32 1. THE THEORY OF CONVERGENCE
5. Sequences and series of functions In what follows, it is assumed that x RN , and x a means that the Euclidean distance between x and a tends∈ to zero,→x a 0. | − |→
5.1. Pointwise convergence. Consider a sequence of functions un(x) (real or complex valued), n = 1, 2,.... The sequence un is said to converge pointwise to a function u on a set D if { }
lim un(x)= u(x) , x D. n→∞ ∀ ∈
Similarly, the series un(x) is said to converge pointwise to a function u(x) on D RN if the sequence of partial sums converges pointwise to u(x): ⊂ P n
Sn(x)= uk(x) , lim Sn(x)= u(x) , x D. n→∞ ∀ ∈ k=1 X
5.2. Trigonometric series. Consider a sequence an C where n = 0, 1, 2,.... The series { } ⊂ ± ± ∞ inx ane , x R ∈ n=−∞ X is called a complex trigonometric series. Its convergence means that sequences of partial sums m 0 + inx − inx Sm = ane , Sk = ane n=1 n=−k X X converge and, in this case, ∞ inx + − ane = lim Sm + lim Sk m→∞ k→∞ n=−∞ X ∞ ∞ Given two real sequences an and b , the series { }0 { }1 ∞ 1 a0 + an cos(nx)+ bn sin(nx) 2 n=1 X 1 is called a real trigonometric series. A factor 2 at a0 is a convention related to trigonometric Fourier series (see Remark below). For all x R for which a real (or complex) trigonometric series con- verges, the sum∈ defines a real-valued (or complex-valued) function of x. 5. SEQUENCES AND SERIES OF FUNCTIONS 33
Since partial sums for a trigonometric Fourier series are 2π periodic, the sum is also 2π periodic. For example the series − − ∞ ∞ einx cos(nx) f(x)= =1+2 n2 + 1 n2 + 1 n=−∞ n=1 X X converges for all real x by the comparison test: cos(nx) 1 1 | | n2 + 1 ≤ n2 + 1 ≤ n2 1 because n2 < . The sum of the series is a 2π periodic function, f(x + 2π)= f(x).∞ − P For any real sequence an that converges to zero monotonically, the trigonometric series { } ∞
g(x)= an cos(nx) n=1 X defines a 2π periodic function for all x = 2πm, where m is any inte- ger. The convergence− is established by Dirichlet’s6 test (see Exercises to the previous section). So, g(2πm) does not exist and, for all other x, g(x + 2π)= g(x).
Trigonometric Fourier series. Suppose that f(x) is a real function on [ π,π] such that − π f(x) dx − Z π exists either as the Riemann integral (e.g., f is continuous) or as an improper Riemann integral, converging absolutely, that is, π f(x) dx < . − | | ∞ Z π (e.g., f(x)= x −p, 0
(Fej´er’s theorem) If f is defined on [ π,π), extended to the • whole real axis periodically − f(t + 2π)= f(t) , and satisfies the aforementioned integrability conditions on [ π,π], then its Fourier series converges for any x at which f−has the right and left limits and, in this case, the sum of the Fourier series is equal to 1 lim f(t)+ lim f(t) 2 t→x+ t→x− In particular, iff is continuous on [π,π], then its Fourier • series converges to a 2π periodic extension− of f for any x = π + 2πm with m being− any integer, and, if f( π) = f(π6), then the Fourier series converges to the midpoint− of the6 jump 1 discontinuity, 2(f(π)+ f( π)), at x = π + 2πm. Not every convergent trigonometric− series is a Fourier series. • In other words, the sum of a trigonometric series may not satisfy the aforementioned integrability conditions required to define the Fourier coefficients. For example, the trigonometric
series ∞ sin(nx) ln(n) n=2 converges for all real Xx by Dirichlet’s test. However, this series is not a Fourier series.
∞ 5.3. Power series. Let an be a sequence of complex numbers. The { }0 series ∞ n anz n=0 is called a power series. Put X
1 n R = , α = lim sup an α n→∞ | | p 4E.T. Whittaker and G.N. Watson, A course of modern analysis, Chapter 9 5. SEQUENCES AND SERIES OF FUNCTIONS 35
The upper limit may take values in the extended system of real num- bers. If α = 0, then R = and, if α = , then R = 0. The quantity R is called the radius of convergence∞ of the∞ power series. This term is justified by the following property of R.
n Theorem 5.1. A power series anz converges absolutely if z < R and diverges if z > R. | | | | P Indeed, by the root test
n n n z lim sup an z = z lim sup an = z α = | | n→∞ | || | | | n→∞ | | | || | R the power seriesp converges if z /R < 1p or z < R and diverges if z > R. In the complex plane,| a| power series| converges| in the interior of| | the disk of radius R centered at the origin. For the boundary points z = R, the series may converge or diverge. The radius of converges of the| | series ∞ zn n n=1 X is −1 −1 R = lim sup √n n = lim √n n = 1 . n→∞ n→∞ On the boundary of the unit disk z = eiθ, 0 θ 2π, the series was shown above to converge conditionally for all 0≤<θ<≤ 2π (for all points of the boundary except z = 1) It follows from Corollary 3.1 that Corollary 5.1. If
a +1 lim n = α →∞ | | n an | | n 1 then the radius of convergence of the power series anz is R = α, where R = 0 if α = 0 and R = if α = 0. ∞ P
5.4. Three important questions. Suppose the terms un(x) of a func- tional sequence are continuous, or differentiable, or Riemann integrable. Suppose un(x)= u(x) converges pointwise for all x. Does the sum of the series u(x) inherit properties of the terms of the series? In par- ticular, P
Does the continuity of un implies the continuity of u? • Does the differentiability of un implies the continuity of u? • Does the integrability of un implies the continuity of u? • 36 1. THE THEORY OF CONVERGENCE
Furthermore, can the order of summation of the series and taking the limit, or derivative, or integral with respect to a parameter (variable x) be changed: ∞ ∞ ? lim un(x) = lim un(x) x→a x→a n=1 n=1 X∞ X∞ d ? d u (x) = u (x) dx n dx n n=1 n=1 X X b ∞ ∞ b ? un(x) dx = un(x) dx a n=1 ! n=1 a Z X X Z The answer to all these questions is negative in general! Note also that the same applies to convergent functional sequences. The limits of functional sequences or sums of functional series do not generally inherit properties of terms. Furthermore a computation of the limit, or derivative, or integral of the sum of the series cannot be done by summation of the limits, or derivatives, or integrals of individ- ual terms. Only under some special conditions imposed on terms of a series, the above equalities can hold. These conditions will be discussed in the next section.
Example 1. Put x2 un(x)= , x R , n = 0, 1, 2,... (1 + x2)n ∈ and consider the series un(x). The sequence of partial sums is easy to find using the geometric sum: P n−1 k 2 1 2 1 Sn(x)= x = x + 1 . 1+ x2 − (1 + x2)n−1 k=0 X Therefore 0 , x = 0 lim Sn(x)= u(x)= 2 n→∞ 1+ x , x = 0 6 The terms un(x) are rational functions defined on R and, hence, differ- entiable any number of times, whereas the sum is not even continuous at x = 0.
Example 2. Even if the limit function
u(x)= lim un(x) n→∞ 5. SEQUENCES AND SERIES OF FUNCTIONS 37
0 of a sequence un happens to be differentiable, the sequence un of the derivatives{ does} not necessarily converge to the derivative{u0 (the} operations of lim (or ) and d/dx are not commutative): 0 0 u (x) = lim un(x) P 6 n→∞
Consider a sequence un(x) = sin(nx)/√n. It converges pointwise to u(x) = 0forall x R. A constant function is differentiable everywhere, u0(x) = 0. However,∈ the limit of the sequence of derivatives 0 un(x)= √n cos(nx) does not even exist.
Example 3. Consider the function u(x), x R, that is defined as the limit of the double sequence: ∈ 2n um(x)= lim unm(x)= lim [cos(πxm!)] , n→∞ n→∞
u(x)= lim um(x) . m→∞ If x = p/q is a rational number, then cos2(πxm!) = 1 for m q so ≥ that um(x) = 1 and u(x) = 1. If x is irrational, then the number xm! cannot be an integer for any m so that cos2(πxm!) < 1 for any m so that um(x) = 0 and, hence, u(x). Thus, the limit function is the Dirichlet function: 1 , x Q u(x)= 0 , x∈ / Q ∈ where Q denotes the set of all rational numbers. The terms unm(x) are smooth functions, whereas the limit function u(x) is continuous nowhere and not Riemann integrable on any interval. To prove the latter, recall that, if a function is Riemann integrable, then a sequence of its Riemann sums converges and the limit does not depend on the choice of sample points in the partition. Consider a uniform partition of an interval [a,b]: xk = a + k∆x, k = 0, 1, 2,...,n, ∆x =(b a)/n. Then a sequence of Riemann sums is − n ∗ ∗ Rn(u)= u(x )∆x, xk−1 x xk k ≤ k ≤ k=1 ∗ X where xk, k = 1, 2,...,n, are sample points. Any partition interval ∗ [xk−1,xk] contains rational and irrational numbers. If xk Q for all k, ∗ ∈ then u(xk) = 1 and n ∗ Rn(u)= ∆x = b a, x Q − k ∈ k=1 X 38 1. THE THEORY OF CONVERGENCE
∗ If none of sample points is rational, then u(xk) = 0 and ∗ Rn(u) = 0 , x / Q k ∈ Note one can make this sequence to converge to any number between 0 and b a by a suitable choice of sample points. So, the limit of a sequence− of Riemann sums depends on the choice of sample points and, hence, the Dirichlet function is not Riemann integrable on any interval.
5.5. Exercises.
1 1 2 Find the Fourier series for f(x)= 2 (π x x ) and show that ∞ | |− sin2(nx) f(x)= n2 n=1 X Note that f( π) = f(π) = 0 and, hence, the 2π periodic extension of f is continuous− everywhere, and the Fourier series− converges to the extension everywhere.
2 Show that the trigonometric series ∞ ( 1)n+1 cos(nx) − n n=1 π X converges for all x = 2 + πm where m is any integer. Next, show that it is the Fourier series6 for f(x) = ln 2cos(x/2) . What can be said about the sum of the Fourier series and| f(x)? |
3. Find the radius of convergence of each of the following series ∞ n p anz , an = n , p R ∈ n=1 X∞ n p n anz , an = n c , p R , c> 0 n=1 ∈ X∞ n n c anz , an = , c> 1 . n! n=1 X 4. If any of the series in Problem 3 has a finite radius of convergence, investigate the convergence of the power series for z equal to the radius of convergence. | |