This is a book written primarily for graduate students and early researchers in the fields of Analysis and Partial Differential Equations (PDEs). Coverage of the material is essentially self-contained, extensive and novel with great attention to details and rigour.

Dr Taheri is a Reader in Mathematics as the University of Sussex. His primary research area is in the field of Analysis & PDEs where he has been working for over 15 years. During which time he has published research papers in prestigious journals, made valuable contributions to the field and has taught and conducted research in some of the leading institutions in the world including Oxford, Courant Institute, Max-Planck-Institute Leipzig and Warwick. He heads up the Analysis and PDEs research group in Sussex In June 2014 he was awarded the First University of Sussex Student Led Teaching Prize for "Outstanding and Innovative Postgraduate Teaching in Mathematics".

• 1: Harmonic Functions and the Mean-Value Property


• 2: Poisson Kernels and Green's Representation Formula


• 3: Abel-Poisson and Fejer Means of Fourier Series


• 4: Convergence of Fourier Series: Dini vs. Dirichlet-Jordon


• 5: Harmonic-Hardy Spaces hp(D)


• 6: Interpolation Theorems of Marcinkiewicz and Riesz-Thorin


• 7: The Hilbert Transform on Lp(T) and Riesz's Theorem


• 8: Harmonic-Hardy Spaces hp(Bn)


• 9: Convolution Semigroups; The Poisson and Heat Kernels on Rn


• 10: Perron's Method of Sub-Harmonic Functions


• 11: From Abel-Poisson to Bochner-Riesz Summability


• 12: Fourier Transform on S0(Rn); The Hilbert-Sobolev spaces Hs(Rn)