Fourier and east Squares Convergence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 7, 28
Outline
et s look at the original Fourier sin series and Fourier Cosine Series. These are ( ( 2 iπ iπ FS(x = f (x, sin x sin x and FC(x = < f, > + ( ( 2 iπ iπ f (x, cos x cos x. To understand the convergence of the Fourier Sine Series on [, ] we extend f from [, ] to [, 2] as an odd function and then use the convergence analysis we have already completed on the interval [, 2] to infer convergence of the Fourier Sine Series on [, ]. Then to understand the convergence of the Fourier Cosine Series on [, ] we extend f from [, ] to [, 2] as an even function and then use the convergence analysis we have already completed on the interval [, 2] to infer convergence of the Fourier Cosine Series on [, ].
et f be defined only on the interval [, ]. Extend f to be an odd function f o on [, 2] as follows: f o (x = { f (x, x, f (2 x, < x 2. Then extend f o periodically as usual to ˆf o. The Fourier coefficient for the sin terms are then 2 f o (t sin t dt = f (t sin t dt + 2 ( f (2 t s Consider the second integration. Making the change of variable y = 2 t, we find 2 ( f (2 t sin t dt = ( iπ ( f (y sin (2 y ( dy
But iπ (2 y = 2iπ iπ y and since sin function is 2π periodic, we have ( iπ ( f (y sin (2 y dy = ( f (y sin ( y ( dy = f (y sin (y dy. This is the same as the first integral. Hence, we have shown 2 f o (t sin t dt = 2 f (t sin t dt. The terms corresponding to the cos parts will all then be zero. The argument is straightforward. For i >, 2 f o (t cos t dt = + 2 ( iπ ( f (2 t cos t ( iπ f (t cos t dt. dt
Consider the second integration. Making the change of variable y = 2 t, we find 2 ( f (2 t cos t dt = ( iπ ( f (y cos (2 y ( dy. Again, iπ (2 y = 2iπ iπ y and since cos term 2π periodic and cos is an even function, we have 2 ( f (2 t cos t dt = f (y cos y dy f (y cos y dy which is the negative of the first integral. So all of these coefficients are zero. The case i = is 2 f o (tdt = because f is odd on this interval.
Thus, all the cos based terms in the Fourier series vanish. The Fourier series on the interval [, 2] of the odd extension f o becomes the standard Fourier sine series on the interval [, ] of the function f. We know this converges to f (x at each point x where f is differentiable and converges to the average 2 (f (x + + f (x at each point x where f satisfies lim y x ± f (y exists. Note because the sin functions are always at the endpoints and, this series must converge to at those points. et s look at the Fourier cos series. et f be defined only on the interval [, ]. Extend f to be an even function f e on [, 2] as follows: f e (x = { f (x, x, f (2 x, < x 2. Then extend f e periodically as usual to ˆf e.
The Fourier coefficient for the sin terms are now 2 f e (t sin t dt = f (t sin t dt + 2 f (2 t sin t dt. Consider the second integration. Making the change of variable y = 2 t, we find 2 f (2 t sin t dt = = ( iπ f (y sin ( iπ f (y sin ( y (2 y ( dy dy. However, sin is an odd function and thus the second integral is the negative of the first and these coefficients vanish.
Next, consider the first Fourier cos coefficient. This is 2 2 f e (tdt = 2 = 2 f (tdt + f (tdt + 2 f (2 tdt f (y( dy = f (tdt. Now let s look at the other cos based coefficients. We have 2 f e (t cos t dt = f (t cos t dt + 2 f (2 t cos t dt.
Consider the second integration. Making the change of variable y = 2 t, we find 2 f (2 t cos t dt = = ( iπ f (y cos (2 y ( dy ( iπ f (y cos (2 y dy Again, iπ (2 y = 2iπ iπ y and since cos term 2π periodic and cos is an even function, we have 2 ( f (2 t cos t dt = f (y cos y dy This is the same as the first integral.
Thus, the Fourier series on the interval [, 2] of the even extension f e becomes the standard Fourier cosine series on the interval [, ] of the function f < f, ˆv i > ˆv i = < f (x, > + ( ( 2 iπ iπ f (x, cos x cos x. i= We know this series converges to f (x at each point x where f is differentiable and converges to the average 2 (f (x + + f (x at each point x where f satisfies lim y x ± f (y exists.
In addition to the pointwise convergence and uniform convergence of sequences and series of functions, we can add the idea of least squares convergence which is the same as convergence with respect to 2 on C([a, b]. Definition et {u n } be a sequence of integrable functions defined on the interval [a, b] We say S n converges in least squares to the integrable function u on [a, b] if given a positive ɛ, there is a positive integer N so that b n > N = a u n(t u(t 2 dt < ɛ. This is the same as saying given a positive ɛ, there is a positive integer N so that n > N = u n u 2 < ɛ. We denote this by u n 2 u.
We can also define the least squares convergence of a series of functions as follows: Definition et {u n } be a sequence of integrable functions defined on the interval [a, b] and let {S n } be the associated sequence of partial sums. We say S n converges in least squares to the integrable function S on [a, b] if given a positive ɛ, there is a positive integer N (depending on ɛ only so that b n > N = a S n (t S(t 2 dt < ɛ. This is the same as saying given a positive ɛ, there is a positive integer N so that n > N = S n S 2 < ɛ. We denote this by S n 2 S.
et s do some estimates. Assume we have a function f extended periodically on the interval [, 2] to ˆf as usual with Fourier series S(x = < f, > ( ( ( 2 iπ iπ + f (x, sin x sin x + 2 ( iπ f (x, cos and using b = < f, >, b n = 2 ( a n = 2 iπ f (x, sin x as usual, we can write ( S(x = b + a i sin x x ( iπ f (x, cos x and ( iπ + b i cos x. ( iπ cos x Now if we assume f exists in [, 2], the Fourier series of f converges to f at each point and ( ( ( iπ iπ f (x = b + a i sin x + b i cos x.
et S n be the n th partial sum of the series above. Then assuming 2 f 2 (xdx is finite, we have ( < f b a i sin x ( f b a j sin x j= ( iπ + b i cos x, ( jπ + b j cos x > ( ( iπ As usual, we let u i (x = sin x iπ and v i (x = cos x with v (x =. Then, we can rewrite this as < f b v (x f b v (x (a i u i (x + b i v i, (a j u j (x + b j v j (x > j=
Thus, we find, after a lot of manipulation f b v (x (a i u i (x + b i v i (x 2 = < f, f > 2b < f, v > +b 2 < v, v > 2 a i < f, u i > 2 b i < f, v i > + ai 2 < u i, u i > + bi 2 < v i, v i > i= because all the cross terms vanish. Hence, since < f, v i >= b i and < f, u i >= a i we have f b v (x (a i u i (x + b i v i 2 = < f, f > b 2 ai 2 b 2 i
We conclude that b 2 + (ai 2 + bi 2 f 2 This tells us that the series of positive terms, b 2 + (a2 i + b 2 i converges. et s look at the derivative of this series next. et s assume f is differentiable on [, 2] and we extend f periodically as well. We can calculate the Fourier series of f like usual. Now this series converges to f, if we assume f exists on [, 2]. Then we know the Fourier series of f converges pointwise to f (x at each point x. However, we can calculate the derivative Fourier series directly.
et the Fourier series of f (x be T (x. Then T (x = 2 < f, > ( ( ( iπ iπ + f (x, sin x sin x + ( iπ f (x, cos x The first coefficient is 2 < f, > = (f (2 f ( 2 Now let s consider the other Fourier coefficients carefully. We can rewrite each coefficient using integration by parts to find, ( iπ f (x, cos x = 2 = (f (x cos ( iπ x 2 ( iπ f (x, cos + iπ 2 dx x ( iπ f (x sin x dx
( iπ f (x, sin x = 2 = (f (x sin ( iπ x 2 ( iπ f (x, sin iπ 2 dx x ( iπ f (x cos x A little thought shows we can rewrite this as ( ( iπ f (x, cos x = ( 2iπ f (2 cos = ( f (2 f ( + iπ a i ( ( iπ f (x, sin x = ( 2iπ f (2 sin f ( cos f ( sin dx ( iπ ( π + iπ a i iπ b i
Now if we assume f (2 = f (, these reduce to 2 < f, > = (f (2 f ( = ( 2 iπ f (x, cos x = ( f (2 f ( + iπ a i = iπ a i ( iπ f (x, sin x = iπ b i Hence, if f is periodic, we find ( T (x = πi b i sin x + πi ( iπ a i cos x. This is the same result we would have found if we differentiated the Fourier series for f term by term. So we conclude that the Fourier series of f can be found be differentiating the Fourier series for f term by term and we know it converges to f (x at points where f exists.
We can apply the derivation we did above for f to the series expansion for f we have just found. Assuming f is integrable, we find ˆb 2 + (âi 2 + ˆb i 2 f 2 where here ˆb =, â i = πi b i and ˆb i = πi a i. Hence, we have π 2 2 i 2 (ai 2 + bi 2 f 2 We are almost at the point where we can see circumstances where the Fourier series expansion of f converges uniformly to f on the interval [, 2]. We assume f is continuous and periodic on [, 2] and that f exists. Further assume f is integrable. Hence, we know the Fourier series of f exists and converges to f (x.
From our earlier arguments, if f has Fourier series ( ( ( iπ iπ f (x = b + a i sin x + b i cos x. then we know for all n π 2 2 i 2 (ai 2 + bi 2 f 2 et s look at the partial sums of absolute values. b + a i sin x + b i cos x = b + = b + ( iπ a i sin x + ( iπ i a i i sin x + b i cos x ( iπ i b i i cos x.
et T n denote the n th partial sum here. Then, the difference of the n th and m th partial sum for m > n gives m ( iπ m ( T m (x T n (x = i a i i sin x iπ + i b i i cos x. i=n+ i=n+ Now apply our analogue of the Cauchy - Schwartz inequality for series. m ( T m (x T n (x i a i ( iπ m ( sin i x + i b i ( x iπ cos i i=n+ i=n+ m m ( i 2 a i 2 iπ 2 sin x + i=n+ m i=n+ m i=n+ i 2 b i 2 i 2 a i 2 i 2 i=n+ m i 2 i=n+ m i=n+ ( iπ 2 cos x i 2 + m i=n+ i 2 b i 2 m i=n+ i 2
Now each of the front pieces satisfy i 2 a 2 i 2 π 2 f 2, i 2 bi 2 2 π 2 f 2 Hence, T m(x T n (x m i=n+ 2 f π ( iπ (a i sin m i=n+ x i 2. + m i=n+ ( x iπ (b i cos Since the series (/i 2 converges, this says the sequence of partial this says (T n satisfies the UCC for series and so the sequence of partial sums converges to a function T which by uniqueness of limits must be f.
et s summarize this result. Theorem Given f on [, 2], assume f is continuous with f ( = f (2 and f exists. Further f is integrable on [, 2]. Then the Fourier series of f converges uniformly to f on [, 2].
Homework 3 (( 3. Show the sequence of functions w N (x = sin N + π 2 t are mutually orthogonal with length /2 on [, 2]. 3.2 Find the odd periodic extension ˆf o and even periodic externsion ˆf e for f (t = t + 2 t 3 on [, 6]. Draw several cycles. 3.3 the function f is defined on [, ] = [, 3]. { 2, x 2 f (x =, 2 < x 3 Find its periodic extension to [, 6], odd periodic extension to [, 6] and its even periodic extension to [, 6]. Find the first four terms of the Fourier sin series on [, 3]. Find the first four terms of the Fourier cos series on [, 3].