Jenkinson joined Jack Wilshere, Alex Oxlade-Chamberlain, Kieran Gibbs and Aaron Ramsey as one of five British players to sign long-term deals with the Gunners earlier this week.
And the full-back admits he sees a bright future at the Emirates Stadium, suggesting the deal proves the club have faith in him.
“Obviously I’m over the moon - really happy,” he told Arsenal’s official website. “I see my future at this football club and I’m delighted they see things the same way.
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“People here clearly have belief in me, and now I have to make sure I prove them right.”
Jenkinson also spoke of his pride at the progress he has made since joining from Charlton Athletic last summer, but acknowledges he still has a lot of developing to do.
“I’m obviously really pleased with my progress over the past 18 months,” he said.
“Of course there have been some challenging times along the way, and I’ve had to work really hard, but I’m pleased with where I am now and I’m so happy with the new contract.
"Now I just need to take every day as it comes, working and training hard.”
All five of the Gunners who signed new long-term deals are British, and Jenkinson believes the fact that Arsenal are building a side with a home-grown core is something which should be championed.
“A big thing has been made of the fact that we’re all British boys, and obviously that’s great,” he added. “The guys who signed at the same time as me are all excellent young players and close friends of mine.
"I’m delighted not just for myself but for everyone else – it’s great for the club as a whole, as well as for the players. We want to be playing with the best and believe we can do that here.
“We’re an English club and I think the core of the squad should reflect that. That’s why I’m delighted the other boys have signed up with me.
"Arsenal has such a vast scouting system that it can pick up great players from numerous other countries, and that will always be the case, but I think it’s fantastic and appropriate to have an English core.
"It’s something the [English] fans can relate to.”