Some properties and applications of convolution algebras

Abstract We consider the classical double convolution product and the Duhamel product for functions in two variables and study their some properties and applications. Namely, we investigate *-generators of the Banach algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:msubsup> <m:mi>C</m:mi> <m:mrow> <m:mi>x</m:mi> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> <m:mo>:=</m:mo> <m:mfenced open="{" close="}"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>C</m:mi> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mfenced open="(" close=")"> <m:mrow> <m:mi>t</m:mi> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">]</m:mo> <m:mo>×</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mfenced> <m:mo>:</m:mo> <m:mi>f</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> <m:mspace width="thinmathspace"/> <m:mspace width="thinmathspace"/> <m:mtext>for some</m:mtext> <m:mspace width="thinmathspace"/> <m:mspace width="thinmathspace"/> <m:mi>g</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mi>C</m:mi> <m:mrow> <m:mi>t</m:mi> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> </m:mrow> </m:mfenced> <m:mo>.</m:mo> </m:math> $$C^{(n)}_{xy}:=\left\{f\in C^{n}\left(t[0,1]\times t[0,1]\right) :f(x,y)=g(xy)\,\,\text{for some}\,\,g\in C^{(n)}_{t[0,1]}\right\} . $$ Also we describe the maximal ideal space of the Banach algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo stretchy="false">(</m:mo> <m:msubsup> <m:mi>C</m:mi> <m:mrow> <m:mi>x</m:mi> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> <m:mo>,</m:mo> <m:mo>⊛</m:mo> <m:mo stretchy="false">)</m:mo> </m:math> $ (C^{(n)}_{xy},\circledast) $ with respect to the Duhamel product ⊛ defined by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mo stretchy="false">(</m:mo> <m:mi>f</m:mi> <m:mo>⊛</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:mfrac> <m:msup> <m:mi mathvariant="normal">∂</m:mi> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mi mathvariant="normal">∂</m:mi> <m:mi>x</m:mi> <m:mi mathvariant="normal">∂</m:mi> <m:mi>y</m:mi> </m:mrow> </m:mfrac> <m:mrow> <m:munderover> <m:mo>∫</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:munderover> </m:mrow> <m:mrow> <m:munderover> <m:mo>∫</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>y</m:mi> </m:mrow> </m:munderover> </m:mrow> <m:mi>f</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>−</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo>−</m:mo> <m:mi>τ</m:mi> <m:mo stretchy="false">)</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mi>τ</m:mi> <m:mo stretchy="false">)</m:mo> <m:mtext>d</m:mtext> <m:mi>τ</m:mi> <m:mspace width="thinmathspace"/> <m:mrow> <m:mtext>d</m:mtext> </m:mrow> <m:mi>t</m:mi> <m:mo>.</m:mo> </m:math> $$(f\circledast g)(x,y)=\frac{\partial^{2}}{\partial x\partial y} \int \limits_{0}^{