Grundlegende Eigenschaften der polynomischen Operationen. Zweite Mitteilung

Mazur, Stanisław; Orlicz, Władysław

doi:10.4064/sm-5-1-179-189

Cited by 103 publications

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“…The following two theorems were proved by Mazur and Orlicz [12] and [13], and in greater generality by Djoković [5]. (…”

Section: Background Resultsmentioning

confidence: 92%

A general functional equation and its stability

Baker

2005

Proc. Amer. Math. Soc.

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Abstract. Suppose that V and B are vector spaces over Q, R or C and α 0 , β 0 , . . . , αm, βm are scalar such thatthen each f k is a "generalized" polynomial map of "degree" at most m − 1.In case V = R n and B = C we show that if some f k is bounded on a set of positive inner Lebesgue measure, then it is a genuine polynomial function.Our main aim is to establish the stability of ( * ) (in the sense of Ulam) in case B is a Banach space.We also solve a distributional analogue of ( * ) and prove a mean value theorem concerning harmonic functions in two real variables. Blanket assumptionsThroughout this paper F = Q, R or C, V and B are vector spaces over F, m ∈ N, and α 0 , β 0 , . . . , α m , β m are given members of F. Our aim is to study the functional equationwhere the functions f k : V → B may be thought of as the unknowns. The stability of ( * ) (in the sense of Ulam) is of principal interest.Background results

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“…The following two theorems were proved by Mazur and Orlicz [12] and [13], and in greater generality by Djoković [5]. (…”

Section: Background Resultsmentioning

confidence: 92%

A general functional equation and its stability

Baker

2005

Proc. Amer. Math. Soc.

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“…The following theorem was proved by Mazur and Orlicz [1,2] and in greater generality by Djoković (see [3]). Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, the stability question of functional Equation (1) is whether there is an exact solution of (1) near each solution of inequality (2). If the answer is affirmative with inequality (3), we would say that the Equation (1) is stable.…”

Section: Introductionmentioning

confidence: 99%

Stability of a Monomial Functional Equation on a Restricted Domain

Lee

2017

Mathematics

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“…Mazur and Orlicz have pointed out the relationship between bilinear (and multilinear) transformations and polynomial transformations (cf. [5], p. 59). Polynomial transformations have also been studied by Banach (cf.…”

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confidence: 99%

“…It is a consequence of [5] ( §11, p. 179) and Principle A (p. 59) that continuity at a single point implies continuity at every point. Thus F(f, g) is continuous.…”

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confidence: 99%

Bilinear transformations in Hilbert space

Murray¹

1939

Trans. Amer. Math. Soc.

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Grundlegende Eigenschaften der polynomischen Operationen. Zweite Mitteilung

Cited by 103 publications

References 0 publications

A general functional equation and its stability

A general functional equation and its stability

Stability of a Monomial Functional Equation on a Restricted Domain

Bilinear transformations in Hilbert space

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