TAILIEUCHUNG - Functional Equations: Electronic Edition 2007

Contructive Problems, Binary and other base, Constructing functions by iterations, Approximating with linear functions, Extremal element method, Multiplicative Cauchy Equation, As the main contents of the document "Functional Equations: Electronic Edition 2007". Each of your content and references for additional lectures will serve the needs of learning and research. | Functional Equations Titu Andreescu lurie Boreico Electronic Edition 2007 17 Chapters and 199 Problems With Solution Content 1. Chapter 1 Contructive Problems 1 2. Chapter 2 Binary and other base 19 3. Chapter 3 Constructing functions by iterations 21 4. Chapter 4 Approximating with linear functions 24 5. Chapter 5 Extremal element method 26 6. Chapter 6 Multiplicative Cauchy Equation 28 7. Chapter 7 Substitutions 30 8. Chapter 8 Fixed Point 42 9. Chapter 9 Additive Cauchy Equation 44 10. Chapter 10 Polynomial Equation 55 11. Chapter 11 Interation and Recurrence Relation 65 12. Chapter 12 Polynomial Recurrence and Continuity 73 13. Chapter 13 Odd and Even Part of Function 78 14. Chapter 14 Symmetrization and Addition Variable 80 15. Chapter 15 Functional Inequality 82 16. Chapter 16 Miscellaneous 84 17. Chapter 17 Solution To All Problems 90 Enjoy Constructive Problems This problems involve explicit construction of functions or inductive arguments. Problem 1. Let k be an even positive integer. Find the number of all functions f N0 N0 such that f f n n k for any n G No. Solution. We have f n k f f f n f n k and it follows by induction on m that f n km f n km for all n m G No. Now take an arbitrary integer p 0 p k 1 and let f p kq r where q G No and 0 r k 1. Then p k f f p f kq r f r kq. Hence either q 0 or q 1 and therefore either f p r f r p k or f p r k f r p. In both cases we have p r which shows that f defines a pairing of the set A 0 1 . k . Note that different functions define different pairings of A. Conversely any pairing of A defines a function f No No with the given property in the following way. We define f on A by setting f p r f r p k for any pair p r of the given pairing and f n f q ks for n k 1 where q and s are respectively the quotient and the remainder of n in the division by k. Thus the number of the functions with the given property is equal to that of all pairings of the set A. It is easy to see that this number k is equal to . . k 2 Remark. The

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