Adkins guided the Saints to back-to-back promotions but was sacked following the 2-2 draw at Chelsea earlier this month, with executive chairman Nicola Cortese describing the timing as "perfect".
Southampton travel to Manchester United on Wednesday four points clear of the relegation zone and Pochettino remains conscious of what the squad had achieved before his arrival.
"We have our own work ethic and our own philosophy as to where we want to take the club," he said.
"We have goals and ambition, we want to work for the team to achieve those goals.
"However, that does not mean we want to be anything contrary to what was being done before, it is not about that.
"It is more about our philosophy, how we want to take the team forward and how we want to achieve those goals."
New signing Vegard Forren could make his debut at Old Trafford after joining from Molde and Pochettino hopes the 24-year-old will adjust quickly to the pace of the Premier League.
"We are very pleased Forren has decided to come here. He is a player who is going to cover certain necessities of the squad," said the Argentinian.
"The most important aspect now is he adapts well to the team, is up to speed and will be able to compete with us."
One target who eluded Southampton was Philippe Coutinho as the Inter Milan midfielder headed to Liverpool instead.
"I am not frustrated about not being able to sign him," said Pochettino, who worked with the 20-year-old on loan at Espanyol last season.
"I knew there was a small chance he would come here, but overall I have not had much time to consider new signings.
"I am very happy with the squad which is available to me.
"We all know who Coutinho is, and that he is a quality player. However, there are other factors when we want to sign a player."
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