Analysis SEP Summer 25 Problem. (a)suppose (a n ) n= is a sequence of nonnegative real numbers such that n= a n <, construct a sequence (b n ) n=, such that b n, a n b n, but still n= b n < ; (b)suppose (a n ) n= is a sequence of nonnegative real numbers such that n= a n =, construct a sequence (b n ) n=, such that b n, a n b n, but still n= b n = Problem 2. (Jan 24) (a)let f be a differentiable function defined on [,]. Assume that f = f 2. Show that if f () >, then f () > 4, if f () <, then f ( ) < 4 ; (b)let ε >. Find a differentiable function g defined on [-,], such that g (x) = x g(x) 2, g(), but g(x) ε for x [,] Problem 3. (Jan 24) Suppose f n is a sequence of continuous functions defined on [,], such that f n (x) f n+ (x) for any n, x [,]. Suppose f n (x) f (x) pointwise and f (x) is continuous. Show that the convergence is uniform. Problem 4. (Jan 22) Let (a n ) n= be a sequence of real numbers and let b n = n 6 n k=k 5 a k (a)prove or disprove: If a n converges, then b n converges. (b)prove or disprove: If b n converges, then a n converges. Hint: n 6 n k= k5 converges. Problem 5. Suppose (a n ) n= is a sequence of nonnegative real numbers, denote s n = n k= a k. Show that n= a n < if and only if a n+ n= +s n < Problem 6. (Jan 2) For λ >, set F(λ) = e λx4 +λx 6 dx Prove there exists constants A and C >, such that F(λ) = A Problem 7. (Jan 25) Prove that the integral f (a) = converges for a and f is continuous on [, ) λ 4 sin(x 2 + ax) dx x + E(λ) where E(λ) C Problem 8. (Jan 25) Let g be a non-constant differentiable real function on a finite interval [a, b], with g(a) = g(b) =. Show that there exists c (a,b) such that g 4 b (c) > (b a) 2 g(t) dt a Problem 9. Let f : be a convex function, let E = {x : f is not differentiable at x}. Show that E is at most countable. Problem. Let f : n be a bounded Lebesgue measurable function, define Show that M f is Borel measurable. M f (x) = sup r> vol(b r (x)) B r (x) f (y) dy λ 2
Problem. Let f C[,], suppose for any x [,], there exists ε > (may depend on x ), such that f (x ) f (x) for any x (x ε,x + ε) [,]. Prove f is a constant. Hint: Try to prove f ([,]) is at most countable. Problem 2. (a)does there exist a function f :, such that f is discontinuous at all x Q and continuous at all x Q? (b)does there exist a function f :, such that f is discontinuous at all x Q and continuous at all x Q? (c)does there exist f C(), such that f (Q) Q, and f ( Q) Q Problem 3. (Aug 2) Let f n C([,]), and for each fixed x [,], f n (x) converges as a sequence of numbers to a finite number f (x). Show that for each ε >, there exists a non-empty interval (a,b), and an integer p such that sup x (a,b) f n (x) f (x) < ε for any n > p. Problem 4. (Jan 24) (a)let χ Q be the characteristic function of the rationals in [,]. Does there exist a sequence of functions f n : [,] such that f n converges to χ Q pointwise? (b)can the sequence f n in (a) be taken to be continuous? Problem 5. Let f : n be a continuous function. We say f is differentiable at x n, if there exists p n f (x) f (x, such that lim ) p (x x ) x x x x =. Let E be the set of x such that f is differentiable at x, prove that E is a Borel set. Problem 6. We say a Borel measure µ on n is doubling, if there exists C >, such that µ(b 2r (x)) Cµ(B r (x)) for all r >, x n. Let < α < n, show that µ = x α dx is doubling. Problem 7. Let ( f n ) n= be a sequence of measurable functions on [,], such that f n f pointwise. Suppose there exists some function Φ : [, ) [, ), with lim t Φ(t) =, such that sup n Φ( f n (x))dx <. Prove f n f dx Problem 8. (Aug 2) Assume that E [,2π] and its Lebesgue measure is positive. (a)show that, for any sequence t n of real numbers, (b)let a n and b n satisfy Prove that a n,b n as n lim cos(nx +t n )dx = E lim a n cos(nx) + b n sin(nx) = x E Problem 9. (Jan 2) (a)give an example of functions f n L ([,]), n =,2, with the properties:. lim f n (x) = for any x [,] 2. f n (x) dx = 2 for any n =,2, (b)show that if the f n L ([,]) are as in part (a), then lim f n (x) dx = Problem 2. (Jan 25) Let f L 2 ([,]) satisfy tn f (t)dt = n+2 for n =,,2,. Must then f (t) = t a.e? 2
Problem 2. (a)let ( f n ) n, f L p ( n ) with p <. Suppose f n f in L p loc (n )(i.e, in L p (K) for every compact set K n ). suppose also f n, f to be equi-measurable in the sense that mes({x : f n (x) > t}) = mes({x : f (x) > t}) for every n and t. Show that f n f in L p ( n ). (b)if we weaken the equi-measurability condition to the following: mes({x : f n (x) > t}) = mes({x : f m (x) > t}), for any n,m,t, do we still have convergence in L p ( n )? Problem 22. (Aug 27) ecall the following definition: a function g : (,) has bounded variation if sup sup N 2 <x N < <x < Find the exponents p for which the function f : (, ) has bounded variation. g(x ) g(x 2 ) + + g(x N ) g(x n ) < f (x) = x p sin(/x) Problem 23. (Aug 26) Fix a function g L ( n ) such that g(x)dx =. Denote g ε (x) = ε g(ε x). Consider an operator T ε f (x) = g ε (y) f (x y)dy (a)prove there exists a constant C > such that T ε f p C f p, for all ε and p < (b)prove that lim ε T ε f p = for any f L p ( n ) with p < Problem 24. Let f : [,] be a measurable function. Suppose for any g L 2 ([,]), we have f g L ([,]). Show that f L 2 ([,]). Problem 25. (Aug 22) Prove that sinx e tx x dx = π 2 arctant, for any t >. Problem 26. (Jan 25) (a) If f C([,]) and the distributional derivative f of f on (.) is in L ((,)), prove that f () f () = f (x)dx (b)let p (, ] and let F C([,]) be such that for each f F we have f L ((,)) and f L p ((,)) with f the distributional derivative of f on (,). Prove or disprove: F is precompact in C([,]). Problem 27. (Jan 2) Let D d, d 2 be a convex set with smooth boundary D so that the origin belongs to the interior of D. For every y D, let α(x) be the angle between the position vector x and the outer normal vector n(x). Let ω d be the surface area of unit sphere in d. Compute ω d D cos(α(x)) x d dσ(x) Problem 28. Let H be a Hilbert space, and let {x n } n= be a sequence in H such that x n converges weakly to x H. Show there exists a subsequence x n j such that converges to x strongly. N N j= Problem 29. Let ( f n ) n, f L 2 ([,]), such that f n f weakly. Let φ : be a nonnegative convex function. Show the following version of Fatou s lemma: φ( f (x))dx lim inf φ( f n (x))dx Hint:Use previous problem. Problem 3. Let f : be measurable and periodic with period. If f L ([,]), and α an irrational number, show that N N k= f (kα + x) f (u)du in L ([,]). x n j 3
Problem 3. (Jan 2) (a)let f be a continuously differentiable function on 2. Assume at each point. f > f x x 2 Show that if f (x,x 2 ) = f (x,x 2 ), then x x < x 2 x 2 unless (x,x 2 ) = (x,x 2 ). (b)let φ : 2 2 be defined by φ(x,x 2 ) = (x + sin( x 2 + x 2 4 ),x 2 + cos( x 4 + x 2 2 )) Show that φ is a diffeomorphism from 2, that is, a smooth map with smooth inverse. Evaluate the partial direvatives of φ at the point (,). Hint:Part (a) can be used to prove injectivity. For surjectivity, try to show the image is both open and closed in 2. Problem 32. (Jan 2) Let K be a continuous function on the square [,] [,] and let g be a continuous function on [,]. Show that there is a unique continuous function f on [,], such that f (x) = x K(x,y) f (y)dy + g(x) Problem 33. (Jan 24) Suppose f is a non-constant function holomorphic on a neighborhood of the closed unit disk D and such that f is constant on D. Prove that f has at least one zero in D. Problem 34. (Jan 25) Let f : D D be holomorphic with f () = and f () = a >. (a) Prove that f (z) az z 2 ( z ). (b)prove that f (D) contains {z D : z < r a }, for some r a > depending only on a. Problem 35. (Jan 23) Let f : D D be a holomorphic function such that f ( 2 ) = and f () = 2 5. Show that f ( 2 ) >. Problem 36. (Aug 2) Let f be a holomorphic function in the unit disk D. (a)suppose that f (z) M for all z D. Prove that f (z) M( z ) for all z D (b)suppose that f () = and that D f (x + iy) 2 dxdy = A 2 <. Show that Problem 37. (Aug 29) Let F : C C be an entire holomorphic function. f (z) Alog( z ) (a) Prove that if F(x + iy) 2 dxdy <, then F(z) = for all z C (b)suppose for each w C, the equation F(z) = w has at most solution, prove there are complex numbers a and b such that F(z) = az + b. Problem 38. (Aug 27) Let Ω C be an open set containing the point. Suppose f : Ω Ω is a holomorphic mapping, with f () = and f () =. Suppose f has Taylor expansion f (z) = z + n=2 a nz n at. Define the iteration f (z) = f (z), f 2 (z) = f ( f (z)) and then by induction f k (z) = f ( f k (z)). (a) Show that if f (z) = z + a n z n + O(z n+ ), then f k (z) = z + ka n z n + O(z n+ ). (b)show that if Ω is bounded and connected, then f (z) = z for all z Ω. 4
Problem 39. (Aug 22) ecall that a distribution T on has order if for each compact subset K of, T,φ C K max x φ(x), supp φ K, and φ C c () Let < a n+ < b n+ < a n < b n, for n. Let χ [an,b n ] be the characteristic function of [a n,b n ] and f = n= c n χ [an,b n ], c n Assume f L (). Prove that the distribution f has order if and only if c n < Problem 4. (Jan 24) Consider the following operator A f = x + logx Is A bounded as an operator from L 2 [,] to L 2 [,]? Is it compact? x f (t)dt Problem 4. (Aug 24) Let F D () be a distribution and F,φ = for every test function φ D() such that / supp(φ). Take a test function φ such that φ ( j) () = for any integer j. Is it true that F,φ =? Problem 42. (Aug 23) Give an example of a Hilbert space H and a bounded linear operator A acting from H to H such that the spectrum of A is equal to {z C : z }. Problem 43. (Aug 2) (a)what are the necessary and sufficient conditions on λ n > for the set {(x,x 2, ) l 2 (N) : x n λ n, n} to be compact in l 2 (N). Here l 2 denotes the Hilbert space of square summable sequences with inner product (x,y) = n= x ny n, where x = (x,x 2, ),y = (y,y 2, ). (b)what are the necessary amd sufficient conditions on µ n > for the set to be compact in l 2 (N)? Problem 44. (Jan 29) For f L 2 (,), define T [ f ](x) = x f (x). {(x,x 2, ) l 2 x n (N) : 2 n µ n 2 } (a)show that T is a continuous self-adjoint operator on L 2 (,). (b)show that T does not have any eigenvalues. (c)for which λ C is the operator T λi not invertible? Problem 45. (Jan 25) Construct a function f L 2 ( 2 ), with distributional derivative f L 2 ( 2 ), that diverges to infinity(lim x r f (x) = ) at every rational point r in the unit square. Problem 46. Find all functions f L ( d ) with the property that f f = f. Find all functions f L 2 ( d ) with the property that f f = f. Problem 47. (Aug 24) (a)show there is a distribution U on such that for all test function φ Cc () that vanish identically near : φ(x) U,φ = x 4 dx (b)show that there is no distribution V on such that for all test functions φ Cc () that vanish identically near : V,φ = φ(x)e x dx 5