Multivariate linear integral equations
$begingroup$
In the univariate case, linear integral equations have the form (0):
$$ f(x) = lambda phi(x) - int_a^b K(x,y) phi(y) dy $$
where $ a < x,y < b $ and $K:[a,b]times[a,b] to mathbb R$ is the integral kernel
Could anyone indicate good references (accessible textbooks or papers) discussing the practical and theoretical issues presented by multivariate generalizations? For example (1):
$$ f(mathbf x) = int_a^b K(mathbf x,y) phi(y) dy $$
with $x in mathcal Dsubset mathbb R^n,a<y<b $ and kernel $K:mathcal Dtimes[a,b] to mathbb R$
(note that the second-kind equation would not make sense since $phi$ is univariate)
or (2):
$$ f(mathbf x) = lambda phi(mathbf x) - int_mathcal D K(mathbf x,mathbf y) phi(mathbf y) dmathbf y $$
with $mathbf {x,y} in mathcal D $ and kernel $K:mathcal Dtimesmathcal D to mathbb R$
My personal understanding is that case (2) shares the same Fredholm-Riesz theory as the univariate case (0). I am not so sure about case (1)?
functional-analysis multivariable-calculus integral-equations
asked Oct 29 '18 at 17:13
phaedophaedo
12610
$endgroup$
add a comment |
$begingroup$
In the univariate case, linear integral equations have the form (0):
$$ f(x) = lambda phi(x) - int_a^b K(x,y) phi(y) dy $$
where $ a < x,y < b $ and $K:[a,b]times[a,b] to mathbb R$ is the integral kernel
Could anyone indicate good references (accessible textbooks or papers) discussing the practical and theoretical issues presented by multivariate generalizations? For example (1):
$$ f(mathbf x) = int_a^b K(mathbf x,y) phi(y) dy $$
with $x in mathcal Dsubset mathbb R^n,a<y<b $ and kernel $K:mathcal Dtimes[a,b] to mathbb R$
(note that the second-kind equation would not make sense since $phi$ is univariate)
or (2):
$$ f(mathbf x) = lambda phi(mathbf x) - int_mathcal D K(mathbf x,mathbf y) phi(mathbf y) dmathbf y $$
with $mathbf {x,y} in mathcal D $ and kernel $K:mathcal Dtimesmathcal D to mathbb R$
My personal understanding is that case (2) shares the same Fredholm-Riesz theory as the univariate case (0). I am not so sure about case (1)?
functional-analysis multivariable-calculus integral-equations
asked Oct 29 '18 at 17:13
phaedophaedo
12610
$endgroup$
add a comment |
$begingroup$
In the univariate case, linear integral equations have the form (0):
$$ f(x) = lambda phi(x) - int_a^b K(x,y) phi(y) dy $$
where $ a < x,y < b $ and $K:[a,b]times[a,b] to mathbb R$ is the integral kernel
Could anyone indicate good references (accessible textbooks or papers) discussing the practical and theoretical issues presented by multivariate generalizations? For example (1):
$$ f(mathbf x) = int_a^b K(mathbf x,y) phi(y) dy $$
with $x in mathcal Dsubset mathbb R^n,a<y<b $ and kernel $K:mathcal Dtimes[a,b] to mathbb R$
(note that the second-kind equation would not make sense since $phi$ is univariate)
or (2):
$$ f(mathbf x) = lambda phi(mathbf x) - int_mathcal D K(mathbf x,mathbf y) phi(mathbf y) dmathbf y $$
with $mathbf {x,y} in mathcal D $ and kernel $K:mathcal Dtimesmathcal D to mathbb R$
My personal understanding is that case (2) shares the same Fredholm-Riesz theory as the univariate case (0). I am not so sure about case (1)?
functional-analysis multivariable-calculus integral-equations
asked Oct 29 '18 at 17:13
phaedophaedo
12610
$endgroup$
In the univariate case, linear integral equations have the form (0):
$$ f(x) = lambda phi(x) - int_a^b K(x,y) phi(y) dy $$
where $ a < x,y < b $ and $K:[a,b]times[a,b] to mathbb R$ is the integral kernel
Could anyone indicate good references (accessible textbooks or papers) discussing the practical and theoretical issues presented by multivariate generalizations? For example (1):
$$ f(mathbf x) = int_a^b K(mathbf x,y) phi(y) dy $$
with $x in mathcal Dsubset mathbb R^n,a<y<b $ and kernel $K:mathcal Dtimes[a,b] to mathbb R$
(note that the second-kind equation would not make sense since $phi$ is univariate)
or (2):
$$ f(mathbf x) = lambda phi(mathbf x) - int_mathcal D K(mathbf x,mathbf y) phi(mathbf y) dmathbf y $$
with $mathbf {x,y} in mathcal D $ and kernel $K:mathcal Dtimesmathcal D to mathbb R$
My personal understanding is that case (2) shares the same Fredholm-Riesz theory as the univariate case (0). I am not so sure about case (1)?
functional-analysis multivariable-calculus integral-equations
functional-analysis multivariable-calculus integral-equations
asked Oct 29 '18 at 17:13
phaedophaedo
12610
asked Oct 29 '18 at 17:13
phaedophaedo
12610
edited Nov 4 '18 at 0:52
phaedo
asked Oct 29 '18 at 17:13
phaedophaedo
12610
asked Oct 29 '18 at 17:13
phaedophaedo
12610
asked Oct 29 '18 at 17:13
phaedophaedo
12610
12610
add a comment |
add a comment |
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