Site to Site Tunnel

You have a double NAT which could pose a problem.  What are the LAN IPs and netmask of the Peer site:
It can not use 192.168.1.0; 10.0.0.0 255.0.0.0 or any other segment option that would include your 10.1.1.0/24.

Yes, You still can setup a site to site by configuring the linksys to forward ports 500 (IPSEC),4500 (IPSEC NAT-T) to the WAN side of the Checkpoint.
On the other hand, do you have some devices that connect to the linksys and you want other devices protected by the Checkpoint?
Or is the linksys serves as a wireless access point as well?

Linksys is my personal lan router .My workplace want me to work from home using their appliance i.e checkpoint appliance.I am trying to plug it into one of the lan ports of linksys so that i do not have  to remove my linksys and also keep my company happy but question is can we do it ? Has anyone ttried this kind of setup?

Yes it is doable.  The only Network related thing to address is you must make sure that the IPs used on your Linksys are different then those at your work.
Actually since the appliance at your end will be initiating the VPN, you might not need to open any ports on the linksys, but make sure that IPSEC passthrough is enabled.
Did you or your employer setup the appliance?  You might fair better by not using a 10..x.x.x based network behind the appliance. Look at using some in the higher ranges of 192.168.x.y where x is in the 200's. Or 172.x.s.i where x is between 22 and 30 (out of the normal range of 16-31).
Check with your VPN/Networking folks to make sure the IP ranges you pick are not in use internally or in the location to which you need access.
My guess is that most of your setup relies on DHCP so switching DHCP configuration on the Linksys or on the appliance will not cause you too much trouble/hustle.

my lan is using 192.168.1.0 subnet on Linksys

my work is using 10.1.1.0 /24 subnet on checkpoint LAN and I cannot change this .Checkpoint Wan is using  192.168.1.10 (this is connected to linksys)

I setup checkpoint appliance ...
I checked the log on appliance ,it says tunnel is established but i am not able to ping anything on destination (10.0.0.0/8) not able to connect o mail server etc...

MAILSERVER is 10.94.16.8

You can not use 10.1.1.x as the LAN side on the Checkpoint appliance!
This is where your problems are.  You have an IP overlap.
The tunnel might get setup between the Checkpoint appliance and the office. But when you try to access anything in the office where the IP is 10.1.1.x those packets are seen by your computer as local to its segment since it is also has an IP on 10.1.1.x.

Reconfigure your Checkpoint appliance to use for example a 172.25.245 x network and you should be set with the workstation behind the Checkpoint appliance accessing resources in the office.

i am using Hide-NAT on the appliance to avoid overlap

You are double NAtted. I.e. the linksys NATs, then your checkpoint NATs.
Is it an option for test out a possible IP overlay conflict even with the steps you've taken? Alter the LAN IPs to be on the 172.16-31 network block.  Trying to stay away from the same segments on the LAN from the one on the linksys and the one at the office.
See if that makes a difference,  Is placing the Appliance front and center replacing the linksys an option?
 

so double Natt does not work for VPN ?

A VPN can be established through a double NAT, but the Segments used in the VPN path can not exist on both sides.
Your EDGE-X appliance has a 192.168.1.x IP address for the WAN.
Your Linksys LAN IP segment is also on the same 192.168.1.x.

Change one of the LAN networks to something other than 192.168.1.x and see if that fixes.

because EDGE-X  WAN is connected to one of the ports on the LAN side of Linksys that is why they are on the same LAN and that is how I wanted as I want Edge to be behind Linksys

Are you testing your VPn setup while the appliance is behind a linksys?
You may need to disable the check on the linksys that deals with same interface traffic which is how your VPN connection is seen.
comin out of the lan going out of the wan and then tries to come back in through the wan.

This might be what is preventing the VPN from establishing in addition to the same IP segment overlay.

The setup you outline can work as long as there are no IPs that could be in conflict.  The use of the 192.168.1.x is such a condition that may pose a problem.
Your Linksys will advertise the 192.168.1.x as a LAN in the VPN which your router already sees as its WAN.