Math 321 Final Examination April 1995 Notation used in this exam: N 1 π (1) S N (f,x) = f(t)e int dt e inx. 2π n= N π (2) C(X, R) is the space of bounded real-valued functions on the metric space X, equipped with the metric ρ(f,g)=sup{ f(x) g(x) : x X}. 1. [12 marks] Define carefully: (a) The outer Lebesgue measure of a subset, E, ofr. (b) The σ-field of Borel subsets of R. (c) The set, R(α), of real-valued functions on [a, b] which are Riemann integrable with respect to the increasing function α. 2. [12 marks] Calculate: (a) (b) 2 xd ( x 2 +[x] ).([x] is the greatest integer less than or equal to x.) 1 π 2 2π t 1 e int dt. n= π 3. [15 marks] Give examples of the following. (Briefly justify your answers.) (a) A sequence {f n } of Riemann integrable functions on [, 1] which converges pointwise to zero but for which f n (x) dx does not converge to zero as n. (b) A sequence of polynomials which converges uniformly on [, 1] to f(x) =e x. (c) A function f:[, 2] R which is not Riemann integrable with respect to α(x) = x, but is with respect to β(x) =[x]. 4. [7 marks] Let f n and f be continuous real-valued functions on [, 1] (n N) such that for each x in [, 1], f n (x) decreases to f(x) asn.provethat lim f n (x) dx = f(x) dx. 5. [6 marks] Let E be a countable subset of R. Prove that the Lebesgue measure of E is zero. 6. [16 marks] For each natural number p define a subset, A p,ofc([ 1, 1], R) by { } N A p = f:[ 1, 1] R : f(x) = a n x pn for some n N, a,...,a n R. n= For each p in N, find the closure of A p in C([ 1, 1], R).
7. [16 marks] Let (a n ) be a sequence of real numbers that converges to a (a R). Let (φ n ) be a sequence of bounded, continuous, real-valued functions on R which converges uniformly to a function φ. Let f n C([, 1], R) satisfy the differential equation f n (x) =φ n (f n (x)) x [, 1]; f n () = a n. Prove there is a subsequence (f nk ) which converges uniformly to a solution, f, of f (x) =φ (f(x)) x [, 1]; f() = a. 8. [16 marks] Let f: R C be continuous and suppose f(x +2π) =f(x) for all x. (a) If f(n) converges (to a finite limit), prove that {S n (f,x) :n N} converges to f(x) uniformly on [ π, π]. (b) If n f(n) converges (to a finite limit), prove that f has a continuous derivative. (c) Suppose conversely that f is continuous. (i) Show that f(n) =(in) 1 f (n). (ii) Show that lim n n f(n) =.
Math 321 Final Examination April 1996 1. [2 marks] True or false? If true give a proof; if false a counterexample. (In part (c) you may use any standard properties of the cosine function, provided you state them clearly.) (a) If f: R R and lim f(x +( 1)n /n) =f(x) thenf is continuous at x. (b) The function g(x) =x 2 is uniformly continuous on R. (c) The function h(x) = cosx is uniformly continuous on R. 2. [2 marks] (a) State the Mean Value Theorem. (b) The functions f(t), g(t) are continuous and differentiable on R, and satisfy f() =, g() = 1, and f (t) =g(t), g (t) = tf(t), for all t R. (i) Let a =inf{t :g(t) =}. Explain briefly why a>. (ii) Prove that f is monotonic increasing on [,a]. (iii) Prove that g(t) 1fort [,a]. (iv) Prove that f(t) t for t [,a]. 3. [2 marks] Let f(x) =x, α(x) =β(x)+γ(x), where β(x) =x 2 and γ(x) = {, if x<1, 1, if x 1. Calculate the Riemann-Stieltjes integral 2 fdα. 4. [2 marks] Let f(t) =t 2,for π t π. (a) Find the Fourier series for f. (b) What can you say about the convergence of the Fourier series for f? ( 1) n (c) Using your answers to (a) and (b) evaluate. n 2 n=1
5. [2 marks] (a) State a theorem which relates uniform convergence of sequences of functions and integration. (b) For n N let f n :[, 1] R be defined by 1 f n (x) ={ if x is rational, x = p/q in lowest terms, and q n, q otherwise. For (i) (iv) below, give reasons for your answers. (i) Calculate the functions f 1, f 2, f 3. (ii) Is f n Riemann integrable on [, 1]? (iii) Let f be the pointwise limit of the sequence f n. Do the functions f n converge uniformly to f on [, 1]? (iv) Is the function f Riemann integrable on [, 1]?
Math 321 Final Examination April 1997 To obtain full marks you must provide full justification for your answers. 1. [16 marks] (a) State Taylor s Theorem (without proof). (b) Let f(x) = sinπx. Compute a Taylor polynomial approximation P (x) tof at a =1/4that yields a value at x =7/32 correct to 3 decimals, i.e., f(x) P (x) <.5. Is the approximation P (x) too large or too small? (The question does not ask that you actually compute P (7/32).) 2. [15 marks] Let f(x) = 2n +1x(1 x 2 ) n. (a) Determine the largest n= set S R on which f is well-defined. (b) On what intervals is the convergence of the sum uniform, and on what intervals is it non-uniform? (c) Where is f continuous? (d) Where is f differentiable? 3. [22 marks] For each of the following statements, first define the underlined words and then determine whether the statement is TRUE or FALSE. If true, give a brief reason or reference; if false, give a counterexample. (a) If f is differentiable on [a, b], then f has the intermediate value property. (b) Given a bounded function f and a monotonically increasing function α on [a, b], b the Riemann-Stieltjes integral fdα exists if f has only a finite number of a discontinuities. (c) If {f n } C(R) converges uniformly on R,then{f n } is uniformly equicontinuous. 4. [15 marks] Let f(x) =xe ix/2 and g(x) =xe x/2 on [ π, π]. (a) Compute Fourier series expansions of the form c n e inx for both f and g. (b) With reference to appropriate theorems, investigate the mean, pointwise, and uniform convergence of the two series in (a).
5. [32 marks] Let B n be the unit ball in R n, n 2, { B n = x =(x 1,...,x n ) R n : r = ( x 2 1 + + ) 1/2 } x2 n 1, and let S n 1 = B n. Let ω = dx 1 dx n bethevolumeforminr n and V n = ω. B n Define the (n 1)-form α = n ( 1) i 1 x i dx 1 dx i dx n, i=1 where dx i means that dx i is omitted, and let A n 1 = α. S n 1 (a) Show that dα = nω. (b) State Stokes Theorem for differential forms and show that A n 1 = nv n. (c) Explain geometrically why A n 1 deserves to be interpreted as the area of S n 1. Letting s = ( x 2 1 + + x 2 n 1) 1/2, R = { x R n : r>}, ands = { x R n : s>}, define the forms and β = r n α E n 1 (R), n 1 γ = s n+1 ( 1) i 1 x i dx 1 dx i dx n 1 E n 2 (S), i= ν = s n+1 dx 1 dx n 1 E n 1 (S), n 1 δ = s 2 i=1 x i dx i E 1 (S). (d) Prove that β and γ are closed. (e) Show that δγ = ν and that for f E (S), n 1 d(fγ)= i=1 x i f dx i ν +( 1) n f x n γdx n. (f) Prove that β is exact on S by showing that β = d(fγ)wheref( x) =( 1) n g(x n /r) and y g(y) = (1 z 2 ) (n 3)/2 dz, 1 y 1. 1 (g) Is β exact on R? (h) Given a parametrization (ξ 1 ( u),...,ξ n 1 ( u)) of B n 1, use the parametrization x i ( u, θ) = { sin θξi ( u), i < n, cos θ, i = n, ( θ π)
π of B n to show that V n = I n V n 1,whereI n = sin n θdθ. Note: By an integration by parts argument (not required!), ( ) n 1 2 n! 2 /n!, if n is odd, 2 I n = ( ) n 2 n πn!/! 2, if n is even. 2 (i) Deduce that the volume of the n-dimensional unit ball is ( n 1 2 n π (n 1)/2 V n = ( 2 n π n/2 / 2 )!/n!, if n is odd, )!, if n is even.
Calculators may be used. Math 321 Final Examination April 1998 1. [1 marks] State carefully: (a) The definition of the σ-field of Borel subsets of R (which we will denote by B). You may assume that we know what a σ-field is. (b) The definition of an equicontinuous family of functions in C(X, C). (X is a metric space.) (c) The Stone-Weierstrass Theorem (real-valued case). 2. [1 marks] Let f be a Riemann integrable function on [, 1] and define F (x) = f(t) dt for x [, 1]. Prove that F is continuous on [, 1]. NOTE: Recall that Riemann integrable functions are assumed to be bounded. Note also that this result does not follow from the fundamental theorem of calculus. You should give an ε-δ argument. ( 1/2 3. [1 marks] If f is Riemann integrable on [, 1], recall that f 2 = f(t) dt) 2. (a) If f n and f are Riemann integrable functions on [, 1] such that (f n )converges uniformly to f, show that lim f n f 2 =. (b) Give an example of continuous functions f n and f on [, 1] such that lim f n f 2 = but (f n ) does not converge uniformly to f. 4. [15 marks] True or False. If true give a proof. If false provide a counterexample. (a) A countable subset of R has Lebesgue measure. (b) Every Borel subset of R is open or closed. (c) If α: R R is an increasing function and µ is a measure on (R, B) such that µ((a, b]) = α(b) α(a) for all a b in R, thenα must be right-continuous. 5. [12 marks] α is a continuous strictly increasing function on [, 2] such that α() =, α(1) = 1,andα(2) = 1. Define β(x) on[, 2] by 3 { 1, if x<1, β(x) = 3, if 1 x 2. 2 Calculate the following Riemann-Stieltjes integrals (you may assume they all exist): x (a) 2 βdα (b) 2 αdβ (c) 2 αdα 6. [12 marks] Let S denote the set of functions in C([ π, π]) of the form f(x) =a sin x+ b sin 2x for some real numbers a and b. Let g(x) =x for x [ π, π]. Find f in S for which g f 2 is smallest.
7. [16 marks] Recall that f(n) = 1 π f(x)e inx dx for f a Riemann integrable function on 2π π [ π, π]. Let f:[ π, π] R be a continuously differentiable function such that f( π) =f(π). (a) [4 marks] Show that f (n) =in f(n). ( π 1/p (b) [8 marks] If f p = f(x) dx) p, show that π n= f(n) π f 2 + 1 f 1. 3 2π Hint: Recall the Cauchy-Schwarz inequality in the form N n= N n a n b n N n= N n a n 2 1/2 N n= N n b n 2 (c) [4 marks] Use (b) to show that the Fourier series for f converges uniformly to f on [ π, π]. 8. [15 marks] Let {f n : n N} and f be continuous real-valued functions on [, 1]. (a) [5 marks] If (f n ) converges uniformly to f, show that for each k {, 1, 2,...}, lim f n (x)x k dx = f(x)x k dx. (b) [1 marks] If for each n, f n () =, f n is differentiable, and {f n} is a uniformly bounded sequence of Riemann integrable functions, show that the following converse to (a) holds: If for each k {, 1, 2,...} one has lim f n (x)x k dx = m k for some m k R, then(f n ) converges uniformly to a function f in C[, 1] for which m k = f(x)x k dx. 1/2.
Math 321 Final Examination April 1999 1. [2 marks] (a) Given functions f,f n : X R (n N), define the two statements (i) f n converges to f pointwise on X, and (ii) f n converges to f uniformly on X. Show by example that (i) does not imply (ii). (b) State both parts of the Fundamental Theorem of Calculus. (c) State the Arzela-Ascoli theorem. Define the terms involved. (d) Given α:[, 1] R nondecreasing, define the phrase, f is Riemann-Stieltjes integrable with respect to α. 2. [2 marks] True or False? (Supply a proof or a counterexample, as appropriate.) (a) Iff, g:[, 1] R are continuous functions satisfying x k f(x) dx = x k g(x) dx for all k =, 1, 2,...,thenf = g. (b) If f (x) > for all x>andf(x) asx,thenf (x) < for all x>. (c) If each f n :[, 1] R is integrable and satisfies both (i) sup f n 1 n, (ii) lim f n(x) exists for each x [, 1], [,1] then f(x) = lim f n(x) is Riemann-integrable and f = lim f n. 3. [1 marks] Consider the sequence of functions f n : R R defined by x f n (x) = 1+nx, n N. 2 Show that (f n ) converges uniformly to a function f, and that the equation f (x) = lim f n (x) is correct if x, but false if x =. 4. [15 marks] Given f:[, ) R, the Laplace transform of f is the function F (s) = lim e st f(t) dt. b Prove the Final Value Theorem, namely, If (i) f is integrable on every interval [,b](b>), and (ii) lim f(t) =L (some L R), t then lim sf (s) exists and equals L. s + (Hint: Study, then extend, the case f(t) =L.) b
5. [15 marks] A certain Riemann-integrable f:[ π, π] C and complex sequence (c k ) obey n f(t) c k e ikt as n. k= n 2 (a) Prove that for any g R[ π, π] wehave 1 π f(t)g(t) dt = 2π π (b) Deduce that c k = f(k), and that c k 2 < +. k= k= c k ĝ(k). 6. [2 marks] Let Ω R n be an open set, with Ω, and let f:ω R m be continuously differentiable. Prove: If f () is one-to-one, then there is an open set U in R n with U Ω such that f is one-to-one on U. One possible approach is to prove and then use these two statements: (i) inf f ()v >, v =1 (ii) ε >, δ > s.t. f(y) f(x) f ()(y x) <ε y x x, y N δ ().
Math 321 Final Examination April 2 1. [18 marks] The functions f n,f:[, 1] R are given; each f n is continuous on [, 1]. (a) Write precise definitions of what it means to say f n converges to f (1) pointwise, (2) uniformly, or (3) in mean square (or 2 ). (b) Six statements (m) (n) can be formed by choosing distinct m, n {1, 2, 3} above. Label each statement True or False; briefly justify your choice with either a proof or a specific counterexample. π 2. [7 marks] Evaluate, with justification, lim n +sinnx 3n sin 2 nx dx. 3. [15 marks] For each n N, the differentiable function f n : R R satisfies f n(t) 321 + t 21 t R and f n () = 2. Prove that there exist f: R R and a subsequence (f nk ) k with this property: for every compact subset C of R, f nk f uniformly on C. Clearly identify the principal theorems and methods you apply. 4. [1 marks] True or False (proof or counterexample): (a) There exists a continuous f: R R whose set of differentiability points consists of exactly one element. (b) In every metric space, every closed and bounded subset is compact. 5. [1 marks] Let f: R 3 R 2 be defined by f(x, y, z) = ( xy 2 + yz 2,x 2 + y 2 + z 2). Show that (, 1) is an interior point for the set range(f) ={f(x, y, z) :x, y, z R}. Include careful statements of any theorems you apply. 6. [2 marks] (a) State the Stone-Weierstrass Theorem (Algebra Form). (b) State Fejer s Theorem concerning the convergence of Fourier series. (c) Given f C[,π]andε>, show that some n N and a,...,a n R obey max x [,π] f(x) (a + a 1 cos(x)+a 2 cos(2x)+ + a n cos(nx)) <ε.
7. [2 marks] Let f C[ π, π] satisfy x [ π, π]. π π f(t) dt =. Define g(x) = (a) Derive a relationship between the Fourier coefficients ĝ(k) and f(k). (b) Show that the series ĝ(k) converges. k= x π f(t) dt, N (c) Let s N (x) = ĝ(k)e ikx.musts N converge to g? If so, how (see Question 1)? k= N Make the strongest statement you can justify; include a detailed proof.
Math 321 Final Examination April 21 1. [15 marks] Let X and Y be metric spaces. (a) What does it mean to say that two subsets A and B of X are separated? (b) Give the definition of a connected subset of X. (c) Let f: X Y be a continuous function. For C X, D Y recall that f(c) ={f(x) :x C}, f 1 (D) ={x X : f(x) D}. Of the two statements below, one is true and the other is false. Give a proof of the true one and an explicit counterexample to the false one. (1) If C X is connected then f(c) is connected. (2) If D Y is connected then f 1 (D) is connected. 2. [2 marks] State any version of the Mean Value Theorem. Let f: R R be twice differentiable at all points in R. Let I =[, 2] and suppose that f(x) 1and f (x) 1 for all x I. (a) Prove that f satisfies f (x) f (y) x y for all x, y I. (b) Hence deduce that f (x) 3 for all x I. 3. [15 marks] State a theorem (the Riemann integrability criterion or (RIC)) which gives conditions for a bounded function f on an interval I =[a, b] to be Riemann- Stieltjes integrable with respect to a monotonic increasing function α. Let I =[ 1, 1] and α and β be given by { if x< α(x) = 2 if x { 1 if x β(x) = 2 if x> (a) Prove that β R(α) and evaluate (b) Prove that α R(α). 1 βdα. 4. [15 marks] Give examples of the following (you need not justify your examples): (a) A sequence of continuous functions f n on [,1] such that f n f pointwise but f is not continuous. (b) A sequence of differentiable functions f n on [,1] which converge uniformly to a differentiable function f, such that f n do not converge to f.
5. [15 marks] State carefully the Stone-Weierstrass theorem (real case). Let I =[a, b] 2 = {(x, y) :a x b, a y b} R 2. Let C(I) denote the set of continuous real-valued functions on I, and B(I) denote the set of real-valued functions on I of the form f(x, y) = n j= k= m a jk e jx2 +ky 2, where a jk R and n and m are positive integers. (a) If I =[, 1] 2 prove that B(I) is a dense subset of C(I). (b) If I =[ 1, 1] 2 prove that B(I) is not a dense subset of C(I). Why does this not contradict the Stone-Weierstrass theorem? 6. [2 marks] Let f(x) = x on [ π, π]. (a) Find the Fourier series c n e inx of the function f. n (b) If S N (x) = c n e inx is the N-th partial sum of the Fourier series, what can n N you say about the convergence of S N to f? Give reasons for your answer. (c) Use the Fourier series of f to calculate the sum (d) Given that r= 1 (2r +1) 2. n 2 = π 2 /6 verify your answer by another method. n=1 (e) Use Parseval s theorem to calculate r= 1 (2r +1) 4.