ABSTRACT
As it has been noticed in Chapter 5, if f is a bounded and continuous function over ℝ d and { T ( t ) } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429262593/03624a6d-a4e5-45fb-ba52-c2b8f08b80c3/content/math15_1.tif"/> is the Gauss-Weierstrass semigroup, then, for each t > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429262593/03624a6d-a4e5-45fb-ba52-c2b8f08b80c3/content/math15_2.tif"/> , the function T ( t ) f https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429262593/03624a6d-a4e5-45fb-ba52-c2b8f08b80c3/content/math15_3.tif"/> can be expressed in an integral form as the convolution of the so-called heat kernelK and the function f, i.e., ( T ( t ) f ) ( x ) = ∫ ℝ d K t ( x , y ) f ( y ) d y , x ∈ ℝ d . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429262593/03624a6d-a4e5-45fb-ba52-c2b8f08b80c3/content/math15_4.tif"/>
