Jesus to the apostles: "You're alright in my book"

Jesus gave authority to and validated the message of his apostles many times.

Before his death

Jesus gave his apostles the authority to do amazing things:

Jesus called his twelve disciples and gave them authority over unclean spirits so they could cast them out and heal every kind of disease and sickness. ... Jesus sent out these twelve, instructing them as follows: “Do not go to Gentile regions and do not enter any Samaritan town. Go instead to the lost sheep of the house of Israel. As you go, preach this message: ‘The kingdom of heaven is near!’ Heal the sick, raise the dead, cleanse lepers, cast out demons.” (Matthew 10:1,5-8)

And indeed, their names are written in heaven:

Then the seventy-two returned with joy, saying, “Lord, even the demons submit to us in your name!” So he said to them, “I saw Satan fall like lightning from heaven. Look, I have given you authority to tread on snakes and scorpions and on the full force of the enemy, and nothing will hurt you. Nevertheless, do not rejoice that the spirits submit to you, but rejoice that your names stand written in heaven.” (Luke 10:17-20)

Jesus said that the Holy Spirit would speak through his apostles after he was gone:

Now as Jesus was going out of the temple courts, one of his disciples said to him, “Teacher, look at these tremendous stones and buildings!” Jesus said to him, “Do you see these great buildings? Not one stone will be left on another. All will be torn down! ... You must watch out for yourselves. You will be handed over to councils and beaten in the synagogues. You will stand before governors and kings because of me, as a witness to them. First the gospel must be preached to all nations.When they arrest you and hand you over for trial, do not worry about what to speak. But say whatever is given you at that time, for it is not you speaking, but the Holy Spirit.” (Mark 13:1-2,9-11)

He gave them the authority to forgive sins:

When Jesus came to the area of Caesarea Philippi, he asked his disciples, “Who do people say that the Son of Man is?” They answered, “Some say John the Baptist, others Elijah, and others Jeremiah or one of the prophets.” He said to them, “But who do you say that I am?” Simon Peter answered, “You are the Christ, the Son of the living God.” And Jesus answered him, “You are blessed, Simon son of Jonah, because flesh and blood did not reveal this to you, but my Father in heaven! And I tell you that you are Peter, and on this rock I will build my church, and the gates of Hades will not overpower it. I will give you the keys of the kingdom of heaven. Whatever you bind on earth will have been bound in heaven, and whatever you release on earth will have been released in heaven.” (Matthew 16:13-19)

After his resurrection

The evening after his resurrection, more on the authority to forgive sins:

On the evening of that day, the first day of the week, the disciples had gathered together and locked the doors of the place because they were afraid of the Jewish leaders. Jesus came and stood among them and said to them, “Peace be with you.” When he had said this, he showed them his hands and his side. Then the disciples rejoiced when they saw the Lord. So Jesus said to them again, “Peace be with you. Just as the Father has sent me, I also send you.” And after he said this, he breathed on them and said, “Receive the Holy Spirit. If you forgive anyone’s sins, they are forgiven; if you retain anyone’s sins, they are retained.” (John 20:19-23)

How can he give them this authority? Because he has all authority, to give to whomever he pleases. And it pleased him much to give it to his chosen apostles, whom he promised he would remain present with even after ascending into heaven:

So the eleven disciples went to Galilee to the mountain Jesus had designated. When they saw him, they worshiped him, but some doubted. Then Jesus came up and said to them, “All authority in heaven and on earth has been given to me. Therefore go and make disciples of all nations, baptizing them in the name of the Father and the Son and the Holy Spirit, teaching them to obey everything I have commanded you. And remember, I am with you always, to the end of the age.” (Matthew 28:16-20)

Moments before his ascension, he promised once again that they would receive the Holy Spirit and that they would preach powerful truths about him throughout the world:

So when they had gathered together, they began to ask him, “Lord, is this the time when you are restoring the kingdom to Israel?” He told them, “You are not permitted to know the times or periods that the Father has set by his own authority. But you will receive power when the Holy Spirit has come upon you, and you will be my witnesses in Jerusalem, and in all Judea and Samaria, and to the farthest parts of the earth.” After he had said this, while they were watching, he was lifted up and a cloud hid him from their sight. (Acts 1:6-9)

All verses are quoted from the NET.