In this article, a new family of graph wavelets, abbreviated LocLets for Localized graph waveLets, is introduced. These wavelets are localized in the Fourier domain on subsets of the graph Laplacian spectrum. LocLets are built upon the Spectral Graph Wavelet Transform (SGWT) and adapt better to signals that are sparse in the Fourier domain than standard SGWT. In fact, as a refinement of SGWT, LocLets benefits from the Chebyshev's machinery to ensure the LocLets transform remains an efficient and scalable tool for signal processing on large graphs. In addition, LocLets exploits signals sparsity in various ways: compactness, efficiency and ease of use of the transform are improved for sparse signals in the Fourier domain. As typical examples of such sparse signals, there are smooth and highly non-smooth signals. For these latter signals, their mixtures or even a wider class of signals, it is shown in this paper that LocLets provide substantial improvements in standard noise reduction tasks compared to advanced graph-wavelet based methods.