Results 1 to 1 of 1

Math Help - Convolution of Functions on Lp norm

  1. #1
    Junior Member
    Joined
    Nov 2009
    From
    Pocatello, ID
    Posts
    59

    Convolution of Functions on Lp norm

    Let 1\leq p\leq\infty. Show that if f\in L^p(\mathbb{R}^d) and g\in L^1(\mathbb{R}^d), then f\ast g\in L^p(\mathbb{R}^d) and also that:

    ||f\ast g||_{L^p(\mathbb{R}^d)}\leq ||f||_{L^p(\mathbb{R}^d)}||g||_{L^1(\mathbb{R}^d)}.

    It goes on to give a hint that says for 1<p<\infty write:

    |(f\ast g)(x)|\leq\int_{\mathbb{R}^d}[|f(y)|\cdot|g(x-y)|^{1/p}]|g(x-y)|^{1/q}dy.

    where q is the conjugate exponent of p, and apply Holder's Inequality.
    Last edited by CaptainBlack; September 23rd 2010 at 10:34 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Convolution of functions
    Posted in the Differential Geometry Forum
    Replies: 13
    Last Post: May 15th 2011, 10:40 AM
  2. Norm Of The Convolution Operator
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: August 15th 2010, 08:33 AM
  3. convolution of exp(-a*norm(x)^2) and exp(-b*norm(x)^2) ?
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: August 14th 2010, 11:30 AM
  4. Norm of a sequence of functions
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 30th 2009, 01:03 AM
  5. space of all continous real functions and a norm
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 31st 2009, 12:09 PM

Search Tags


/mathhelpforum @mathhelpforum