This paper presents a novel approach to tracking-by-detection by exploiting the circulant structure of the kernel matrix. The key idea is that when training a classifier with all subwindows of an image (dense sampling), the kernel matrix becomes circulant, allowing for efficient learning and detection using the Fast Fourier Transform (FFT). This approach enables closed-form solutions for training and detection with various kernels, including Gaussian and polynomial kernels, and results in a tracker that is competitive with state-of-the-art methods, with significantly faster performance and simpler implementation. The paper shows that the kernel matrix in dense sampling is circulant for unitarily invariant kernels, which allows the use of FFT to compute the kernel matrix and perform learning and detection efficiently. This leads to a fast and exact solution for Kernel Regularized Least Squares (KRLS) and fast detection of all subwindows. The method also provides efficient computation of non-linear kernels, such as dot-product and radial basis function (RBF) kernels, using FFT-based techniques. The proposed tracker is implemented with a few lines of code and runs at hundreds of frames-per-second. It is evaluated on 12 challenging videos, showing competitive performance compared to state-of-the-art trackers. The results demonstrate that the proposed method achieves high precision and accuracy, particularly in sequences with large scale changes. The method is also shown to be effective in scenarios where traditional tracking methods struggle, such as when dealing with occlusions or changes in scale. The paper concludes that the circulant structure of the kernel matrix is a powerful tool for improving the efficiency and performance of tracking-by-detection systems. The theoretical framework presented allows for the development of fast and accurate tracking algorithms that can be applied to a wide range of computer vision tasks.This paper presents a novel approach to tracking-by-detection by exploiting the circulant structure of the kernel matrix. The key idea is that when training a classifier with all subwindows of an image (dense sampling), the kernel matrix becomes circulant, allowing for efficient learning and detection using the Fast Fourier Transform (FFT). This approach enables closed-form solutions for training and detection with various kernels, including Gaussian and polynomial kernels, and results in a tracker that is competitive with state-of-the-art methods, with significantly faster performance and simpler implementation. The paper shows that the kernel matrix in dense sampling is circulant for unitarily invariant kernels, which allows the use of FFT to compute the kernel matrix and perform learning and detection efficiently. This leads to a fast and exact solution for Kernel Regularized Least Squares (KRLS) and fast detection of all subwindows. The method also provides efficient computation of non-linear kernels, such as dot-product and radial basis function (RBF) kernels, using FFT-based techniques. The proposed tracker is implemented with a few lines of code and runs at hundreds of frames-per-second. It is evaluated on 12 challenging videos, showing competitive performance compared to state-of-the-art trackers. The results demonstrate that the proposed method achieves high precision and accuracy, particularly in sequences with large scale changes. The method is also shown to be effective in scenarios where traditional tracking methods struggle, such as when dealing with occlusions or changes in scale. The paper concludes that the circulant structure of the kernel matrix is a powerful tool for improving the efficiency and performance of tracking-by-detection systems. The theoretical framework presented allows for the development of fast and accurate tracking algorithms that can be applied to a wide range of computer vision tasks.