Get An Introduction to the Theory of Functional Equations and PDF

By Marek Kuczma (auth.), Attila Gilányi (eds.)

ISBN-10: 3764387483

ISBN-13: 9783764387488

ISBN-10: 3764387491

ISBN-13: 9783764387495

Marek Kuczma used to be born in 1935 in Katowice, Poland, and died there in 1991.

After completing highschool in his domestic city, he studied on the Jagiellonian college in Kraków. He defended his doctoral dissertation less than the supervision of Stanislaw Golab. within the 12 months of his habilitation, in 1963, he got a place on the Katowice department of the Jagiellonian collage (now college of Silesia, Katowice), and labored there until his death.

Besides his a number of administrative positions and his remarkable instructing task, he complete first-class and wealthy medical paintings publishing 3 monographs and one hundred eighty medical papers.

He is taken into account to be the founding father of the prestigious Polish tuition of useful equations and inequalities.

"The moment 1/2 the name of this booklet describes its contents competently. most likely even the main committed professional should not have concept that approximately three hundred pages might be written with reference to the Cauchy equation (and on a few heavily comparable equations and inequalities). And the ebook is not at all chatty, and doesn't even declare completeness. half I lists the necessary initial wisdom in set and degree idea, topology and algebra. half II offers information on ideas of the Cauchy equation and of the Jensen inequality [...], specifically on non-stop convex services, Hamel bases, on inequalities following from the Jensen inequality [...]. half III bargains with comparable equations and inequalities (in specific, Pexider, Hosszú, and conditional equations, derivations, convex services of upper order, subadditive features and balance theorems). It concludes with an expedition into the sphere of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews)

"This ebook is a true vacation for the entire mathematicians independently in their strict speciality. you can think what deliciousness represents this ebook for practical equationists." (B. Crstici, Zentralblatt für Mathematik)

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Extra resources for An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality

Example text

21) m(B) 1 m(Gn ) me (A) + n , n ∈ N. As n → ∞, we obtain hence m(B) me (A). On the 2 other hand, A ⊂ B, whence me (A) me (B) = m(B). 3. For every set A ⊂ RN there exists a measurable set B such that A ⊂ B and for every measurable set Z such that A ⊂ Z we have m(B \ Z) = 0. Proof. 1. 1 mi (B \ Z) = 0. 2 m(B \ Z) = 0. , if Y ⊂ B \Z, then Y ∈ L and m(Y ) = 0. 8. Every analytic set is Lebesgue measurable. 54 Chapter 3. 2 Linear transforms As is well known, the Lebesgue measure in RN is invariant under translations.

Thus card Z card A > ℵ0 . Moreover, the function f | Z is one-to-one. 2 Z = D ∪ B with D ⊂ Dd and card B ℵ0 . 2) It follows that card D > ℵ0 and the function f | D is one-to-one. Take x0 , x2 ∈ D , x0 = x2 . There exist open balls K0 , K2 ⊂ z such that xi ∈ 1 Ki , d(Ki ) < , i = 0, 2, and f (cl K0 ) ∩ f (cl K2 ) = ∅, since f is continuous. Let I 2 be the family of all sequences I = {in } , in = 0, 2. 8. Cantor-Bendixson theorem 41 define a sequence of points {xi1 , xi1 i2 , . . in , . } and a sequence of open balls {Ki1 , Ki1 i2 , .

Proof. Consider first the case where m(A) < ∞. 7) so that me f (A) = mi f (A) < ∞. 7). If m(A) = ∞, then there exist pairwise disjoint measurable sets An such that A = ∞ ∞ An , m(An ) < ∞ , n ∈ N (cf. 1). Then f (A) = n=1 f (An ), and f (An ) ∈ L , n ∈ N, by the first part of the proof. 1 m f (A) = mi f (A) = |det L| mi (A) = |det L| m(A). 3) holds. 2. Let f : RN → RN be the transform f (x) = ax + b , x ∈ RN , where a ∈ R \ {0} , b ∈ RN . Then, for every set A ⊂ RN , we have N me f (A) = me (aA + b) = |a| me (A) , mi f (A) = mi (aA + b) = |a|N mi (A) , and, if A ∈ L, then also f (A) ∈ L, and N m f (A) = m(aA + b) = |a| m(A).

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An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality by Marek Kuczma (auth.), Attila Gilányi (eds.)


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