MATH 5463 - Real Analysis II, Section 001 - Spring 2009
TR 1:30-2:45 a.m., 115 PHSC

Instructor: Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu

Office Hours: Tue 3:30-4:30 p.m., Wed 2:30-3:30 p.m., or by appointment.

Prerequisite: 5453 (Real Analysis I).

Text: G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edition, Wiley-Interscience, 1999.
The course will cover parts of chapters 3, 6, 7, 8, 9.

Homework:

• Homework 1, due Thu, Feb 5.
• Homework 2, due Thu, Feb 12.
• Homework 3, due Thu, Feb 19.
• Homework 4, due Thu, Feb 26.
• Homework 5, due Thu, Mar 5.
• Homework 6, due Thu, Apr 2.
• Homework 7, due Thu, Apr 9.
• Homework 8, due Thu, Apr 16.
• Homework 9, due Thu, Apr 23.
• Homework 10, due Tue, May 5.

Please take a couple of minutes to fill out your evaluation of the course:

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• Lecture 1 (Tue, Jan 20): Differentiation on Euclidean space: goal: defining pointwise derivative of a complex (signed, positive) measure with respect to the Lebesgue measure on Rⁿ, Wiener's covering lemma (Lemma 3.15), locally integrable functions, average value of a locally integrable function over a ball, joint continuity of A_r(x) in r>0 and x∈Rⁿ (pages 95-96 of Sec. 3.4).
• Lecture 2 (Thu, Jan 22): Differentiation on Euclidean space (cont.): Hardy-Littlewood maximal function, the (Hardy-Littlewood) Maximal Theorem (Theorem 3.17), limsup of a real-valued function of a real variable, almost-everywhere convergence of averages of locally integrable functions to the value of the function as the radius of the ball tends to 0 (Theorem 3.18) (pages 96-97 of Sec. 3.4).
  Reading assignment: regular measures, condition for regularity of fdm for f in L⁺(Rⁿ), Theorem 3.22 (pages 99-100 of Sec. 3.4).
• Lecture 3 (Thu, Jan 29): Differentiation on Euclidean space (cont.): almost-everywhere convergence of averages of locally integrable functions to the value of the function as the radius of the ball tends to 0 (Theorem 3.18) - finishing the proof, Lebesgue set of a function, the complement of the Lebesgue set of a locally integrable function has Lebesgue measure zero (Theorem 3.20), definition of a nicely shrinking family of Borel sets, The Lebesgue Differentiation Theorem (Theorem 3.21) (pages 97-98 of Sec. 3.4).
• Lecture 4 (Tue, Feb 3): Functions of bounded variation: a.e.-differentiability of increasing functions (Theorem 3.23), total variation of a funciton, BV functions, total variation of a function on a closed interval, examples, representing a real-valued BV function as a difference of two bounded increasing functions (Lemma 3.26) (pages 101-102 of Sec. 3.5).
• Lecture 5 (Thu, Feb 5): Functions of bounded variation (cont.): properties of functions of bounded variation (Theorem 3.27), Jordan decomposition of a real-valued BV function, positive and negative variations of a real-valued BV function, NBV (normalized BV) functions, Lemma 3.28, bijective correspondence between complex Borel measures on R and NBV functions on R, |μ_F|=μ_TF (Theorem 3.29), conditions on NBV functions so that the corresponding Borel measures are mutually singular (resp. absolutely continuous) with respect to m (Proposition 3.30), absolutely continuous functions, absolute continuity implies uniform continuity, an NBV function F is AC iff μ_F<<m (Proposition 3.32) (pages 103-105 of Sec. 3.5).
• Lecture 6 (Tue, Feb 10): Functions of bounded variation (cont.): statements about L¹ functions (Corollary 3.33), F is AC([a,b]) implies that F is BV([a,b]) (Lemma 3.34; think about the proof), the Fundamental Theorem of Calculus for Lebesgue integrals (Theorem 3.35); discrete and continuous measures, μ=μ_d+μ_ac+μ_sc, corresponding decomposition of NBV functions: F=F_d+F_ac+F_sc, examples; Lebesgue-Stieltjes integrals, "integration by parts" formula for Lebesgue-Stieltjes integrals (Theorem 3.36) (read the proof from the book) (pages 105-107 of Sec. 3.5).
• Lecture 7 (Thu, Feb 12): Basic theory of L^p spaces: definition of L^p spaces, proof that L^p is a vector space, triangle equality does not hold for p<1, Holder's inequality (Theorem 6.2), conjugate exponents, Minkowski's inequality (Theorem 6.5), ||.||_p is a norm for p≥1, norm, normed vector space, defining metric through norm, norm topology (pages 181-183 of Sec. 6.1, pages 151-152 of Sec. 5.1).
• Lecture 8 (Tue, Feb 17): Basic theory of L^p spaces (cont.): completeness of L^p for 1≤p<∞ (Theorem 6.6), the space of simple functions supported on sets of finite measure is dense in L^p for 1≤p<∞ (Theorem 6.7) (page 183 of Sec. 6.1).
• Lecture 9 (Thu, Feb 19): Basic theory of L^p spaces (cont.): ||f||_∞, essential supremum, the space L^∞, properties of ||f||_∞ and L^∞ (Theorem 6.8), ∞ is a conjugate exponent to 1 (page 184 of Sec. 6.1).
• Lecture 10 (Tue, Feb 24): Basic theory of L^p spaces (cont.): Propositions 6.9-6.12 on inclusions among L^p spaces and bounds for L^p-norms (pages 185-186 of Sec. 6.1).
• Lecture 11 (Wed, Feb 25): Some useful inequalities: Chebyshev inequality, boundedness of integral operators on L^p spaces (Theorem 6.18), Minkowski inequality for integrals (Theorem 6.19 - only a sketch, no proof) (pages 193-193 of Sec. 6.3).
  Positive linear functionals on C_c(X): brief review of some concepts from topology, support of a function, C(X) and C_c(X), uniform norm on C_c(X), positive linear functionals (pages 116-117 of Sec. 4.1, 121-122 of Sec. 4.2, 128 of Sec. 4.4, 131 of Sec. 4.5, 211-212 of Sec. 7.1).
• Lecture 12 (Thu, Feb 26): Positive linear functionals on C_c(X) (cont.): "continuity" property of positive linear functionals (Proposition 7.1), outer regular, inner regular and regular Borel measures, Radon measures (page 212 of Sec. 7.1).
• Lecture 13 (Tue, Mar 3): Positive linear functionals on C_c(X) (cont.): Riesz representation theorem for linear functionals on C_c(X) - statement and beginning of the proof (pages 212-213 of Sec. 7.1).
• Lecture 14 (Wed, Mar 4): Positive linear functionals on C_c(X) (cont.): Riesz representation theorem for linear functionals on C_c(X) - continuation of the proof (pages 213-214 of Sec. 7.1).
• Lecture 15 (Thu, Mar 5): Positive linear functionals on C_c(X) (cont.): Riesz representation theorem for linear functionals on C_c(X) - end of proof; remarks on delta measures and linear functionals on L^p (pages 214-215 of Sec. 7.1).
• Lecture 16 (Tue, Mar 24): Regularity and approximation theorems: Radon measures are inner regular on all of its σ-finite sets (Proposition 7.5); regularity of σ-finite Radon measures and of Radon measures on σ-compact spaces (Corollary 7.6); for any σ-finite Radon measure μ, Borel set E and ε>0 there exist an open U and a closed F with F⊂E⊂U and μ(U\F)<ε (Proposition 7.7(a)) (page 216 of Sec. 7.2).
• Lecture 17 (Thu, Mar 26): Hour exam 1.
• Lecture 18 (Tue, Mar 31): Regularity and approximation theorems (cont.): every Borel measure on an LCH space X (in which every open set is σ-compact) that is finite on compact sets is regular and hence Radon (Theorem 7.8), C_c(X) is dense in L^p(μ) for a Radon measure μ and 1≤p<∞ (Proposition 7.9, read the proof from the book), Urysohn's Lemma and Tietze Extension Theorems (Theorems 4.32 and 4.34 - only the statements) (page 217 of Sec. 7.2).
• Lecture 19 (Thu, Apr 2): Regularity and approximation theorems (cont.): Lusin's Theorem (Theorem 7.10) (pages 217-218 of Sec. 7.2).
  Preliminaries on Fourier anaysis: notations, common function spaces, multiindices, Taylor's formula and product rule for functions in C^k(Rⁿ), construction of nonzero smooth compactly supported functions, Schwartz space, ∂^αf belongs to L^p for any multiindex α and any p∈[1,∞] if f is in the Schwartz space (pages 235-237 of Sec. 8.1).
• Lecture 20 (Tue, Apr 7): Preliminaries on Fourier anaysis (cont.): a characterization of the Schwartz space (Proposition 8.3), translation operator τ_y (y∈Rⁿ), definition of uniform continuity through the uniform norm, every C_c function is uniformly continuous (Lemma 8.4), continuity of the translation in the L^p norm for 1≤p<∞ (Proposition 8.5), translation is not continuous in the L^∞ norm (pages 237-238 of Sec. 8.1).
• Lecture 21 (Thu, Apr 9): Preliminaries on Fourier anaysis (cont.): linear space C₀ of functions that vanish at infinity, C₀ is the closure of C_c in the uniform metric (Proposition 4.35) (page 132 of Sec. 4.5).
  Convolutions: definition of convolution, elementary properties of convolutions (Proposition 8.6), Young's inequality (Proposition 8.7); existence, boundedness, and uniform continuity of f*g if f∈L^p and g∈L^q where p and q are conjugate exponents, for 1<p<∞, f*g is in C₀ (Proposition 8.8) (pages 239-241 of Sec. 8.2).
• Lecture 22 (Tue, Apr 14): Convolutions (cont.): differentiability properties of convolution of an L^p function and a C^k function (Proposition 8.10), the Schwartz space is closed under convolution (Proposition 8.11), functions on the n-torus Tⁿ and convolution for such functions, definition of φ_t(x), convergence properties of f*φ_t(x) as t→0+ for φ∈L¹ (Theorem 8.14) (pages 241-243 of Sec. 8.2).
• Lecture 23 (Thu, Apr 16): Convolutions (cont.): convergence properties of f*φ_t(x)(x) as t→0+ for f in L^p, φ decaying fast enough, and x in the Lebesgue set of f (Theorem 8.15) (pages 243-244 of Sec. 8.2).
• Lecture 24 (Tue, Apr 21): Convolutions (cont.): approximate identity {φ_t}, density of the space of smooth compactly supported functions and the Schwartz space in L^p for 1≤p<∞ and in C₀ (Proposition 8.17), the C^∞ Urysohn's Lemma (Proposition 8.18) (page 245 of Sec. 8.2).
  The Fourier transform: definition of a Fourier transfrom of an L¹(Rⁿ) function, properties of the Fourier transform of a function with translated argument, Fourier transform of a convolution, of a function multiplied by x^α, of a derivative of a function (Theorem 8.22(a)-(e)) (page 249 of Sec. 8.3).
• Lecture 25 (Thu, Apr 23): The Fourier transform: Riemann-Lebesgue Lemma (Theorem 8.22(f)), the Fourier transform of a function from the Schwartz class is in the Schwartz class (Theorem 8.23), the Gaussian is an eigenfunction of the Fourier transform (Proposition 8.24), Fourier Inverstion Theorem (Theorem 8.26), if the Fourier transform of an L¹ function is zero, then the function is zero almost everywhere (Corollary 8.27), the Fourier transform is an isomorphism of the Schwartz class onto itself (Corollary 8.27), the Plancherel Theorem (Theorem 8.29), extending the Fourier transform to L²(Rⁿ) by using the Plancherel Theorem, the Hausdorff-Young inequality (Proposition 8.30) (pages 250-253 of Sec. 8.3).
  Distributions: test functions, convergence in the space of test functions, distributions as continuous linear functionals on the space of test functions, meaning of continuity, distribution corresponding to locally integrable functions, remark - two locally integrable functions are equal a.e. iff they are equal as distributions (pages 281-283 of Sec. 9.1).
• Lecture 26 (Tue, Apr 28): Distributions (cont.): Borel measures finite on compact sets define distributions; for a fixed multiindex α and a fixed point x₀, F(φ)=(∂^αφ)(x₀) defines a distribution; example - distribution defined by the delta measure; definitions of convergence of distributions; for a locally integrable function f, f_t converges to a constant times δ (Proposition 9.1); definition of equality of two distributions on an open set, Proposition 9.2, definition of a support of a distribution; definition of a derivative of a distribution, multiplication of a distribution and a C^∞ function, translation τ_yF of a distribution; examples - Heaviside function and its derivatives, distribution coming from a Borel measure μ_F (pages 283-285 of Sec. 9.1).
• Lecture 27 (Thu, Apr 30): Hour exam 2.
• Lecture 28 (Tue, May 5): Distributions (cont.): for a increasing right continuous function F considered as a distribution, F'=μ_F (where μ_F is the associated Borel measure), examples; the principal value of 1/x as a distribution, Sohotsky's formula.
• Lecture 29 (Thu, May 7): Distributions (cont.): examples of multiplying a smooth function by a distribution: ψδ=ψ(0)δ for a smooth function ψ; an example showing that it is impossible to define multiplication of distributions that is commutative and associative.
  Tempered distributions: goal - to define Fourier transform of a distribution; using Schwartz functions as test functions, tempered distributions, examples of tempered distributions (compactly supported distributions are tempered, locally integrable functions decaying fast enough are tempered distributions, e^ix is a tempered distribution, e^x is not a tempered distribution but e^xcos(e^x) is a tempered distribution), operations on tempered distributions, L^p(Rⁿ) functions are tempered distributions, the Fourier transform of the Heaviside distribution is equal to P(1/x)+(1/2)δ, the Fourier transform of the delta distribution is equal to 1, Heisenberg uncertainty relation in quantum mechanics.

Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also expected. You are fully responsible for the material covered in each class, whether or not you attend. Make-ups for missed exams will be given only if there is a compelling reason for the absence, which I know about beforehand and can document independently of your testimony (for example, via a note or a phone call from a doctor or a parent).

Quizzes: Short pop-quizzes will be given in class at random times; your lowest quiz grade will be dropped. Often the quizzes will use material that has been covered very recently (even in the previous lecture), so you have to make every effort to keep up with the material and to study the corresponding sections from the book right after they have been covered in class.

Homework: Weekly homework assignments will be announced in class, and they will be posted on the course web page. Shortly after the due date, I will send solutions to the homework problems. I expect you to work on the problems for yourself or together with other students when they are assigned. I'll be happy to discuss any aspect of the homework with you during office hours or by appointment. After having worked on the problems yourself, you should then compare your work with my suggested solutions to the homework problems when they are posted. All homework should be written on a 8.5"×11" paper with your name clearly written, and should be stapled. Please take the homework very seriously. Solving problems is an essential part of the learning process.

Exams: There will be two in-class midterms and a (comprehensive) final.
Tentative dates for the midterms are March 5 (Thursday) and April 23 (Thursday).
The final is scheduled for May 14 (Thursday), 1:30-3:30 p.m.
All tests must be taken at the scheduled times, except in extraordinary circumstances.
Please do not arrange travel plans that prevent you from taking any of the exams at the scheduled time.

Grading: Your grade will be determined by your performance on the following coursework:

Academic calendar for Spring 2009.

Course schedule for Spring 2009.

Policy on W/I Grades : Through February 27, you can withdraw from the course with an automatic "W". In addition, From March 2 to May 8, you may withdraw and receive a "W" or "F" according to your standing in the class. Dropping after April 6 requires a petition to the Dean. (Such petitions are not often granted. Furthermore, even if the petition is granted, I will give you a grade of "Withdrawn Failing" if you are indeed failing at the time of your petition.) Please check the dates!

The grade of "I" (Incomplete) is not intended to serve as a benign substitute for the grade of "F". I only give the "I" grade if a student has completed the majority of the work in the course (for example everything except the final exam), the coursework cannot be completed because of compelling and verifiable problems beyond the student's control, and the student expresses a clear intention of making up the missed work as soon as possible.

Academic Misconduct: All cases of suspected academic misconduct will be referred to the Dean of the College of Arts and Sciences for prosecution under the University's Academic Misconduct Code. The penalties can be quite severe. Don't do it!
For details on the University's policies concerning academic integrity see the Student's Guide to Academic Integrity. See also the Academic Misconduct Code, which is a part of the Student Code .

Students With Disabilities: The University of Oklahoma is committed to providing reasonable accommodation for all students with disabilities. Students with disabilities who require accommodations in this course are requested to speak with the instructor as early in the semester as possible. Students with disabilities must be registered with the Office of Disability Services prior to receiving accommodations in this course. The Office of Disability Services is located in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD only 405-325-4173.