Deep learning in k-space has demonstrated great potential for image reconstruction from undersampled k-space data in fast magnetic resonance imaging (MRI). However, existing deep learning-based image reconstruction methods typically apply weight-sharing convolutional neural networks (CNNs) to k-space data without taking into consideration the k-space data's spatial frequency properties, leading to ineffective learning of the image reconstruction models. Moreover, complementary information of spatially adjacent slices is often ignored in existing deep learning methods. To overcome such limitations, we have developed a deep learning algorithm, referred to as adaptive convolutional neural networks for k-space data interpolation (ACNN-k-Space), which adopts a residual Encoder-Decoder network architecture to interpolate the undersampled k-space data by integrating spatially contiguous slices as multi-channel input, along with k-space data from multiple coils if available. The network is enhanced by self-attention layers to adaptively focus on k-space data at different spatial frequencies and channels. We have evaluated our method on two public datasets and compared it with state-of-the-art existing methods. Ablation studies and experimental results demonstrate that our method effectively reconstructs images from undersampled k-space data and achieves significantly better image reconstruction performance than current state-of-the-art techniques. Source code of the method is available at https://gitlab.com/qgpmztmf/acnn-k-space.