<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Let (G,+) be a locally compact abelian Hausdorff group, 𝓀 is a finite automorphism group of G, κ = card𝒦 and let µ be a regular compactly supported complex-valued Borel measure on G such that <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2018-0004_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>μ</mi><mo>(</mo><mtext>G</mtext><mo>)</mo><mo>=</mo><mfrac><mn>1</mn><mi>κ</mi></mfrac></mrow></math><tex-math>$\mu ({\rm{G}}) = {1 \over \kappa }$</tex-math></alternatives></inline-formula>. We find the continuous solutions f, g : G → ℂ of the functional equation
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2018-0004_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><munder><mo>∑</mo><mrow><mi>k</mi><mo>∈</mo><mi>𝒦</mi></mrow></munder><mrow><munder><mo>∑</mo><mrow><mi>λ</mi><mo>∈</mo><mi>𝒦</mi></mrow></munder><mrow><mrow><msub><mo>∫</mo><mtext>G</mtext></msub><mrow><mtext>f</mtext><mo>(</mo><mtext>x</mtext><mo>+</mo><mtext>k</mtext><mo>⋅</mo><mtext>y</mtext><mo>+</mo></mrow></mrow><mi>λ</mi><mo>⋅</mo><mtext>s</mtext><mo>)</mo><mtext>d</mtext><mi>μ</mi><mo>(</mo><mtext>s</mtext><mo>)</mo><mo>=</mo><mtext>g</mtext><mo>(</mo><mtext>y</mtext><mo>)</mo><mo>+</mo><mi>κ</mi><mtext>f</mtext><mo>(</mo><mtext>x</mtext><mo>)</mo><mo>,</mo><mi> </mi><mtext>x</mtext><mo>,</mo><mtext>y</mtext><mo>∈</mo><mtext>G</mtext><mo>,</mo></mrow></mrow></math><tex-math>$$\sum\limits_{k \in {\cal K}} {\sum\limits_{\lambda  \in {\cal K}} {\int_{\rm{G}} {{\rm{f}}({\rm{x}} + {\rm{k}} \cdot {\rm{y}} + } \lambda  \cdot {\rm{s}}){\rm{d}}\mu ({\rm{s}}) = {\rm{g}}({\rm{y}}) + \kappa {\rm{f}}({\rm{x}}),\,{\rm{x}},{\rm{y}} \in {\rm{G}},} } $$</tex-math></alternatives></disp-formula>
in terms of k-additive mappings. This equations provides a common generalization of many functional equations (quadratic, Jensen’s, Cauchy equations).</p></abstract>

Let (G,+) be a locally compact abelian Hausdorff group, 𝓀 is a finite automorphism group of G, κ = card𝒦 and let µ be a regular compactly supported complex-valued Borel measure on G such that μ(G)=1κ$\mu ({\rm{G}}) = {1 \over \kappa }$. We find the continuous solutions f, g : G → ℂ of the functional equation
∑k∈𝒦∑λ∈𝒦∫Gf(x+k⋅y+λ⋅s)dμ(s)=g(y)+κf(x), x,y∈G,$$\sum\limits_{k \in {\cal K}} {\sum\limits_{\lambda  \in {\cal K}} {\int_{\rm{G}} {{\rm{f}}({\rm{x}} + {\rm{k}} \cdot {\rm{y}} + } \lambda  \cdot {\rm{s}}){\rm{d}}\mu ({\rm{s}}) = {\rm{g}}({\rm{y}}) + \kappa {\rm{f}}({\rm{x}}),\,{\rm{x}},{\rm{y}} \in {\rm{G}},} } $$
in terms of k-additive mappings. This equations provides a common generalization of many functional equations (quadratic, Jensen’s, Cauchy equations).

On a quadratic type functional equation on locally compact abelian groups

Department of Mathematics, Faculty of Sciences

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"On a quadratic type functional equation on locally compact abelian groups"}

Let (G,+) be a locally compact abelian Hausdorff group, 𝓀 is a finite automorphism group of G, κ = card𝒦 and let µ be a regular compactly supported...

On a quadratic type functional equation on locally compact abelian groups

Published Online: Sep 10, 2018

Page range: 46 - 55

Received: May 10, 2017

DOI: https://doi.org/10.2478/ausm-2018-0004

Keywords
automorphism group, 𝓀-quadratic functional equation, Cauchy equation

© 2018 Hajira Dimou et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On a quadratic type functional equation on locally compact abelian groups

Published Online: Sep 10, 2018

Page range: 46 - 55

Received: May 10, 2017

DOI: https://doi.org/10.2478/ausm-2018-0004

Keywordsautomorphism group, 𝓀-quadratic functional equation, Cauchy equation

© 2018 Hajira Dimou et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
automorphism group, 𝓀-quadratic functional equation, Cauchy equation