More Series Convergence

Transcription

1 More Series Convergence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University December 4, 218 Outline Convergence Analysis for Fourier Series Revisited

2 et s look at the Fourier Sine and Fourier Cosine series expansions again. You can see that applying the separation of variables techniuque, the boundary constraints for the partial differential equation model lead us to second order linear differential equations of the form u) + Θu = plus boundary conditions. Note this technique always give us the eigenvalue - eigenfunction problem u) = Θu and depending on the boundary conditions, we have been finding the eigenfunctions are sin or cos orthogonal families when u) = u. This is the simplest example of a Stürm - iouville problem SP) whose eigenfunctions are the nice families we have seen. If is a different SP, the eigenfunctions could be Bessel functions, egendre polynomials, aguerre polynomials or others. The eigenfunctions for these models are also mutually orthogonal and if they are the family {wn} then we would look for expansions of the data function f in terms of this orthogonal family, f = n Anwn where the coefficients An would be the generalized Fourier coefficients of f associated with the orthogonal family {wn} as defined below. Definition et {wn} be a mutually orthogonal family of continuous functions in C[a, b]). Note wn 2 is the length of each function. The family {ŵ}n defined by ŵn = 1/ wn 2) wn is a mutually orthogonal sequence of functions of length one in C[a, b]). The generalized Fourier coefficient of f with respect to this family is An = 1 wn 2 2 < f, wn >= 1 b wn 2 f s)wns)ds 2 a For our usual unx) = sinnπx/) and vnx) = cosnπx/) families, on the interval [, ], note the families Unx) = sinnπx a/)b a)) and Vnx) = cosnπx a)/b a)) are mutually orthogonal on [a, b] and so the Fourier coefficients of f on [a, b] would be

3 An = Bn = 1 Un Vn 2 2 b a b a f s)uns)ds = 1 b Un 2 2 a f s)vns)ds = 1 b Un 2 2 a nπ f s) sin nπ f s) cos )) s a ds )) s a ds If we assume f j) is Riemann integrable for j = 1, 2 and 3, we can prove estimates for the Fourier coefficients. Theorem For j = 1, 2 or 3, if f j) exists and is Riemann Integrable on [, ], f j 1 ) = f j 1 ), then we have the estimates ) j A 2 i 2 f j) 2 2, ) j Bi 2 2 f j) 2 2 and the series iπ )j 1 Ai sin iπ ) and Bi iπ )j 1 cos iπ ) converge uniformly on [, ] to continuous functions. We let A n and Bn be the usual Fourier sine and cosine coefficients on [, ]. Case 1 f is integrable on [, ] with f ) = f ): Since f ) = f ), the periodic extension ˆf is continuous on [, 2] and ˆf ) = ˆf 2). So we can apply our previous results to ˆf : et A, n and B n, be the Fourier sin and cosine coefficients for f on [, ]. We have 2 A i) 2 f 2 2, 2 B i ) 2 f 2 2. where A, i = iπ B i and B, i = iπ A i. From this we established the estimates 2 B i ) 2 f 2 2, 2 A i ) 2 f 2 2

4 Then, using the Cauchy - Schwartz Inequality, we found for the partial sums T s n of the Fourier sine series on [, ] that ) iπ iπ Tmx) s Tn s x) = A i sin = A i ) A i 2 iπ 2 sin A i 2 ) )) iπ iπ sin etting the partial sums of the Fourier cosine series be Tn c, we find Tmx) c Tn c x) Bi 2 Since the Fourier sine and Cosine Series derivative arguments gave the estimates 2 A i ) 2 f 2 2, 2 B i ) 2 f 2 2 we have T mx) s Tn s x) f 2 2 T mx) c Tn c x) f 2 2 Since the series 2 i 2 π converges, this says the sequence of partial 2 this says Tn s ) and Tn c ) satisfies the UCC for series.

5 So the sequence of partial sums converges uniformly to continuous functions T s and T c on [, ] which by uniqueness of limits must be f in each case. Case 2 f is integrable on [, ] and f ) = f ). In this case the extension ˆf satisfies ˆf ) = ˆf ) and is continuous. For this case, we need to find the Fourier coefficients of f. A i and B i. The usual arguments find 2 A i ) 2 f 2 2, 2 B i ) 2 f 2 2. The arguments presented in Case 1) can then be followed almost exactly. Assuming f exists on [, ], consider the Fourier Sine series for f on [, ]. Recall, for the even and odd extensions of f, f o and f e, we know the Fourier sine and cosine coefficients of f o, A i,o, and f e, B i,o, satisfy A,2 i,o = B,2,o = 1 2 B,2 i,o = 1 iπ f o s) sin s ) f o s)ds = B, = 1 iπ f o s) cos s ) ds = A, i = 2 = B, i = 2 f s)ds, i = ) iπ f s) sin s ds, i 1 ) iπ f s) cos s ds, i 1

6 For i > 1 iπ f o, sin = 1 = 1 ) = 1 f o x) sini iπ ) 2 2 = iπ B, i = A i 2 f x) sin iπ x) dx 2 f o x) iπ cosi iπ iπ x) dx = f o x) iπ cosi iπ ) dx ) ) iπ f o x), cos If we assume f is integrable, this leads to coefficient bounds. We look at f o = < f, f o > +2 + j=1 =< f o, f o > +2 + i 4 π 4 4 A i ) 2 )) iπ A i sin j 2 π 2 A i A j A i iπ sin A i, f o iπ f o, sin iπ sin f o, sin ) iπ, sin j 2 π 2 j=1 ) ) ) jπ, sin ) iπ ) )) jπ A j sin >

7 Substituting in for < f o, sin iπ ) >, we have f o = < f o, f o > Hence, implying )) iπ A i sin i 4 π 4 4 A i ) 2, f o j=1 j 2 π 2 i 4 π 4 4 A i ) 2 f o 2 2 = 2 f 2 2 i 4 π 4 4 A 2 i 2 f 2 2 )) jπ A j sin > A similar analysis for the cosine components leads to i 4 π 4 4 Bi 2 2 f 2 2 et Tn denote the n th partial sum of n iπ Ai sin iπ ) on [, ]. Then, the difference of the n th and m th partial sum for m > n gives Tmx) Tnx) = = iπ A i A i ) iπ sin ) iπ sin )) iπ

8 Now apply our analogue of the Cauchy - Schwartz inequality for series. ) ) Tmx) Tnx) A i iπ iπ sin i 4 π 4 ) 4 A i 2 iπ 2 sin i 4 π 4 4 A i 2 Since i 4 π 4 4 A i ) 2 f 2 2. We have Tmx) Tnx) f 2 2. Since the series 2 i 2 π converges, this says the sequence of partial 2 this says Tn) satisfies the UCC for series and so the sequence of partial sums converges uniformly to a function T. The same sort of argument works for n iπ Ai cos iπ ) on [, ]. So there is a function Y which this series converges to uniformly on [, ]. Case 3 f is integrable on [, ]: We have ˆf is continuous and ˆf ) = ˆf. So we can apply our previous work to f. The calculations are quite similar and are left to you.

9 et s put all that we know about the convergence of Fourier Series together now. Assume f has continuous derivatives f and f on [, ] and assume these f and f are zero at both and. The Fourier Sine and Cosine series for f then converge uniformly on [, ] to continuous functions S and C. We also know the Fourier Sine and Fourier Cosine series converge to f on [, ]. Further, we know the Derived Series of the Fourier Sine and Fourier Cosine series converge uniformly on [, ] due to our estimates. If we assume the continuity of higher order derivatives, we can argue in a similar fashion to show that term by term higher order differentiation is possible. et s check the conditions of the derivative interchange theorem applied to the sequence of partial sums T s n ) and T c n ) for the Fourier Sine and Cosine Series. 1. T s n and T c n are differentiable on [, ]: True. 2. T s n ) and T c n ) are Riemann Integrable on [, ]: True as each is a polynomial of sines and cosines. 3. There is at least one point t [, ] such that the sequence T s n t)) and T c n t) converges. True as these series converge on [, ]. 4. Tn s ) unif W s and Tn c ) unif W c on [, ] where both limit functions W s and W c are continuous. True because of our Theorem. The conditions of the theorem are thus satisfied and there are functions U s and U c on [, ] so that Tn s unif U s on [, ] with U s ) = W s and Tn c unif U c on [, ] with U c ) = W c. Since limits are unique, we then have U s = f with f = W s. and U c = f with f = W c ; i.e. we can take the derivative of the Fourier Sine and Fourier Cosine Series termwise.

10 That is f t) = f t) = An sin nπ ) ) nπ = An cosnπ ) Bn cos nπ ) ) = nπ Bn sinnπ ) Now assume f also has a continuous third derivative on [, ]. The arguments we just presented can be used with some relatively obvious modifications to show f t) = f t) = ) nπ An cosnπ ) = nπ ) An sinnπ ) = n2 π 2 An sinnπ ) n2 π 2 An cosnπ ) It is possible to do this sort of analysis for more general functions f but to do so requires we move to a more general notion of integration called ebesgue integration. This is for future discussions! Homework If f and g are two nonzero continuous functions on [a, b] which are orthogonal, prove f and g are linearly independent If {f1,..., fn} are nonzero continuous functions on [a, b] which are mutually orthogonal, prove it is a linearly independent set If wn)n 1 is an orthogonal family of continuous functions on [a, b] prove wn)n 1 is a linearly independent set If f is continuous on [, ] and wn)n 1 is an orthonormal family of continuous functions on [, ] prove < f, wn >)2 f Is t in the span of {1, t, t 2,..., t n,...} for t?