequence AN has period L = 2 lcm{qj}. Suppose that there exists a J ∈ {1, . . . , L} such that AJAJ+1 = 0. Then TNf does not converge to SNf uniformly.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Theorem (continued)

Let f be a piecewise C2 function on [−π, π] having a finite number of jump discontinuities at a1 < a2 < ... < am ∈ (−π, π), and suppose f(aj±) and f ′(aj±) exist and are finite. (b) Suppose that the aj are rational multiples of π, aj = pj

qj π (in

lowest terms), so that the sequence AN has period L = 2 lcm{qj}. Suppose that there exists a J ∈ {1, . . . , L} such that AJAJ+1 = 0. Then TNf does not converge to SNf uniformly. For (b), the AN are in this case periodic, say of period L. Then for N = kL + J, there are x where |TNf(x)−SNf(x)| ≥ m min{AN, AN+1}2−C N = m min{AJ, AJ+1}2−C N

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Theorem

Suppose x = 2aπ, where a is an irrational number, and suppose that there exists J ∈ {1, ..., L} such that AJAJ+1 = 0. Then {TNf(x)} fails to converge.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Theorem

Suppose x = 2aπ, where a is an irrational number, and suppose that there exists J ∈ {1, ..., L} such that AJAJ+1 = 0. Then {TNf(x)} fails to converge.

Proof.

For x = 2πa, a irrational, second denominator Lemma immediately gives an infinite number of k (which depend on x) for which the denominator with N = kL + J is bounded. Then previous lemma gives a lower bound for the numerator at these points.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

Indicator function of the interval [−1, 1].

−3 −2 −1 1 2 3 −0.5 0.5 1 1.5

Figure: T20f

−3 −2 −1 1 2 3 −0.5 0.5 1 1.5

Figure: T30f

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

A situation where the hypotheses are not satisfied

Define f1 by f1(x) =           

3 πx + 1

if x ∈ [−π, −2π

3 ) 3 2πx + 1

if x ∈ ( −2π

3 , 0)

if x ∈ (0, 2π

3 ) 3 πx − 1

if ∈ ( 2π

3 , π].

which is a function with jumps of 1 evenly spaced. The graphs

• f S20f1 and T20f1 are shown in figure 2. Here, ANAN+1 = 0 for

all N and our theorems do not apply; indeed it is difficult to distinguish the graphs. Recall: αN =

m

• j=1

dj cos Naj, βN =

m

• j=1

dj sin Naj, AN =

• αN2 + βN2, φN = arctan

βN αN

• Charles Moore

Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

A situation where the hypotheses are not satisfied

−3 −2 −1 1 2 3 −1.5 −1 −0.5 0.5 1 1.5

Figure: S20f

−3 −2 −1 1 2 3 −1.5 −1 −0.5 0.5 1 1.5

Figure: T20f

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

A small change in the last example; hypotheses now hold

Define f2 by f2(x) =           

3 πx + 1

if x ∈ [−π, −2π

3 ) 3 4πx + 1 2

if x ∈ ( −2π

3 , 0)

if x ∈ (0, 2π

3 ) 3 πx − 1

if ∈ ( 2π

3 , π].

This has evenly spaced jumps, but the jumps are not equal, and it is easy to check that Theorem 2 does apply. The next frame gives the graphs of S20f2 and T20f2.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results Notation and lemmas Main results Some examples

A small change in the last example; hypotheses now hold

−3 −2 −1 1 2 3 −1.5 −1 −0.5 0.5 1 1.5

Figure: S20f

−3 −2 −1 1 2 3 −1.5 −1 −0.5 0.5 1 1.5

Figure: T20f

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results

Iteration of δ2

Consider the iterated transform, T 2

n . Suppose f is C2 except for

• ne jump, and x is of the form x = 2πj

k , k odd, where j and k

have no common factors.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results

Iteration of δ2

Consider the iterated transform, T 2

n . Suppose f is C2 except for

• ne jump, and x is of the form x = 2πj

k , k odd, where j and k

have no common factors. Claim: T 2

n f(x) has subsequences which do not converge to

f(x).

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results

Iteration of δ2

Consider the iterated transform, T 2

n . Suppose f is C2 except for

• ne jump, and x is of the form x = 2πj

k , k odd, where j and k

have no common factors. Claim: T 2

n f(x) has subsequences which do not converge to

f(x). Recall: Tnf(x) has three limit points, f(x) and f(x) ±

α2 sin2 (x/2) α sin (x/2)+2β cos (x/2), where α = [f(π) − f(−π)]/π and

β = [f ′(π) − f ′(−π)]/π.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results

Iteration of δ2

The claim follows from:

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results

Iteration of δ2

The claim follows from:

Lemma

Suppose an is a sequence and l, k are fixed positive integers with k ≥ 2, such that an → a if n = l + mk and m = 1, 2, . . . and an → a + b for n = l + mk, m = 1, 2, . . . . Then T(an) → a + b

2

for n = k + ml as m → ∞.

• Proof. Notice that for n = l + mk,

T(an) =an − (an+1 − an)(an − an−1) (an+1 − an) − (an − an−1) → (a + b) − [a − (a + b)][(a + b) − a] [a − (a + b)] − [(a + b) − a] = a + b 2.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results

Shanks transformations – ε algorithm

Analyzing ε4(SNf(x)) seems to be much more difficult.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results

Shanks transformations – ε algorithm

Analyzing ε4(SNf(x)) seems to be much more difficult. ε−1(n) := 0 for every n, ε0(n) := Snf(x)

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results

Shanks transformations – ε algorithm

Analyzing ε4(SNf(x)) seems to be much more difficult. ε−1(n) := 0 for every n, ε0(n) := Snf(x) εk+1(n) = εk−1(n + 1) +

1 εk(n+1)−εk(n).

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results

Shanks transformations – ε algorithm

Analyzing ε4(SNf(x)) seems to be much more difficult. ε−1(n) := 0 for every n, ε0(n) := Snf(x) εk+1(n) = εk−1(n + 1) +

1 εk(n+1)−εk(n).

ε4(n) = Tn+2 + 1 [

1 Tn+2−Tn+2 ] + [ 1 Tn+3−Tn+2 ] − [ 1 Tn+2−Tn+1 ]

Suppose f is C2 with one jump, x = 2πj

k , where j and k have no

common factors and k is odd. There are subsequences along which Tn(x) → f(x) +

α2 sin2(x/2) α sin(x/2)+2β cos(x/2).

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results

Shanks transformations – ε algorithm

Analyzing ε4(SNf(x)) seems to be much more difficult. ε−1(n) := 0 for every n, ε0(n) := Snf(x) εk+1(n) = εk−1(n + 1) +

1 εk(n+1)−εk(n).

ε4(n) = Tn+2 + 1 [

1 Tn+2−Tn+2 ] + [ 1 Tn+3−Tn+2 ] − [ 1 Tn+2−Tn+1 ]

Suppose f is C2 with one jump, x = 2πj

k , where j and k have no

common factors and k is odd. There are subsequences along which Tn(x) → f(x) +

α2 sin2(x/2) α sin(x/2)+2β cos(x/2).

Direct computation shows that curiously, the stray subsequences of Tnf(x) which diverged from f(x) are transformed back to the correct limit.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results

The Lubkin transform

Let ρNf(x) = SN+1f(x)−SNf(x)

SNf(x)−SN−1f(x). The Lubkin transform of SNf(x) is

TNf(x) := SNf(x) + (SN+1f(x) − SNf(x))(1 − ρN+1f(x)) 1 − 2ρN+1f(x) + ρNf(x)ρN+1f(x) .

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results

The Lubkin transform

Let ρNf(x) = SN+1f(x)−SNf(x)

SNf(x)−SN−1f(x). The Lubkin transform of SNf(x) is

TNf(x) := SNf(x) + (SN+1f(x) − SNf(x))(1 − ρN+1f(x)) 1 − 2ρN+1f(x) + ρNf(x)ρN+1f(x) .

Theorem (with Boggess, Bunch, 2008)

Suppose that f ∈ C2([−π, π]) and that f(−π) = f(π). Consider the sequence TNf(x) formed by applying the Lubkin transform to the sequence SNf(x). Then TNf(x) fails to converge to f(x) at every x of the form x = 2πa, where a ∈ (− 1

4, 1 4) is irrational.

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results

f(x) = x and the Lubkin transform

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Figure: Fifty terms of the Fourier series

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Figure: Lubkin transform– 50th term

Charles Moore Acceleration of Fourier Series

Introduction Functions with multiple jumps Comments, further results

Thanks again to the organizers for the invitation and thank you for listening.

Charles Moore Acceleration of Fourier Series

Appendix References

References I

A.C. Aitken, On Bernoulli’s numerical solution of algebraic equations, Proc. Roy. Soc. Edinburgh 46 (1926), 289-305.

• E. Abebe, J. Graber, and C. N. Moore, Fourier Series and

the δ2 Process, J. Comput. Appl. Math. 224 (2009), no. 1, 146-151.

• J. Boggess, E. Bunch, C. N. Moore, Fourier series and the

Lubkin W-transform, Numer. Algorithms 47 (2008), 133-142.

• C. Brezinski, Accélération de la Convergence en Analyse

Numérique, Springer-Verlag, Berlin, Heidelberg, New York. 1977.

Charles Moore Acceleration of Fourier Series

Appendix References

References II

• C. Brezinski, Extrapolation algorithms for filtering series of

functions, and treating the Gibbs phenomenon, Numer. Algorithms 36 (2004), 309-329.

• C. Brezinski. and M. Redivo-Zaglia, Extrapolation Methods.

Theory and Practice. North-Holland, Amsterdam, 1991. J-.P Delahaye, Sequence Transformations, Springer Series in Computational Mathematics 11, Springer Verlag, Berlin,

• J. E. Drummond, Convergence speeding, convergence and

summability, J. Comput. Appl. Math. 11 (1984), 145-159.

• A. Ya. Khinchin, Continued Fractions, The University of

Chicago Press, Chicago, 1964,

Charles Moore Acceleration of Fourier Series

Appendix References

References III

• C. N. Moore, Acceleration of Fourier Series, J. Analysis, 17

(2009), 1-20.

• D. Shanks, Non-linear transformations of divergent and

slowly convergent sequences, J. Math. Phys. 34 (1955), 1-42.

• A. Sidi, Practical Extrapolation Methods: Theory and
• Applications. Cambridge University Press, 1st Edition,

2003.

• D. A. Smith, and W. F

. Ford, Numerical Comparisons of Nonlinear Convergence Accelerators, Math. Comp. 38, no. 158 (1982), 481-499.

Charles Moore Acceleration of Fourier Series

Appendix References

References IV

• R. Tucker, The δ2-Process and related topics, Pacific J.
• Math. 22 , no. 2 (1967), 349-359.
• R. Tucker, The δ2-Process and related topics II, Pacific J.
• Math. 28 , no. 2 (1969), 455-463.
• J. Wimp, Sequence Transformations and Their

Applications, Academic Press, New York. 1981. P . Wynn, Transformations to accelerate the convergence of Fourier series, in: Gertrude Blanch Anniversary Volume (Wright Patterson Air Force Base, Aerospace Research Laboratories, Office of Aerospace Research, United States Air Force, 1967), 339-379.

Charles Moore Acceleration of Fourier Series

Appendix References

References V

• A. Zygmund, Trigonometric Series, Second Edition.

Cambridge University Press, Cambridge, 1959.

Charles Moore Acceleration of Fourier Series