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W2093485221 abstract "Let<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$ id=E1><mml:mi>G</mml:mi></mml:math>be a Hausdorff topological locally compact group. Let<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$M(G)$ id=E2><mml:mi>M</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>denote the Banach algebra of all complex and bounded measures on<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$ id=E3><mml:mi>G</mml:mi></mml:math>. For all integers<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$ngeq 1$ id=E4><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:math>and all<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$muin M(G)$ id=E5><mml:mi>μ</mml:mi><mml:mo>∈</mml:mo><mml:mi>M</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>, we consider the functional equations<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$int_{G}f(xty)dmu(t)=sum_{i=1}^{n}g_{i}(x)h_{i}(y)$ id=E6><mml:msub><mml:mo>∫</mml:mo><mml:mi>G</mml:mi></mml:msub><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mi>t</mml:mi><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>d</mml:mi><mml:mi>μ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>y</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$x,yin G$ id=E7><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>G</mml:mi></mml:math>, where the functions<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$f$ id=E8><mml:mi>f</mml:mi></mml:math>,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=${g_{i}}$ id=E9><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math>,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=${h_{i}}$ id=E10><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math>:<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$Grightarrow mathbb{C}$ id=E11><mml:mi>G</mml:mi><mml:mo>→</mml:mo><mml:mi>ℂ</mml:mi></mml:math>to be determined are bounded and continuous functions on<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$ id=E12><mml:mi>G</mml:mi></mml:math>. We show how the solutions of these equations are closely related to the solutions of the<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$mu$ id=E13><mml:mi>μ</mml:mi></mml:math>-spherical matrix functions. When<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$ id=E14><mml:mi>G</mml:mi></mml:math>is a compact group and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$mu$ id=E15><mml:mi>μ</mml:mi></mml:math>is a Gelfand measure, we give the set of continuous solutions of these equations." @default.
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W2093485221 date "2004-01-01" @default.
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W2093485221 title "On Cauchy-type functional equations" @default.
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