Question

Prove lagrange's interpolation in wch sum of lagrange coefficient is equal to one

Answer 1

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Sep 26th 2011, 09:38 AM#1

mathematicalbagpiper 



Junior MemberJoinedNov 2009FromPocatello, IDPosts59

Sum of Lagrange Basis Polynomials is 1

Show that for any number nn and real number xx that

∑i=0nli(x)=1∑i=0nli(x)=1. 

My attempt at a proof is that we should argue by induction, taking n=1n=1 as a base case, which is easy to show:

x−x1x0−x1+x−x0x1−x0x−x1x0−x1+x−x0x1−x0

=x1−x+x−x0x1−x0=x1−x+x−x0x1−x0

=x1−x0x1−x0=1=x1−x0x1−x0=1

So now let's assume that ∑i=0nli(x)=1∑i=0nli(x)=1. Then:

∑i=0n+1li(x)∑i=0n+1li(x)
=∑i=0nli(x)+ln+1(x)=∑i=0nli(x)+ln+1(x)
=(x−xn+1)1????+ln+1(x)=(x−xn+1)1????+ln+1(x)

And it's finding out what that fraction is going to be over (as I know that sum is not just 1 due to the (xi−xn+1)(xi−xn+1) terms that will be in the bottom. I think once I figure that one out the proof should fall out nicely (unless of course I'm making it way too hard and it's easier to prove directly).

Answer 2

Answer:

$\color{pink}{((❥Ɑղʂⱳҽɾ:-}$

ʟᴀɢʀᴀɴɢᴇ ᴛʜᴇᴏʀᴇᴍ ɪs ᴏɴᴇ ᴏғ ᴛʜᴇ ᴄᴇɴᴛʀᴀʟ ᴛʜᴇᴏʀᴇᴍs ᴏғ ᴀʙsᴛʀᴀᴄᴛ ᴀʟɢᴇʙʀᴀ. ɪᴛ sᴛᴀᴛᴇs ᴛʜᴀᴛ ɪɴ ɢʀᴏᴜᴘ ᴛʜᴇᴏʀʏ, ғᴏʀ ᴀɴʏ ғɪɴɪᴛᴇ ɢʀᴏᴜᴘ sᴀʏ ɢ, ᴛʜᴇ ᴏʀᴅᴇʀ ᴏғ sᴜʙɢʀᴏᴜᴘ ʜ ᴏғ ɢʀᴏᴜᴘ ɢ ᴅɪᴠɪᴅᴇs ᴛʜᴇ ᴏʀᴅᴇʀ ᴏғ ɢ. ᴛʜᴇ ᴏʀᴅᴇʀ ᴏғ ᴛʜᴇ ɢʀᴏᴜᴘ ʀᴇᴘʀᴇsᴇɴᴛs ᴛʜᴇ ɴᴜᴍʙᴇʀ ᴏғ ᴇʟᴇᴍᴇɴᴛs. ᴛʜɪs ᴛʜᴇᴏʀᴇᴍ ᴡᴀs ɢɪᴠᴇɴ ʙʏ ᴊᴏsᴇᴘʜ-ʟᴏᴜɪs ʟᴀɢʀᴀɴɢᴇ. ɪɴ ᴛʜɪs ᴀʀᴛɪᴄʟᴇ, ʟᴇᴛ ᴜs ᴅɪsᴄᴜss ᴛʜᴇ sᴛᴀᴛᴇᴍᴇɴᴛ ᴀɴᴅ ᴘʀᴏᴏғ ᴏғ ʟᴀɢʀᴀɴɢᴇ ᴛʜᴇᴏʀᴇᴍ ɪɴ ɢʀᴏᴜᴘ ᴛʜᴇᴏʀʏ, ᴀɴᴅ ᴀʟsᴏ ʟᴇᴛ ᴜs ʜᴀᴠᴇ ᴀ ʟᴏᴏᴋ ᴀᴛ ᴛʜᴇ ᴛʜʀᴇᴇ ʟᴇᴍᴍᴀs ᴜsᴇᴅ ᴛᴏ ᴘʀᴏᴠᴇ ᴛʜɪs ᴛʜᴇᴏʀᴇᴍ ᴡɪᴛʜ ᴛʜᴇ ᴇxᴀᴍᴘʟᴇs. sᴇᴛ ᴛʜᴇᴏʀʏ sᴇᴛs ᴍᴀᴛʜs ғɪɴɪᴛᴇ ᴀɴᴅ ɪɴғɪɴɪᴛᴇ sᴇᴛs sᴜʙsᴇᴛ ᴀɴᴅ sᴜᴘᴇʀsᴇᴛ ʟᴀɢʀᴀɴɢᴇ ᴛʜᴇᴏʀᴇᴍ sᴛᴀᴛᴇᴍᴇɴᴛ ᴀs ᴘᴇʀ ᴛʜᴇ sᴛᴀᴛᴇᴍᴇɴᴛ, ᴛʜᴇ ᴏʀᴅᴇʀ ᴏғ ᴛʜᴇ sᴜʙɢʀᴏᴜᴘ ʜ ᴅɪᴠɪᴅᴇs ᴛʜᴇ ᴏʀᴅᴇʀ ᴏғ ᴛʜᴇ ɢʀᴏᴜᴘ ɢ. ᴛʜɪs ᴄᴀɴ ʙᴇ ʀᴇᴘʀᴇsᴇɴᴛᴇᴅ ᴀs; |ɢ| = |ʜ| ʙᴇғᴏʀᴇ ᴘʀᴏᴠɪɴɢ ᴛʜᴇ ʟᴀɢʀᴀɴɢᴇ ᴛʜᴇᴏʀᴇᴍ, ʟᴇᴛ ᴜs ᴅɪsᴄᴜss ᴛʜᴇ ɪᴍᴘᴏʀᴛᴀɴᴛ ᴛᴇʀᴍɪɴᴏʟᴏɢɪᴇs ᴀɴᴅ ᴛʜʀᴇᴇ ʟᴇᴍᴍᴀs ᴛʜᴀᴛ ʜᴇʟᴘ ᴛᴏ ᴘʀᴏᴠᴇ ᴛʜɪs ᴛʜᴇᴏʀᴇᴍ. ᴡʜᴀᴛ ɪs ᴄᴏsᴇᴛ? ɪɴ ɢʀᴏᴜᴘ ᴛʜᴇᴏʀʏ, ɪғ ɢ ɪs ᴀ ғɪɴɪᴛᴇ ɢʀᴏᴜᴘ, ᴀɴᴅ ʜ ɪs ᴀ sᴜʙɢʀᴏᴜᴘ ᴏғ ɢ, ᴀɴᴅ ɪғ ɢ ɪs ᴀɴ ᴇʟᴇᴍᴇɴᴛ ᴏғ ɢ, ᴛʜᴇɴ; ɢʜ = { ɢʜ: ʜ ᴀɴ ᴇʟᴇᴍᴇɴᴛ ᴏғ ʜ } ɪs ᴛʜᴇ ʟᴇғᴛ ᴄᴏsᴇᴛ ᴏғ ʜ ɪɴ ɢ ᴡɪᴛʜ ʀᴇsᴘᴇᴄᴛ ᴛᴏ ᴛʜᴇ ᴇʟᴇᴍᴇɴᴛ ᴏғ ɢ ᴀɴᴅ ʜɢ = { ʜɢ: ʜ ᴀɴ ᴇʟᴇᴍᴇɴᴛ ᴏғ ʜ } ɪs ᴛʜᴇ ʀɪɢʜᴛ ᴄᴏsᴇᴛ ᴏғ ʜ ɪɴ ɢ ᴡɪᴛʜ ʀᴇsᴘᴇᴄᴛ ᴛᴏ ᴛʜᴇ ᴇʟᴇᴍᴇɴᴛ ᴏғ ɢ. ɴᴏᴡ, ʟᴇᴛ ᴜs ʜᴀᴠᴇ ᴀ ᴅɪsᴄᴜssɪᴏɴ ᴀʙᴏᴜᴛ ᴛʜᴇ ʟᴇᴍᴍᴀs ᴛʜᴀᴛ ʜᴇʟᴘs ᴛᴏ ᴘʀᴏᴠᴇ ᴛʜᴇ ʟᴀɢʀᴀɴɢᴇ ᴛʜᴇᴏʀᴇᴍ. ʟᴇᴍᴍᴀ : ɪғ ɢ ɪs ᴀ ɢʀᴏᴜᴘ ᴡɪᴛʜ sᴜʙɢʀᴏᴜᴘ ʜ, ᴛʜᴇɴ ᴛʜᴇʀᴇ ɪs ᴀ ᴏɴᴇ ᴛᴏ ᴏɴᴇ ᴄᴏʀʀᴇsᴘᴏɴᴅᴇɴᴄᴇ ʙᴇᴛᴡᴇᴇɴ ʜ ᴀɴᴅ ᴀɴʏ ᴄᴏsᴇᴛ ᴏғ ʜ. ʟᴇᴍᴍᴀ : ɪғ ɢ ɪs ᴀ ɢʀᴏᴜᴘ ᴡɪᴛʜ sᴜʙɢʀᴏᴜᴘ ʜ, ᴛʜᴇɴ ᴛʜᴇ ʟᴇғᴛ ᴄᴏsᴇᴛ ʀᴇʟᴀᴛɪᴏɴ, ɢ ∼ ɢ ɪғ ᴀɴᴅ ᴏɴʟʏ ɪғ ɢ ∗ ʜ = ɢ ∗ ʜ ɪs ᴀɴ ᴇǫᴜɪᴠᴀʟᴇɴᴄᴇ ʀᴇʟᴀᴛɪᴏɴ. ʟᴇᴍᴍᴀ : ʟᴇᴛ s ʙᴇ ᴀ sᴇᴛ ᴀɴᴅ ∼ ʙᴇ ᴀɴ ᴇǫᴜɪᴠᴀʟᴇɴᴄᴇ ʀᴇʟᴀᴛɪᴏɴ ᴏɴ s. ɪғ ᴀ ᴀɴᴅ ʙ ᴀʀᴇ ᴛᴡᴏ ᴇǫᴜɪᴠᴀʟᴇɴᴄᴇ ᴄʟᴀssᴇs ᴡɪᴛʜ ᴀ ∩ ʙ = ∅, ᴛʜᴇɴ ᴀ = ʙ.

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