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The law of large numbers and a functional equation

100%

Sablik M.

Annales Polonici Mathematici

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1998

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tom 68

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nr 2

165-175

EN

We deal with the linear functional equation (E) $g(x) = ∑^r_{i=1} p_i g(c_i x)$, where g:(0,∞) → (0,∞) is unknown, $(p₁,...,p_r)$ is a probability distribution, and $c_i$'s are positive numbers. The equation (or some equivalent forms) was considered earlier under different assumptions (cf. [1], [2], [4], [5] and [6]). Using Bernoulli's Law of Large Numbers we prove that g has to be constant provided it has a limit at one end of the domain and is bounded at the other end.

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Mean values for vector valued functions and corresponding functional equations

63%

Ger R., Sablik M.

Commentationes Mathematicae

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2013

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tom 53

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nr 2

EN

Although, in general, a straightforward generalization of the Lagrange mean value theorem for vector valued mappings fails to hold we will look for what can be salvaged in that situation. In particular, we deal with Sanderson's and McLeod's type results of that kind (see [9] and [7], respectively). Moreover, we examine mappings with a prescribed intermediate point in the spirit of the celebrated Acz\'el's theorem characterizing polynomials of degree at most 2 (cf. [1]).

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Errata list for "Mean values for vector valued functions and corresponding functional equations" (Commentationes Mathematicae 53(2) (2013), 355-369)

63%

Ger R., Sablik M.

Commentationes Mathematicae

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2014

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tom 54

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nr 1

Ograniczanie wyników

2 Commentationes Mathematicae

1 Annales Polonici Mathematici

3 Sablik M.

2 Ger R.

1 2014

1 2013

1 1998

The law of large numbers and a functional equation

Mean values for vector valued functions and corresponding functional equations

Errata list for "Mean values for vector valued functions and corresponding functional equations" (Commentationes Mathematicae 53(2) (2013), 355-369)